Algebraic Geometry Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Sept. 6, 2007
Christian Haesemeyer :
4 p.m. in SEO 636
Abstract
If the Picard group of a Noetherian ring R is equal to that of
R[t], then it stays the same for polynomial rings over R in any number of
variables. Bass asked if the same is true for the Grothendieck group of
vector bundles, or more generally for any algebraic K-group. I will report
on joint work with Cortinas, Walker and Weibel that answers this question;
it turns out that the cohomology of the du Bois complexes appears as a
summand in the K-groups, and classical computations of du Bois invariants of
semiquasihomogeneous surface singularities over the rationals provide
examples for which the answer is \"no\".
Sept. 13, 2007
Dano Kim :
4 p.m. in SEO 636
Sept. 20, 2007
Luca Scala :
4 p.m. in SEO 636
Sept. 24, 2007
Paolo Stellari :
4 p.m. in LC A5
Emanuele Macri :
5 p.m. in LC A5
Oct. 4, 2007
Laurentiu Maxim :
4:30 p.m. in SEO 636
Abstract
The Chern-Hirzebruch-Serre signature theorem asserts that in the category of closed oriented
manifolds the topological signature is multiplicative in fibrations with trivial monodromy
action. In this talk I will survey various extensions of this result to the singular setting,
including the so-called ``stratified multiplicative property'' for Hodge invariants of complex algebraic varieties. This is joint work with S. Cappell, A. Libgober and J. Shaneson.
Oct. 11, 2007
Christian Schnell :
4 p.m. in SEO 636
Abstract
In any family of projective varieties, the integral cohomology groups of the smooth fibers
form a local system over the base $B$. In general, this local system is defined only over an
open subset of $B$, because some fibers may be singular, and its behavior near the ``boundary''
contains information about the original family.
Associated to the local system, there is also a vector bundle with a flat connection. In 1970,
Deligne showed in a more general setting how to construct a ``canonical extension'' for this
type of bundle, using the monodromy of the local system, and his construction has since
played a role in Hodge theory. In the talk, we answer the natural question of what happens
to the local system itself at the boundary.
Oct. 15, 2007
Anton Leykin :
3 p.m. in SEO 712
Abstract
The existing methods for numerical algebraic geometry give a way to decompose an affine complex variety $X$ into irreducible components. The collection of numerical presentations for these components corresponds to minimal primes associated to the defining ideal $I=I(X)$ of the variety.
We propose a method to find embedded components of $I$. Moreover, we give a numerical description of the scheme Spec$(I)$ by means of {\em numerical primary decomposition}. This description, in particular, solves the ideal membership problem for the ideal $I$.
The main ingredient is the construction of a {\em deflated variety} in a higher-dimensional ambient space, which is related to higher Nash blowups.
Oct. 18, 2007
Melissa Liu :
4 p.m. in SEO 636
Abstract
The Nekrasov's partition function is computed by localization on framed moduli spaces of torsion-free sheaves on $\mathbb{P}^2$. The Seiberg-Witten prepotential is computed by period integrals of an algebraic curve. The Nekrasov's conjecture (proved in various versions by Nakajima-Yoshioka, Nekrasov-Okounkov, Braverman-Etingof) relates the above two objects. We will discusss generalization of the Nekrasov's partition function and the Nekrasov's conjecture for other toric surfaces. This is a joint work in progress with Elizabeth Gasparim.
Oct. 25, 2007
Frank Sottile :
4 p.m. in SEO 636
Abstract
Current continuation methods for finding all solutions to systems of
polynomial equations first compute all complex solutions, and then sieve
them to find the real solutions. This method is not optimal in that
number of paths to be followed may not reflect the actual number of real
solutions. This problem is particularly acute for fewnomial systems, a
class of systems whose number of real solutions is typically much
smaller than their number of complex solutions.
Recent work has established a new bound for the number of real
solutions to a system of fewnomials, by transforming the system of
polynomials into an equivalent system of master functions on a
hyperplane complement, called the gale dual system. Sturmfels observed
that the method used to establish those bounds, the Khovanskii-Rolle
Theorem, could be the basis of a continuation algorithm to compute all
real solutions, which has the additional feature that the path
continuation only follows real solutions.
In this talk, I will sketch the main ideas in this new algorithm.
This will also include a sketch of the proof of these new fewnomial
bounds, and some of the continuation issues which arisen in an
implementation of the algorithm. We remark that the complexity of this
algorithm depends on the ambient (real dimension) and the fewnomial
bound, and not on the number of complex solutions. The implementation of
the algorithm is joint work with Daniel J. Bates, while the fewnomial
bounds and reduction to Gale systems is work with Frédéric Bihan and Bates.
Nov. 1, 2007
Alexey Ovchinnikov :
4 p.m. in SEO 636
Abstract
A linear differential algebraic group is given
by algebraic differential equations. We shall discuss how
one can recover such a group knowing its category of finite
dimensional representations. This is done in the language
of Tannakian categories. We shall also see how to express
certain properties of these groups using the language of
representations.
Nov. 8, 2007
Joe Harris :
4 p.m. in SEO 636
Abstract
There has been a tremendous amount of recent progress on
the geometry of moduli and parameter spaces of curves, inspired in
part by the minimal model program. I'll discuss some of the
outstanding issues, and describe recent work on these by various
geometers.
Nov. 15, 2007
Dawei Chen :
4 p.m. in SEO 636
Abstract
We run the log minimal model program for the Kontsevich space of
stable maps $\overline{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$ and give
geometric interpretations of all the intermediate spaces. In particular, we
show that one component of the corresponding Hilbert scheme is the flip of
$\overline{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$ over the Chow variety.
Nov. 19, 2007
Tommaso de Fernex :
4 p.m. in SEO 627
Abstract
Valery Alexeev asked whether it is possible to generalize the
definitions of singularities of pairs in a wider context than the
usual one. In this talk I will address this question, discussing how
this can be done in such a way that most of the essential features of
the theory are preserved. This is joint work with Christopher Hacon.
Nov. 29, 2007
Matthew Simpson :
4 p.m. in SEO 636
Abstract
The study of moduli spaces is central to the understanding of the birational
geometry of varieties, classifying both the varieties themselves, and how
they vary in reasonable families. To understand all of the consequences of the
existence of a particular moduli space, one must completely understand the
geometry of the moduli space itself. One open problem in the much-studied
Deligne-Mumford Moduli Spaces of stable genus g pointed curves is to
understand the canonical--or more generally log-canonical--models.
In this talk, we examine the log-canonical models in the genus zero
case, with respect to the standard log canonical divisors $K+cD$ where $K$ is the canonical
class, and $D$ the boundary divisor parameterizing nodal curves. We will show
that a conjectural description of the cone of curves by Fulton implies that
these log-canonical models are isomorphic to Hassett's moduli space of
weighted genus zero stable curves for various weights.
For certain values $c$, we can prove this result unconditionally. For
large $c$, the proof is essentially inductive. For small $c$, we use methods
from geometric invariant theory. We will survey these results and, if given
time, discuss the consequences of the second technique to Fulton's conjecture.
Dec. 6, 2007
Alfred Chen :
4 p.m. in SEO 636
Abstract
Given a variety $X$ of general type, by definition, the $m$-
canonical map is birationally stable for $m$ sufficiently large. For curves,
it is a classical result that $m$-canonical map is an embedding for $m \ge 3
$. For surfaces, it's known that $m$-canonical map is birational for $m \ge
5
$. Only very recently, it is proved that there is a constant $c(n)$
depending only on $\dim X$ such that $m$-canonical map is birational for all
$m \ge c(n)$.
In a recent joint wotk with Meng Chen, we found an explicit bound for
$c(3)$.
More precisely, we are able to prove that for any complex projective
threefold
of general type $X$, one has:
1. $Vol(X) \ge 1/2660$,
2. $P_{12}(X) >0$,
3. $P_{24}(X) \ge 2$,
4. $m$-canonical map is birational for all $m \ge 77$.
We are going to show some more appplications of our method.
Jan. 17, 2008
Chenyang Xu :
4 p.m. in SEO 636
Abstract
We will talk about rationally connected varieties and their
degenerations. This
problem exhibits behavior similar to the problem of existence of rational
points. In characteristic zero we will explain the geometric reason behind
this similarity. In characteristic p, we will talk about results over finite
fields as further evidence.
Jan. 24, 2008
Gueorgui Todorov :
4 p.m. in SEO 636
Jan. 31, 2008
Jason Starr :
4 p.m. in SEO 636
Abstract
Serre's "Conjecture II" says that every torsor for a simply connected,
semisimple algebraic group over a field of "cohomological dimension 2"
has a rational point. Using "rational simple connectedness" -- an
analogue of simple connectedness where continuous maps from the interval
are replaced by morphisms from the projective line -- A. J. de Jong,
Xuhua He and I proved this conjecture when group is split and the field
is the function field of a surface over an algebraically closed field.
Combined with a lot of earlier work by many authors, this settles the
conjecture for function fields of surfaces.
Feb. 7, 2008
Radu Laza :
4 p.m. in SEO 636
Abstract
A result of Clemens and Griffiths says that a smooth cubic threefold can be
recovered from its intermediate Jacobian. I will discuss the possible
degenerations of these abelian varieties, and give a description of the
compactification of the moduli space of cubic threefolds obtained in this
way. I will then compare this compactification to other known
compactifications. The situation is quite similar to that of the moduli
spaces of low genus curves. This is joint work with S. Casalaina-Martin.
Feb. 14, 2008
Karl Schwede :
4 p.m. in SEO 636
Abstract
I will discuss recent joint work with S\'{a}ndor Kov\'{a}cs and Karen
Smith. I will describe a simple new Kempf-like characterization of Du Bois
singularities in the normal Cohen-Macaulay case and explain how it can be
used to show that Cohen-Macaulay (semi-)log canonical singularities are Du
Bois, a conjecture of Koll\'{a}r. As a corollary, we obtain generalizations to
Kodaira-type vanishing theorems.
Feb. 21, 2008
Anders Buch :
4 p.m. in SEO 636
Abstract
The Gromov-Witten invariants of a homogeneous space X give the number
of rational curves of fixed degree that meet three general Schubert
varieties, at least when this number is finite. When there are
infinitely many such curves, then the moduli space of (stable)
parametrizations of the curves is a projective variety. The
K-theoretic Gromov-Witten invariants are the Euler characteristic of
such varieties, and were used by Y.-P. Lee and Givental to define a quantum
K-theory ring of X. I will present structure theorems for this ring
when X is a Grassmann variety of type A, and a formula for the
K-theoretic Gromov-Witten invariants that generalizes earlier work
with Kresch and Tamvakis. This is joint work with L. Mihalcea.
Feb. 28, 2008
Gary Kennedy :
4 p.m. in SEO 636
Abstract
A quasi-ordinary surface f(x,y)=0 is one for which, at each singular
point, there is a theory of Puiseux expansion as for plane curves.
Taking a transverse slice with x=constant, one obtains a singular
plane curve. Its Milnor fiber f(x,y)=e has two sorts of monodromy:
(1) the Milnor monodromy (also called its horizontal monodromy), in
which x is fixed while e varies around a small circle, (2) the
vertical monodromy, in which e is fixed while x varies. In joint work
with my colleague Lee McEwan, we have discovered simple recursive
formulas for both monodromies.
March 6, 2008
Noam Elkies :
4 p.m. in SEO 636
March 11, 2008
Giancarlo Urzua :
3 p.m. in SEO 636
Abstract
I will show a strong relation between Chern and logarithmic Chern
numbers of complex algebraic surfaces. For a given arrangement of
curves, there exist smooth projective surfaces with Chern ratio
arbitrarily close to the logarithmic Chern ratio of the arrangement.
The method is a random p-th root cover which exploits a large scale
behavior of Dedekind sums and negative-regular continued fractions. I
will emphasize that the random hypothesis is necessary for this
limit result.
For a certain large class of arrangements, this construction controls
the topological fundamental group of the new surfaces. I will show how
to obtain simply connected surfaces of general type with high Chern
ratio, coming from complex line arrangements. Their Chern ratio is less
than 8/3, being this upper bound the best possible for lines in the
complex projective plane.
March 20, 2008
Tommaso de Fernex :
4:15 p.m. in SEO 636
April 1, 2008
Claude Sabbah :
4 p.m. in SEO 512
Abstract
The (big) quantum cohomology of a manifold has the structure of a Frobenius
manifold. If the quantum cohomology of the projective space is well
understood, that of the Grassmannian was only known on its small part. I
will
explain recent results, obtained jointly with I. Ciocan-Fontanine and B.
Kim,
giving a method for obtaining the quantum cohomology of the quotient of a
smooth variety under the action of a reductive group from the quantum
cohomology of the quotient by a maximal torus. For the Grassmannian G(r,n),
this construction can be interpreted as the r-fold alternate product, in the
sense of Frobenius manifolds, of the quantum cohomology of the projective
space P(n).
April 3, 2008
Brendan Hassett :
4 p.m. in SEO 636
Abstract
Consider a pair consisting of a smooth projective variety and a
normal-crossings divisor, defined over the function field of a complex curve
B. For a model (X,D)--->B, integral points are sections B--->X meeting D
only over prescribed points of B. We present density results for integral
points on log Fano pairs, e.g., when the normal
bundle of D is effective and nontrivial. We also discuss some open
problems. (joint with Tschinkel)
April 10, 2008
Aaron Bertram :
4 p.m. in SEO 636
Abstract
The Hodge index theorem and Bogomolov inequality allow one
to "bootstrap" from ordinary slope stability to a Bridgeland stability
condition that detects coherent sheaves supported in codimension two
(unlike slope stability). This new stability condition can be used, for
example, to get a perfect analogue of Thaddeus stable pairs for K3 surfaces.
It also begs an important question. Can one bootstrap further, to stability
conditions in arbitrary codimension? (Joint work with Daniele Arcara)
April 15, 2008
Wim Veys :
3 p.m. in SEO 512
Abstract
To a p-adic or complex polynomial f one associates
its p-adic Igusa zeta function, motivic or topological zeta function.
There is an intriguing 'monodromy conjecture', predicting that poles of
these zeta functions induce eigenvalues of the local monodromy of f.
Up to now the conjecture is proven only for polynomials in two variables.
We want to report on quite general results for polynomials f in three
variables, and mention the link with certain configurations of plane curves
and with principal value integrals.
April 17, 2008
Harry Tamvakis :
4 p.m. in SEO 636
Abstract
The cohomology ring of the usual Grassmannian has been
studied extensively for well over 100 years, but the analogous
questions for symplectic and orthogonal Grassmannians X are still
relatively unexplored. After an overview of the relevant history,
I will discuss a Giambelli formula for isotropic Grassmannians,
and the related theory of theta polynomials. The latter are a
combinatorially explicit family of polynomials whose algebra
agrees with the Schubert calculus on X. This is joint work with
Anders Buch and Andrew Kresch.
April 24, 2008
Herb Clemens :
4 p.m. in SEO 636
Abstract
This is a talk about the classical problem of Hodge which proposes
a characterization of cohomology classes carried by algebraic subvarieties
of a complex projective manifold. An inductive strategy for attacking this
problem is proposed, using a somewhat 'larger' group than the intermediate
Jacobian and a generalization of the classical Abel-Jacobi map.
May 1, 2008
Sandor Kovacs :
4 p.m. in SEO 636
Abstract
I will discuss various conjectures and theorems about
extending (logarithmic) differential forms over singularities. The
main issue at hand is to understand the difference between the sheaf
of differentials and its reflexive hull. This is strongly related to
extending differentials to exceptional sets of resolutions of
singularities. I will also explain the motivation for such
applications and a recent application of the results.
Aug. 21, 2008
Jose Ignacio Cogolludo :
11 a.m. in SEO 636
Aug. 29, 2008
Sonja Petrovic :
4 p.m. in SEO 636
Abstract
Given a graph G, any partition of its vertex set induces a coloring on its
edges by recording whether the ends of an edge have been separated by the
partition. The set of edges whose ends have been separated in this way is
called a cut of the graph. These edge colorings induced by partitioning
the vertex set parametrize a toric variety. Its defining ideal, the cut
ideal of G, records algebraic relations among the cuts.
These toric ideals were introduced by Sturmfels and Sullivant who also
posed the problem of relating their properties to the combinatorial
structure of the graph. We will describe a certain class of graphs whose
cut ideals admit squarefree lexicographic Groebner bases. Thus, the
associated semigroup algebras are Cohen-Macaulay, but not Gorenstein in
general.
Sept. 5, 2008
Robert Waelder :
4 p.m. in SEO 636
Sept. 12, 2008
Ana Cristina Lopez Martin :
4 p.m. in SEO 636
Abstract
Atiyah's characterization of vector bundles allowed Tu to give a geometric
description of the moduli spaces of semistable sheaves on smooth elliptic
curves. All those results can be obtained in a very simple way as an application of the
Fourier-Mukai transform on an elliptic curve. In this talk, we will consider the case of
some degenerations of elliptic curves, focussing on Kodaira fibers of type $E_N$. For a cycle
$E_N$ of projective lines, we will show that the unique degree 0 stable sheaves are the line
bundles having degree 0 on every irreducible component and the sheaves O(-1) supported
on one irreducible component. The Fourier-Mukai transform allows then to prove that the
connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the
$r$-th symmetric product of the rational curve with one node.
Sept. 19, 2008
Christian Schnell :
4 p.m. in SEO 636
Abstract
Let $X$ be a smooth complex projective variety of dimension $n$. Results by P. Griffiths and M. Green describe the vanishing
cohomology of a sufficiently ample and $\textit{smooth}$ hypersurface $Y \subseteq X$ in terms of residues of $n$-forms on $X$ with
poles along $Y$. In the talk, I will present a generalization of this to all sufficiently ample hypersurfaces, using filtered $D$-modules.
I will explain the connection with M. Saito's theory of mixed Hodge modules, and an application of the result to the Hodge problem.
Sept. 26, 2008
Dawei Chen :
4 p.m. in SEO 636
Abstract
Consider genus g curves that admit degree d covers to elliptic
curves only branched at one point with a fixed ramification type. The locus
of such covers forms a one-dimensional family that naturally maps into the
moduli space of genus g curves. We produce a combinatorial method to
investigate the geometry of this family. The results can also
be applied to study effective divisors on the moduli space.
Oct. 3, 2008
János Kollár :
4 p.m. in SEO 636
Oct. 10, 2008
Sebastian Casalaina-Martin :
4 p.m. in SEO 636
Abstract
A well known result of Clemens and Griffiths says that a smooth cubic
threefold can be recovered from its intermediate Jacobian. In this talk I will discuss
the possible degenerations of these abelian varieties, and thus give a description of
the compactification of the moduli space of cubic threefolds obtained in this way. The
relation between this compactification and those constructed in the work of
Allcock-Carlson-Toledo and Looijenga-Swierstra will also be considered, and is similar
in spirit to the relation between the various compactifications of the moduli spaces
of low genus curves. This is joint work with Radu Laza.
Oct. 21, 2008
Tatsuki Hayama :
3 p.m. in SEO 636
Abstract
l will talk about the moduli spaces of degenerating Hodge
structures introduced by Kato-Usui recently. This moduli space is also
a partial compactification of some discrete quotient of a period
domain. In the case where the period domain is Hermitian symmetric,
the moduli space is well-known because it is a toroidal partial
compactification introduced by Mumford et al. On the other hand,
without the assumption of the Hermitian symmetric property, the moduli
space is less well understood. I will explain a distinction between
the cases where the period domain is Hermitian symmetric and
otherwise, and introduce my latest result.
Oct. 24, 2008
Ana-Maria Castravet :
4 p.m. in SEO 636
Abstract
We give many examples of extremal divisors, rigid curves, and birational
morphisms with unexpected properties for the Grothendieck--Knudsen moduli
space $\bar M_{0,n}$ of stable rational curves. The basic tool is an
isomorphism between $M_{0,n}$ and the Brill--Noether locus of a very
special reducible curve corresponding to a hypergraph. This is joint work
with Jenia Tevelev.
Nov. 7, 2008
Jesse Kass :
4:30 p.m. in SEO 636
Abstract
A classical result of Riemann computes the multiplicity of the
theta divisor of a non-singular curve at a point. If $x$ is a point of
$\Theta$ that corresponds to a line bundle $L$, then the Riemann singularity
theorem states that: $mult_{x}(\Theta) = h^{1}(X,L)$.
I will talk about extending this theorem to singular integral curves. In
particular, I prove a direct generalization of Riemann's theorem to nodal
integral curves. This result yields a partial answer to a question of Lucia
Caporaso. This work is joint with Sebastian (Yano) Casalaina-Martin.
Nov. 13, 2008
Erik Carlsson :
4 p.m. in SEO 636
Abstract
For a long time people have recognized that there is a (perhaps mysterious) connection between
the cohomology of the Hilbert scheme of points on a surface and 2-d conformal field theory. At a
minimum, the direct sum \bigoplus_n Hilb_n X is isomorphic to some Fock space in CFT, and the
important operators on the CFT side make for valuable auxilliary gadgets on the Hilbert scheme
side. I'll give a geometric construction for the ``vertex operator'' in CFT for any smooth
surface, and show how this can be used for some simple Hilbert scheme calculations.
Nov. 14, 2008
Sam Grushevsky :
4:30 p.m. in SEO 636
Abstract
Motivated by constructions in the Whitham theory (perturbation
theory for integrable systems), we consider meromorphic differentials with
prescribed singular parts and real periods on Riemann surfaces. We use these
differentials to define real-analytic foliations of $M_g$ with complex leaves,
and to give a short proof of the Diaz' bound on the dimension of complete
subvarieties of $M_g$; potential further applications will also be discussed.
No familiarity with integrable systems is assumed. Joint work with Igor
Krichever
Dec. 5, 2008
Andrei Caldararu :
4:30 p.m. in SEO 636
Abstract
The ribbon graph complex has been studied intensely in the past 15 years,
in part due to Kontsevich's success using it to prove Witten's conjecture. While
being a very simple combinatorial object, it encodes data about a complicated
object, the mapping class group of surfaces, which in turn completely encodes all
the topological information about moduli spaces of curves. Tom Bridgeland observed
that there exists a second differential on the space of all graphs, and in work
with Junwu Tu we proved that this differential, along with the old one, makes the
space of ribbon graphs into a bicomplex. In my talk I shall discuss this bicomplex,
and state a conjecture about the degeneration of the associated spectral sequence.
If time allows, I'll try to speculate on the techniques to prove this conjecture,
and digress on potential applications.
Jan. 15, 2009
Sergey Gorchinsky :
4 p.m. in SEO 636
Abstract
We discuss a new type of resolutions, called adelic resolutions,
for a certain class of abelian sheaves on algebraic varieties.
This class includes sheaves of K-groups. Adelic resolutions are
multiplicative and contravariant (in contrast with the Gersten
resolution). There is an explicit quasiisomorphism between the
adelic and Gersten resolutions. In particular, this allows to
describe (higher) products on Chow groups and biextensions over
Chow groups in terms of the adelic resolution.
March 5, 2009
Anne-Sophie Kaloghiros :
4 p.m. in SEO 636
Abstract
Let X be a quartic hypersurface in P^4 with no worse than terminal
singularities. The Grothendieck-Lefschetz theorem states that the Picard rank
of X is 1, i.e. that every Cartier divisor on X is a hyperplane section of X.
However, no such result holds for the group of Weil divisors of X if X is not
factorial. I will bound the rank of the group of Weil divisors of X when X is a Fano
3-fold with terminal Gorenstein singularities. This bound is optimal in the
case of the quartic 3-fold. I will show how to use birational geometry in order
to understand some aspects of the topology of Fano 3-folds with mild
singularities. If time permits, I will show that these methods yield an
"explicit" description of the lattice of Weil divisors and provide some
additional information on the geometry of X .
March 16, 2009
Uli Walther :
3 p.m. in SEO 712
Abstract
If R is a regular ring containing a field then by results of
Huneke-Sharp and Lyubeznik, the local cohomology modules H^i_I(R)
have, for all ideals
I of R and for all integers i, a finite set of associate primes. The
reasons are quite different: D-modules in characteristic 0, the Frobenius
morphism in characteristic p.
Lyubeznik conjectured that these results can be extended to regular rings
of mixed characteristics, and proved it in the unramified local case. The
talk is concerned with the case of a polynomial ring over ZZ.
In the presence of singularities finiteness can be absent, as examples by
Singh, Katzman, and Swanson show. Inspired by the first example of a local
cohomology module with infinitely many associated primes due to Singh, we
give a theorem that shows that if finiteness fails in polynomial rings
over ZZ then it must fail in very strange ways.
Our main tool is an adaptation of the Bockstein morphism from algebraic
topology to local cohomology.
March 19, 2009
Cornelia Yuen :
4 p.m. in SEO 636
Abstract
In their study on the content of the product of two polynomials, A.
Corso, V. Vasconcelos and R. Villarreal found a minimal reduction of a
particular class of Ferrers ideals. Inspired by their work and the work of A.
Corso and U. Nagel, we find a minimal reduction of an arbitrary Ferrers ideal
using a different approach. In this talk, we will give an introduction to
Ferrers ideals and minimal reductions, explain the motivating result, and
present our generalization.This is joint work with Sonja Petrovic.
April 20, 2009
Ignacio Luengo :
4 p.m. in SEO 612
July 28, 2009
J.I.Cogolludo :
2 p.m. in SEO 427
Aug. 27, 2009
Raymond Hemmecke :
4 p.m. in SEO 636
Abstract
Graver bases of matrices were originally introduced in 1975 as optimality certificates in integer programming. Although Graver bases are generally huge already for small matrices, one can show that for N-fold IPs there exists a polynomial time algorithm to solve them. This statement heavily relies on a nice structural result on Graver bases of so-called N-fold matrices found by Santos/Sturmfels and generalized by Hosten/Sullivant. This result leads to the notion of Graver complexity, a finite integer number associated to a matrix. In fact, this number gives the degree of the polynomial agorithm to solve the N-fold IP. In practice, it is extremely challenging to compute the Graver complexity of a given matrix. In an analogous manner, one can introduce the notion of Gr"obner complexity of N-fold IPs and show that both numbers agree for unimodular matrices. It is still an open question, whether in this situation Graver bases and universal Gr"obner bases of the N-fold matrices coincide for any N. In this talk we present the polynomial time algorithm to solve N-fold IPs, introduce Graver and Gr"obner complexity of matrices, and state some challenging open problems.
Sept. 3, 2009
Rahul Pandharipande :
4 p.m. in SEO 636
Abstract
Elements of the tropical vertex group are formal
families of symplectomorphisms of the
2-dimensional algebraic torus. I will talk about commutators in the tropical vertex group and their relationship to quivers and
curve counts. The latter is joint work with Gross and
Siebert.
Sept. 10, 2009
Christian Schnell :
4 p.m. in SEO 636
Abstract
I will present a global construction of the Neron model for degenerating families
of intermediate Jacobians; a classical case would be families of abelian varieties.
The construction is based on Saito's theory of mixed Hodge modules; a nice
feature is that it works in any dimension, and does not require normal crossing
or unipotent monodromy assumptions. As a corollary, we obtain a different proof
for the theorem of Brosnan-Pearlstein and Saito that the closure of the zero locus
of an admissible normal function without singularities remains analytic.
Sept. 17, 2009
Dawei Chen :
4 p.m. in SEO 636
Abstract
We verify the smoothness of the Hilbert component $H_n$ whose general points parameterize a pair of codimension two linear subspaces in $P^n$.
For $n>2$, we show that $H_n$ intersects only one component in the full Hilbert scheme and they intersect transversely. We study the Mori theory of
$H_n$, including its Picard group, stable base locus decomposition of its effective cone and modular interpretations of the resulting models. This
is a joint work with I. Coskun and S. Nollet.
Sept. 24, 2009
Ionut Ciocan-Fontanine :
4 p.m. in SEO 636
Abstract
I will report on joint work in progress with Bumsig Kim and
Davesh Maulik in which we introduce new modular compactifications
for the spaces of maps from (varying) curves to a large class of
GIT quotients. These compactifications carry virtual classes and the
associated integrals give new Cohomological Field Theories.
The main goal of the talk is to explain how our work offers a unifying
perspective for many earlier constructions. For example,
the spaces of stable quotients of Marian, Oprea, and Pandharipande,
and the spaces of stable toric quasimaps are recovered as
special cases.
Oct. 1, 2009
Kyungyong Lee :
4 p.m. in SEO 636
Abstract
The famous n! conjecture can be stated in an elementary
language. In fact it asserts that the dimension of the vector space
spanned by all derivatives of a certain bivariate analogue of the n by n
Vandermonde determinant is equal to n!. Earlier results of Haiman and
Garsia had shown that the n! conjecture implied the Macdonald positivity
conjecture. Later Haiman proved the n! conjecture, and the proof is
closely related to the algebraic and geometric properties of isospectral
Hilbert schemes of points on the plane. I'll discuss how some of the
results in the plane case can or cannot be generalized to the higher
dimensional case.
Oct. 12, 2009
Scott Nollet :
5 p.m. in SEO 636
Abstract
The Noether-Lefschetz theorem says that the general surface in
complex projective three space of degree d > 3 has Picard group generated by the restriction of the hyperplane class. This result was suggested by Noether in the 1800s and proved by Lefschetz in the 1920s. In the mid 1980s and 1990s there was a sudden revival of interest in this topic, including a better understanding of the Noether-Lefschetz locus and various extensions of the theorem itself. I will report on these results, along with some developments from the last few years.
Oct. 15, 2009
Claudia Polini :
4 p.m. in SEO 636
Abstract
The theory of blowup algebras is a central area of commutative
algebra. Blowup algebras are so called because they are related to the
process of blowing up a variety along a subvariety. Using
elimination theory we will find the defining equations of the Rees algebras of
certain classes of ideals. This is joint work with A. Kustin and B. Ulrich.
Oct. 22, 2009
Gregory Pearlstein :
3:30 p.m. in SEO 636
Abstract
I will discuss recent work with Patrick Brosnan
on the algebraicity of the zero loci of normal functions,
and applications to the study of algebraic cycles.
Oct. 29, 2009
Hal Schenck :
4 p.m. in SEO 636
Abstract
For a toric variety X determined by a polyhedral fan P in a lattice N, the (rational) equivariant Chow cohomology is a graded Sym(N) module. We study the Chern classes of the associated reflexive sheaf on Proj(N). The first two Chern classes depend only on the combinatorics of P, but c_3 depends on the geometry of codimension two intersections of facets of P.
Nov. 5, 2009
Srikanth Iyengar :
4 p.m. in SEO 636
Abstract
The problem that gave rise to the research to be reported in
this lecture is the following: Given a (finite) group G and a finite
dimensional topological space X, can G act freely on X? In the early
1980's Gunnar Carlsson, Bill Browder, Steve Halperin, and others found a
number of interesting algebraic obstructions to free actions.
In my talk, I will present certain aspects of recent work in commutative
algebra that is motivated by, and perhaps clarifies, some of their
results. This is based on joint work with Avramov, Buchweitz, and C. Miller,
and reported in our paper "Homology of perfect complexes", arXiv:
math/0609008.
Nov. 12, 2009
Roya Beheshti-Zavareh :
4 p.m. in SEO 636
Abstract
This talk is on the geometry of spaces of rational curves on Fano hypersurfaces.
I will talk about some of the known results on the Kodaira dimension of these spaces. I will also discuss the relation between the birational geometry of spaces of rational curves on a hypersurface and the geometry of
the hypersurface itself.
Nov. 19, 2009
Tatsuki Hayama :
4 p.m. in SEO 636
Abstract
For families of intermediate Jacobians over a curve, there are two constructions of an analytic Neron model: one introduced by Green-Griffiths-Kerr, using admissible normal functions (ANF); the other
introduced by Kato-Nakayama-Usui, using log mixed Hodge theory. In this talk, we will talk about the two constructions, and state our main result: the existence of a map between these Neron models.
Jan. 14, 2010
Mihnea Popa :
4 p.m. in SEO 636
Abstract
I will explain a result, joint with Christian Schnell, saying that if two smooth projective varieties have equivalent derived categories of coherent
sheaves, then their Picard (and Albanese) varieties are isogeneous; in particular the number of independent holomorphic 1-forms is derived invariant.
A consequence of this is that derived equivalent threefolds have the same Hodge numbers.
Jan. 21, 2010
Alina Marian :
4 p.m. in SEO 636
Abstract
I will discuss a conjectural geometric duality involving pairs of moduli
spaces of sheaves on a smooth complex projective surface. I will focus on
the case of a K3 surface, when the moduli spaces have particularly beautiful
geometry.
Jan. 28, 2010
Chris Brav :
4 p.m. in SEO 636
Abstract
We review the relation between the geometry of Kleinian singularities and Dynkin diagrams
of types ADE, recalling in particular the construction of a braid group
action of type A,D, or E on the derived category of coherent sheaves on the
minimal resolution of a Kleinian singularity. By work of Seidel-Thomas,
this action was known to be faithful in type A. We extend this faithfulness
result to types ADE and then promote this to a faithful action of an extended affine braid group of
the appropriate type. Our faithfulness results provide the missing
ingredient for completing Bridgeland's description of spaces of stability
conditions for certain triangulated categories associated to Kleinian
singularities.
This is joint work with Hugh Thomas from the University of New Brunswick.
Feb. 4, 2010
Manish Kumar :
4 p.m. in SEO 636
Abstract
Analogus to Hodge and Tate conjectures, Beilinson conjectured the
surjectivity of the "regulator" map from "higher Chow groups" to the space
of Hodge (resp. Tate) cycles of appropriate weight for a smooth
quasi-projective varieties. I will explain the meaning of these words and
discuss the conjectures for varieties dominated by product of curves. This
is a joint work with Donu Arapura.
Feb. 8, 2010
Laurentiu Maxim :
2 p.m. in SEO 636
Abstract
An old problem in geometry and topology is the computation
of topological and analytical invariants of complex hypersurfaces,
e.g., Betti numbers, Euler characteristic, signature, Hodge-Deligne
numbers, etc. While the non-singular case is easier to deal with, the
singular setting requires a subtle analysis of the intricate relation
between the local and global topological and/or analytical structure
of singularities. In this talk I will explain how to compute
characteristic classes of complex hypersurfaces in terms of local
invariants of singularities. This is joint work with S. Cappell, J.
Schuermann and J. Shaneson.
Feb. 11, 2010
Anton Leykin :
4 p.m. in SEO 636
Abstract
We develop new algorithms to compute generalized
Bernstein--Sato polynomials of Budur--Mustata--Saito and Shibuta for
an arbitrary variety. These lead to computations of log canonical
thresholds, jumping coefficients, and multiplier ideals. The
algorithms have been implemented in the D-modules package of the
computer algebra system Macaulay2. (Joint work with Christine
Berkesch)
Feb. 18, 2010
Yusuf Mustopa :
4 p.m. in SEO 636
Abstract
The dth symmetric product C_d of a smooth projective curve C is a
smooth projective variety which encodes the "degree-d aspect" of the
geometry of C. The subordinate loci on C_d associated to linear series on C
encode the degree-d aspect of maps from C to projective space. In this
talk, I will discuss how these loci govern the cone of effective divisors of
C_d, how some natural divisors on C_d may be characterized as subordinate
loci associated to higher-rank vector bundles, and also a conjectural
description of the effective cone of C_d when C is a general curve of genus
g and d is at least (g/2)+1.
Feb. 25, 2010
Madhav Nori :
4 p.m. in SEO 636
Abstract
This is a report on joint work with V.Srinivas. Motivated
by the splitting principle, we construct a simplicial complex which
has the action of GL(n). This yields a sequence L_n(A) of groups
that is connected to Quillen's higher groups K_n(A) via an
exact sequence. We recover a Suslin of theorem on the Bloch group
and K_3-ind.
March 4, 2010
Manish Patnaik :
4 p.m. in SEO 636
Abstract
The local Riemann-Roch problem on certain simple surfaces may be reformulated in terms
of the problem of comparing various central extensions of loop groups. Using an adelic version
of this construction (over function fields of positive characteristic), we can interpret the
points of certain arithmetic quotients of loop groups as bundles on a surface together with some
information about the second Chern class of the bundle. We will sketch the connection with
Eisenstein series on loop groups. This is joint work with Howard Garland.
March 11, 2010
Alex Oblomkov :
4 p.m. in SEO 636
Abstract
By intersecting a small three-dimensional sphere which surrounds a singular point
of a planar curve, with the curve, one obtains a link in three-dimensional space.
In my talk I explain a conjectural formula for the HOMFLY polynomial of the
link which interprets the polynomial in terms of topology of some natural stratification
on the moduli space of torsion free sheaves on the curve. The talk presents
joint work with Vivek Shende.
March 18, 2010
Paul Hacking :
4 p.m. in SEO 636
Abstract
We use the Strominger-Yau-Zaslow interpretation of mirror
symmetry to describe deformations of surface singularities in terms of
counts of holomorphic curves and discs on a mirror surface. In
particular we prove Looijenga's conjecture on smoothability of cusp
singularities. This is joint work with Mark Gross and Sean Keel.
April 1, 2010
David Smyth :
4 p.m. in SEO 636
Abstract
A modular compactification of M_{g,n} is (roughly) a deformation-open class of singular curves with the property that every one-parameter family of smooth curves has a unique limit contained in that class. A modular compactification is stable if all the curves parametrized have the property that every rational component has three distinguished points. We will present a general classification of modular compactifications of M_{g,n} in terms of simple combinatorial data, which will include Schubert's moduli space of pseudostable curves and Hassett's spaces of weighted pointed stable curves as special cases.
April 15, 2010
Dragos Oprea :
4 p.m. in SEO 636
Abstract
We will explain how Fourier-Mukai and degeneration techniques can be used to prove results about
the expected dimension of the theta loci over moduli spaces of sheaves on K3 surfaces.
Connections with the strange duality theorem for generic K3s will also be outlined. (Based on
joint work with Alina Marian.)
April 19, 2010
Li Li :
4 p.m. in SEO 612
Abstract
I will talk on the multiplicities of singular points on Schubert
varieties in the complete flag variety. We study the multiplicities
using degenerations of the Kazhdan-Lusztig ideals, and give a positive
combinatorial rule for the covexillary case. Then I will define drift
configurations and use them to compare multiplicities with
Kazhdan-Lusztig polynomials. This is joint work with Alex Yong.
July 30, 2010
E.Artal-Bartolo :
11 a.m. in SEO 636
Aug. 26, 2010
Eric Zaslow :
4 p.m. in SEO 636
Abstract
The moment map of the complex projective plane
is a triangle. Generalizing this familiar observation
somewhat, I will describe a correspondence between
equivariant coherent sheaves on
toric varieties and polyhedrally constant
sheaves on vector spaces. Specializing to one
dimension, I will then
describe how to assign a category to a ribbon
graph by appropriately gluing sheaves on
the real line.
The ribbon graph category is conjecturally equivalent to
the Fukaya category of the Riemann surface
described by the graph. A glued
version of the correspondence above allows
us to prove that the ribbon graph category
is equivalent to the category of coherent
sheaves on a "mirror" algebraic curve.
I will develop the necessary mathematics
from a *very* simple example.
This talk is based on joint work with
Bohan Fang, Chiu-Chu Melissa Liu,
Nicolo' Sibilla and David Treumann.
Sept. 2, 2010
Dawei Chen :
4 p.m. in SEO 636
Abstract
We study the geometry of Teichmuller curves parameterizing square-tiled
Riemann surfaces (i.e. covers of elliptic curves with a unique branch
point).
The results can be applied to the following questions in algebraic geometry
and complex dynamics: (a) Produce rigid curves on the moduli
space of pointed rational curves; (b) Bound the cone of effective divisors
on the moduli space of curves; (c) Calculate the Lyapunov exponents of the
Hodge bundle over the moduli space of differentials; (d) Verify the
invariance of Siegel-Veech constants in low dimensional strata.
Sept. 9, 2010
Luca Scala :
4 p.m. in SEO 636
Abstract
By techniques by Danila and Le Potier, the comprehension of
global sections of certain determinant line bundles on moduli spaces
of sheaves over the projective plane can be reduced to the
understanding of the cohomology of symmetric powers of some
tautological line bundles on Hilbert schemes of points on P_2. We will
discuss a work in progress on these symmetric powers, in order to
understand their global sections, and some ideas to get their higher
cohomology.
Sept. 17, 2010
Yuri Tschinkel :
5 p.m. in SEO 636
Abstract
This talk is a continuation of the colloquium.
Sept. 21, 2010
Lawrence Ein :
4 p.m. in SEO 636
Abstract
This is the first of two talks about syzygies, in preparation for Aprodu's lectures.
Sept. 23, 2010
Lawrence Ein :
4 p.m. in SEO 636
Abstract
This is the second of two talks about syzygies, in preparation for Aprodu's lectures.
Sept. 28, 2010
Marian Aprodu :
4 p.m. in SEO 636
Abstract
TBA
Sept. 30, 2010
Marian Aprodu :
4 p.m. in SEO 636
Abstract
TBA
Oct. 5, 2010
Marian Aprodu :
4 p.m. in SEO 712
Abstract
TBA
Oct. 14, 2010
Sabin Cautis :
4 p.m. in SEO 636
Abstract
I will define what it means to have a geometric Lie algebra action and
survey some examples. Such actions can be used to construct braid group
actions and knot invariants.
Oct. 19, 2010
Vikram Mehta :
4 p.m. in SEO 636
Abstract
In characteristic zero, it is known that if $\pi_{et} (X) = 1$, then any
semistable bundle with zero Chern classes is trivial. This follows from
the uniformization theorems and a theorem of Selberg. We prove a
corresponding theorem in characteristic $p$, using the fundamental group scheme of
Nori, Langer's boundedness theorem and Hrushovski's work on the Frobenius
automorphism. (Joint work with Helene Esnault.)
Oct. 21, 2010
Shihoko Ishii :
4 p.m. in SEO 636
Abstract
The aim of this talk is to show that Mather discrepancy is a reasonable
invariant.
By using Mather discrepancy,
we can define a new log-canonical threshold and minimal log discrepancy for an
arbitrary
singularity.
We will show the formula for the log-canonical threshold and minimal log discrepancy
in terms of
arc space, and inversion of adjunction in rather general settings.
Oct. 28, 2010
Mircea Mustata :
4 p.m. in SEO 636
Nov. 4, 2010
Tommaso de Fernex :
4 p.m. in SEO 636
Abstract
The minimal model program has led to the introduction of
several notions of singularities. One can consider these notions in
larger context. Already the simple case of cone singularities offers
an interesting class of examples, and by working in such generality
one can also gain some new insight in global geometry. This talk is
based on joint works with C. Hacon, and with S. Boucksom and C. Favre.
Nov. 11, 2010
Tom Nevins :
4 p.m. in SEO 636
Abstract
The study of point modules over a graded noncommutative
algebra R---an analog of skyscraper sheaves on a projective variety---
has played a central role in classification results in noncommutative
ring theory. If the algebra R is strongly noetherian (i.e. tensoring
with any commutative noetherian algebra gives another noetherian
algebra), connected graded, and generated in degree 1, then its point
modules are parametrized by a projective scheme. By contrast, a
strange new class of algebras, the naive blow-ups, exhibit a puzzling
phenomenon: their point modules cannot be parametrized by any scheme
locally of finite type. I'll explain the resolution of this puzzle.
Namely, there are two parameter spaces, one of which is a fine moduli
space but not a scheme, and the other of which is a projective variety
but only a coarse moduli space. The two moduli spaces are related by
an analog of the Hilbert-Chow morphism. This is joint work with Susan
Sierra.
Nov. 15, 2010
Xiaotao Sun :
2 p.m. in SEO 1227
Abstract
The moduli space of stable bundles of rank r with a fixed determinant over
a smooth projective
curve is a Fano manifold with cyclic Picard group. In this talk, we will determine
all rational curves
of minimal degree on the moduli space and minimal rational curves through generic
point . Then we will
give some applications of the main theorem. Some recent results about minimal
elliptic curves on moduli
spaces will be given if time permits.
Nov. 18, 2010
Laura Matusevich :
4 p.m. in SEO 636
Abstract
We consider torus equivariant ideals in the Weyl algebra, and construct
their quotients. The goal is to see what D-module theoretic properties
(holonomicity, reducibility, etc) pass to the quotient. Our main example
is the case of hypergeometric D-modules: this point of view allows us to
prove theorems about classical hypergeometric differential equations using
known results about their equivariant versions. This is joint work with
Christine Berkesch.
Dec. 2, 2010
Anthony Licata :
4 p.m. in SEO 636
Abstract
For every finite subgroup G of SU(2) there is an associated
infinite dimensional Heisenberg algebra. Nakajima and Grojnowski
constructed representations of these Heisenberg algebras on the cohomology
of Hilbert schemes of points on the corresponding ALE space. We give a
graphical categorification of the Heisenberg algebra, and prove that this
categorification acts on the derived categories of Hilbert schemes. This is
joint work with Sabin Cautis.
Jan. 19, 2011
Qile Chen :
4 p.m. in SEO 1227
Abstract
For the purpose of computing Gromov-Witten invariants, the compactification of the space of relative stable maps with respect to a smooth divisor were introduced and studied using expanded degeneration during the past decade. Starting from the backgrounds and basic definitions, I will introduce a new way of compactification using logarithmic structures in the sense of Kato-Fontaine-Illusie. In particular, this covers many interesting cases, such as the target variety with a simple normal crossings divisor, or a simple normal crossings degeneration of a variety with simple normal crossings singularities. This is in part joint work with Dan Abramovich.
Feb. 7, 2011
Jie Wang :
4 p.m. in SEO 612
Abstract
A central problem in curve theory is to describe algebraic curves in a given projective space with
fixed genus and degree. One wants to know the extrinsic geometry of the curve, i.e information on
the equations defining the curve. Koszul cohomology groups in some sense carry 'everything one
wants to know' about the extrinsic geometry of curves in projective space: the number of equations
of each degree needed to define the curve, the relations between the equations, etc. In this talk, I will
present a new method using deformation theory to study Koszul cohomology of general curves. Using this
method, I will describe a way to determine number of defining equations of a general curve in some special degree
range (but for any genus).
Feb. 9, 2011
Tony Pantev :
4 p.m. in SEO 1227
Abstract
I will discuss the deformation theory of Fourier-Mukai transforms in a general complex analytic setting. Suppose that X and Y are two complex manifolds and P is a coherent sheaf on the product which implements an equivalence between the coherent derived categories of X and Y. Given an arbitrary formal quantization of X we construct a unique quantization of Y such that the Fourier-Mukai transform deforms to an equivalence of the derived categories of the quantizations. Here quantizations are understood in the framework of stacks of algebroids. This is a joint work with D.Arinkin and J.Block.
Feb. 16, 2011
Maksym Fedorchuk :
4 p.m. in SEO 1227
Abstract
The modularity program for the moduli space of stable curves is an attempt to give modular interpretation to log canonical models of $\bar{M}_g$. I will discuss recent work on GIT stability of canonical curves and a flip of the hyperelliptic locus (joint with Jensen), and predictions for the future steps of the program (joint with Alper and Smyth).
Feb. 23, 2011
Greg Pearlstein :
4 p.m. in SEO 1227
Abstract
Let S' be a Zariski-open subset of a complex manifold S, and
let V be a variation of mixed Hodge structure on S'. Suppose that V is
defined over the integers, graded polarizable, and admissible with
respect to S. Let Hdg(V) denote the locus of Hodge classes in V . Then
each component of Hdg(V) extends to an analytic space, finite and
proper over S.
March 2, 2011
Jack Huizenga :
4 p.m. in SEO 1227
Abstract
Consider the following basic problem about multiplication of
polynomials in one variable. Fix a general 3-dimensional subspace V
of the polynomials of degree a, and fix a second degree b. Given a
subspace W of the polynomials of degree b, think of W as occupying
the fraction dim(W)/(b+1) of the space of polynomials of degree b.
For every such subspace W, does the product VW occupy at least as
large a fraction of the polynomials of degree a+b as W does of the
polynomials of degree b? That is, does multiplication by V always
increase the fraction of the space occupied by W?
Surprisingly, the answer to this question is connected to the golden
ratio and its continued fraction expansion. We will further discuss
how this question is connected with semistability and splitting
properties of certain particularly nice vector bundles on $P^2$, known
as Steiner bundles. These bundles can be viewed as natural
generalizations of the tangent bundle. Finally, we will discuss how
these bundles give rise to extremal effective divisors on the Hilbert
scheme of points in $P^2$.
March 9, 2011
Alex Kueronya :
4 p.m. in SEO 1227
Abstract
The volume of a Cartier divisor on an irreducible
projective variety describes the asymptotic rate of growth
of the number of its global sections. As such, it is a
non-negative real number, which happens to be rational
whenever the section ring of the divisor in question is
finitely generated.
In a joint work with Catriona Maclean and Victor Lozovanu
we study the multiplicative semigroup of volumes of
divisors. We prove that this set is countable on the one
hand, on the other hand it contains transcendental
elements.
March 16, 2011
Steve Zelditch :
4 p.m. in SEO 1227
Abstract
Let $X$ be a Riemann surface and let $E_N \to Pic^N$ be the vector bundle
consisting of pairs $(L, s)$ of a line bundle of degree $N$ and a holomorphic section
of
$L$. As a generalization of ``random polynomial" (the case $X = CP^1$), we put a
Gaussian
type probability measure on $E_N$, i.e. we choose $L$ at random and then s at random
from $H^0(X, L)$. We use a Hermitian metric $h_L$ on $L$ and a measure $\nu$ on $X$ to
define
the Gaussian measure on
$E_N$. This probability measure induces a probability measure on the configuration
space $S^N X$ of $N$ points, and we can ask, what is the probability that a given
configuration of $N$ points is the zero set of a random section? The bosonization
formulae of string theory are used to determine the induced
measure on configurations of points. It turns out that as $N \to \infty$, the
configurations concentrate very quickly on a certain equilibrium configuration
determined by $(h, L)$.
No prior knowledge of probability theory (or bosonization formulae) is assumed.
This
result is a generalization of joint work with O. Zeitouni
in the genus zero case and with B. Shiffman on the planar case.
March 28, 2011
Daniel Huybrechts :
4 p.m. in SEO 636
Abstract
Spherical objects form a distinguished discrete set of objects in
the derived category of coherent sheaves on a K3 surface. The subcategory
generated by them is of particular interest for the group of
autoequivalences as well as for the Chow group of the surface. I will in
particular discuss its relation to the Bloch-Beilinson conjecture predicting
that over a number field the Chow group is finite dimensional.
April 5, 2011
Andrew Snowden :
2:30 p.m. in SEO 1227
Abstract
The Segre embedding is the natural embedding of a product of
projective spaces into a single projective space. Despite the
fundamental nature of this map, its syzygies are not very well
understood. I will explain how, by considering all Segre
embeddings simultaneously, the spaces of pth syzygies can be
given certain structure, and that this structure is finitely
generated in a reasonable sense. This implies that there are
only finitely many "forms" of pth syzygies. Also, from this
finiteness result we deduce that a certain generating function,
which records essentially all the information about the pth
syzygies of all Segre embeddings, is rational.
Mark Gross :
4 p.m. in SEO 1227
Abstract
I will describe joint work with Bernd Siebert, providing a
vast generalization of relative Gromov-Witten invariants.
The proper context for these invariants is stable maps
from log smooth curves to log smooth target spaces. This
includes such cases one might frequently want to work with,
such as working relative to a normal crossings divisor,
or relative to the toric boundary of a toric variety.
April 6, 2011
Uwe Nagel :
4 p.m. in SEO 1227
Abstract
The Weak Lefschetz Property (WLP) is a property of
finite-dimensional graded algebras and an abstraction of the
conclusion of the Hard Lefschetz Theorem. Many algebras are expected
to have the WLP. However, this property is rather subtle and
establishing it is often very challenging. We will discuss several
approaches, relating the WLP to semistablity of syzygy bundles, the
postulation of fat points, and counting problems in combinatorics.
April 13, 2011
Dmitri Orlov :
4 p.m. in SEO 1227
Abstract
I am going to talk about triangulated categories in algebra, geometry and
physics, and about differential-graded (DG) enhancements of triangulated
categories. It can be proved that unique DG enhancements exist
for a large class of triangulated categories. This class includes
all derived categories of quasi-coherent sheaves, bounded
derived categories of coherent sheaves, and the category of perfect
complexes on quasi-projective schemes, as well as on a noncommutative
varieties.
This shows that triangulated categories which have a geometric nature
are distinguished among all of triangulated categories, for which
this property does not hold in general.
These results have also applications to the deformation theory of
objects in derived categories, and to homological mirror symmetry.
The talk is based on a joint paper with Valery Lunts.
April 20, 2011
Jenia Tevelev :
4 p.m. in SEO 1227
Abstract
It is well-known that moduli of surfaces of general type can be compactified
by stable semi log canonical surfaces of Kollár and Shepherd-Barron.
However, lack of explicit examples makes the global structure of these
moduli
spaces a bit mysterious. I will report on work in progress (joint with
Giancarlo Urzua
and Paul Hacking), where we investigate the boundary in case of surfaces
with $p_g = q = 0$.
Aug. 31, 2011
Jaya Iyer :
5 p.m. in SEO 612
Remke Kloosterman :
4 p.m. in SEO 612
Sept. 14, 2011
Artie Prendergast-Smith :
4 p.m. in SEO 427
Abstract
I will give an overview of the different versions (due to Morrison,
Kawamata, and Totaro) of the Cone Conjecture in birational geometry, and discuss some
cases in which the conjecture has been proven.
Sept. 21, 2011
Artie Prendergast-Smith :
4 p.m. in SEO 427
Abstract
I will give an overview of the different versions (due to Morrison,
Kawamata, and Totaro) of the Cone Conjecture in birational geometry, and discuss some
cases in which the conjecture has been proven.
Oct. 5, 2011
Aaron Bertram :
4 p.m. in SEO 427
Abstract
Moduli spaces of Bridgeland-stable complexes on P^2 are projective. We know this because they are moduli of representations
of the quiver associated to P^2. On the other hand, we don't know
much about projectivity of moduli for other surfaces. In this talk
I want to pursue an idea of Faltings (from the curve setting) of
using the determinantal line bundle on moduli to prove projectivity
without geometric invariant theory.
Oct. 10, 2011
Anne-Sophie Kaloghiros :
4 p.m. in SEO 427
Abstract
The goal of the Minimal Model Program is to produce "good" representatives
of birational equivalence classes of varieties. If X is a smooth projective
variety, the MMP (conjecturally) produces in a finite number of elementary
steps either a minimal model, or, if X is uniruled, a Mori fibre space.
However, this good representative is not unique. It is natural to ask when
two minimal models or when two Mori fibre spaces are birational.
In the case of Mori fibre spaces, Hacon and McKernan recently proved that
any birational map between Mori fibre spaces may be decomposed into a finite
number of "elementary Sarkisov links". This decomposition is not unique.
Their proof is based on recent advances in Mori Theory. I will present their
argument, and show how to understand/describe relations in the Sarkisov
program. If time permits, I will show more definite applications of this
approach to the case of 3-folds.
Oct. 19, 2011
David Nadler :
4 p.m. in SEO 427
Abstract
I'll discuss joint work with D. Ben-Zvi (Texas) devoted to characters
of categorical loop group representations. From one perspective, such
characters should be adjoint-equivariant sheaves on the loop group itself.
Unfortunately, the adjoint-quotient of a loop group is a fearsome infinite
(both positive and negative) dimensional stack. Fortunately, gauge theory
provides another perspective in which such characters should be sheaves on the
moduli of bundles on an elliptic curve. This leads to many interesting
connections with other parts of representation theory, and in particular a
Langlands dual spectral description in terms of commuting varieties.
Oct. 26, 2011
Emanuele Macri :
4 p.m. in SEO 427
Abstract
In this seminar (based on joint work with A. Bayer, Y. Toda, and A.
Bertram), we will present a conjectural approach to the construction
of Bridgeland stability conditions on the derived category of a higher
dimensional variety. The main ingredient is a generalization to
complexes of the classical Bogomolov inequality for sheaves.
We will also discuss an application of this inequality to the Fujita
Conjecture for threefolds.
Nov. 3, 2011
Mihai Paun :
4 p.m. in SEO 512
Nov. 9, 2011
Yu-Jong Tzeng :
4 p.m. in SEO 427
Abstract
A famous problem in classical algebraic geometry is how many r-nodal curves are there in a linear system |L| on an algebraic surface S. If the line bundle L is sufficiently ample, Gottsche conjectured that the number of r-nodal curves is a universal polynomial of Chern numbers of L and S for any r. This conjecture was proven independently by Tzeng and Kool-Shende-Thomas In this talk we will generalize Gottsche's conjecture and show the numbers of curves with any number of arbitrary isolated singularity on surfaces are also given by universal polynomials. Moreover these polynomials can be combined to form a huge generating series and we will discuss its properties.
Nov. 10, 2011
Sándor Kovács :
4 p.m. in SEO 427
Abstract
In this talk I will review how one figures out the right
definition for a stable variety in higher dimensions. This stability is
independent of GIT and it is only called stable by
analogy. I will mention
many new challenges that arise in higher dimensions that
are absent in the
case of curves. Yet, stable varieties of dimension 1 in
this "new sense"
are the same as stable curves, so this is a direct
generalization.
The talk is aimed at graduate student with some basic
knowledge of algebraic geometry.
Nov. 16, 2011
Aaron Pixton :
4 p.m. in SEO 427
Abstract
I'll discuss various conjectures about divisors on
$\overline{M}_{0,n}$ and describe a counterexample to one of them: it
is not true that every nef divisor is numerically equivalent to an
effective sum of boundary divisors. The counterexample is
combinatorial in nature and is closely related to the (11, 5, 2)
biplane.
Feb. 8, 2012
Brian Lehmann :
4 p.m. in SEO 427
Abstract
A classical way to study a line bundle L is to analyze the map
defined by its sections. I will show how to construct a map that instead
reflects the numerical properties of L. This map is in many ways better
behaved; in particular, it has interesting ramifications for the minimal
model program.
Feb. 15, 2012
Milena Hering :
4 p.m. in SEO 427
Abstract
Section rings of arbitrary line bundles on toric varieties are
polytopal semigroup rings and thus
always finitely generated. A related question is whether the section ring
of
the Serre line bundle on the projectivization of a toric vector bundle is
always finitely
generated. It turns out that this is not the case. We show this by
finding toric vector bundles
whose Cox ring is a polynomial ring over the Cox ring of the blow up of
points in projective
space. The latter is well known not to be finitely generated in general.
This is joint work with José González, Sam Payne and Hendrik Süss.
Arend Bayer :
5 p.m. in SEO 427
Abstract
I will present a construction of a nef divisor for every moduli space of
Bridgeland stable complexes on an algebraic variety. In the case of K3
surfaces, we can use it to prove projectivity of the moduli space,
generalizing a result of Minamide, Yanagida and Yoshioka. Its dependence
on the stability condition gives a systematic explanation for the
compatibility of wall-crossing of the moduli space with its birational
transformations; this phenomenon had first been observed by
Arcara-Bertram. This is based on joint work with Emanuele Macrì.
Feb. 22, 2012
Bhargav Bhatt :
4 p.m. in SEO 427
Abstract
We will discuss the relation between the moduli space of a
product of varieties, and the product of the moduli spaces of the
factors. For (stable) curves, Van Opstall showed that taking products
gives an isomorphism between these two spaces, up to controlled finite
etale covers. We will explain why the same picture exists in all
dimensions provided we replace (stable) curves with (stable)
varieties, as defined by the minimal model program. This is joint work
with Wei Ho, Zsolt Patakfalvi, and Christian Schnell.
Feb. 29, 2012
Marti Lahoz :
4 p.m. in SEO 427
Abstract
Let X be a variety of maximal Albanese dimension.
Chen and Hacon proved that if X has positive holomorphic Euler-characteristic,
then its tricanonical map is birational onto its image; in particular, X is of
general type.
When the Euler characteristic is not positive, we use generic vanishing
techniques to prove that the tetracanonical map of X induces the Iitaka
fibration.
Moreover, if X is of general type, then the tricanonical map is already
birational.
If time permits, I will also construct examples showing that these results are
optimal.
This is a joint work with Zhi Jiang and Sofia Tirabassi.
March 7, 2012
Karl Schwede :
4 p.m. in SEO 427
Abstract
The F-signature of a local ring R of characteristic p > 0 is a real number which reflects the severity of the singularities of R. It was introduced explicitly by C. Huneke and G. Leuschke building upon work of K. Smith and M. Van den Bergh. In this talk, I will explain some of its history, and its recent generalization to the context of pairs. I will also explain connections to the minimal log discrepancy and how the F-signature might be useful for studying geometric questions in the future. Most of what is discussed is joint work with Manuel Blickle and Kevin Tucker.
April 4, 2012
Gabriele La Nave :
4 p.m. in SEO 427
Abstract
It has been widely recognized by now that in order to understand the
geometry of Kaehler manifolds under the Ricci flow one needs to understand a
geometric-analytic version of Mori's Minimal Model Program. Much like in
Perelman's execution of Hamilton's program, one of the major stumbling blocks
to the development/utilization of the Ricci-flow stems from the formation of
finite time singularities.
For the Ricci flow on projective manifolds this manifests itself precisely when
the polarization determined by the moving metric hits an extremal ray so that a
birational operation is necessary.
In recent work, Tian and I proposed an approach to the description of finite
time singularities which relates the flow and its singularity formation to
variation of symplectic reductions of a Kaehler manifold endowed with a
1-dimensional (complex) Hamiltonian torus action, where the Kaehler metric in
the total space satisfies a static elliptic equation of soliton type. I will
explain how this works and how it relates to a Geometric version of the Minimal
Model program.
April 11, 2012
Kevin Tucker :
4 p.m. in SEO 427
Abstract
Multiplier ideals are invariants measuring singularities on complex algebraic varieties with analytic origins. In contrast, test ideals are invariants in positive characteristic defined via Frobenius. We will discuss recent advances further linking these invariants to one another, and in particular a characteristic free description of these two invariants. This is joint work with Manuel Blickle and Karl Schwede.
April 18, 2012
Daniel Erman :
4 p.m. in SEO 427
Abstract
The central idea in Boij-Soederberg Theory is that there is
a connection between free resolutions over the polynomial ring and
sheaf cohomology on projective space. I'll provide motivation for
this connection with an example, looking at properties of the long
exact sequence in sheaf cohomology from a new perspective. Then I'll
describe the construction of a duality pairing that leads to precise
duality results relating free resolutions and sheaf cohomology. This
is joint work with David Eisenbud.
April 25, 2012
Ezra Getzler :
2 p.m. in SEO 427
Aug. 29, 2012
J.I.Cogolludo :
3 p.m. in SEO 427
Ben Antieau :
4 p.m. in SEO 427
Abstract
I will describe how to use algebraic topology and representation theory to construct examples of smooth varieties $X$ over the complex numbers and division algebras $D$ over the function field $\mathbb{C}(X)$ such that $D$ is unramified over $X$, and yet there are no projective maximal orders for $D$ over $X$. This solves an old question of Auslander and Goldman, who showed this cannot happen if $X$ is a curve or a surface. Time-permitting, I will discuss the relevance of this research to the period-index problem.
Sept. 5, 2012
Tommaso de Fernex :
4 p.m. in SEO 427
Abstract
Nash was the first to observe that the space of arcs through the
singularities of a complex variety has finitely many irreducible
components, each of which is naturally associated to a divisorial
valuation of the function field of the variety. Every valuation
arising in this way is essential for the singularity, in the sense
that its center in any resolution of singularities is an irreducible
component over the singular locus. The Nash problem asks whether,
conversely, every essential valuation corresponds to a component of
the space of arcs through the singularities. In this talk I will give
an overview of the history and solution of the problem.
Sept. 12, 2012
Luigi Lombardi :
4 p.m. in SEO 427
Abstract
In this talk I will describe the behavior under derived equivalence of
a twisted version of Hochschild homology. This result is then applied to
study the derived invariance of cohomological support loci, fibrations
onto curves, the Albanese dimension, and certain Hodge numbers of special
classes of irregular varieties.
Sept. 19, 2012
Paul Reschke :
4 p.m. in SEO 427
Abstract
I will discuss cohomological criteria for the condition that an automorphism
of a projective surface has positive entropy. I will then present results
and questions in the study of complex surface dynamics that highlight the
importance of the cohomological interpretation of entropy.
Oct. 3, 2012
Anand Patel :
4 p.m. in SEO 427
Abstract
In this talk we will describe a natural decomposition of the Hurwitz space $\mathcal{H}_{d,g}$ parametrizing simply-branched covers of $\mathbb{P}^1$. By studying the geometry of this decomposition, we will deduce the irreducibility of the Gieseker-Petri locus $\mathcal{GP}^1_{d} \subset \mathcal{M}_{g}$ where $d = \frac{g+2}{2}$. Furthermore, we will explain the role this decomposition plays in establishing upper bounds for slopes of sweeping curves in the $d$-gonal locus.
Oct. 10, 2012
Chenyang Xu :
4 p.m. in SEO 427
Abstract
I will talk about the existence of a good minimal model (i.e., minimal model on which the abundance conjecture
holds) for log canonical pairs in two situations. Using it, we show the
existence of log canonical closure for any lc pairs, the moduli space of
canonical polarized varieties satisfying the valuation criterion of
properness and the existence of log canonical flips. (Joint with Christopher Hacon.)
Oct. 17, 2012
Mike Roth :
4 p.m. in SEO 427
Abstract
If $X$ is a variety of general type defined over a number field $k$, then the
Bombieri-Lang conjecture predicts that the $k$-rational points of $X$ are not
Zariski dense. One way to view the conjecture is that a global condition on
the canonical bundle (that it is ''generically positive'') implies a global
condition about rational points. By a well-established principle in
geometry we should also look for local influence of positivity on the
accumulation of rational points. To do that we need measures of both these
local phenomena.
Let $L$ be an ample line bundle on $X$, and $x\in X(\overline{k})$. By
slightly modifying the usual definition of approximation exponent on
$\mathbf{P}^1$, we define a new invariant $\alpha_{x}(L)\in (0,\infty]$ which
measures how quickly rational points accumulate around $x$, as measured by
$L$.
The central theme of the talk is the interrelations between $\alpha_x(L)$ and
the Seshadri constant $\epsilon_{x}(L)$ which measures the local positivity
of $L$ near $x$. In particular, the classic approximation theorem of Klaus
Roth on $\mathbf{P}^1$ generalizes as an inequality between $\alpha_{x}$ and
$\epsilon_{x}$ valid for all projective varieties. This is joint work with
David McKinnon.
Oct. 24, 2012
Gordon Heier :
4 p.m. in SEO 427
Abstract
We will investigate the structure of projective Kaehler manifolds
based on various curvature assumptions. If the holomorphic sectional
curvature is negative, we will prove positivity theorems for the
canonical line bundle. In the positive curvature case, positive total
scalar curvature will be shown to be a sufficient condition for
uniruledness. This is joint work with S. S. Y. Lu and B. Wong.
Oct. 31, 2012
Majid Hadian :
4 p.m. in SEO 427
Nov. 7, 2012
Dan Edidin :
4 p.m. in SEO 427
Abstract
Given a group acting properly on a smooth variety $X$, we show how to define a family inertial products, Chern classes and operations (Adams, $\lambda$, $\psi$) on the rational $K$-theory of the associated inertia stack $I {\mathcal X}$. We give a conjectural relationship between certain of these inertial operations
and operations on the classical $K$-theory of a resolution of singularities of moduli space of
the cotangent bundle stack $T^*{\mathcal X}$. Finally we give some toric examples where the relationship holds.
This is joint work with Tyler Jarvis and Takashi Kimura.
Nov. 14, 2012
No seminar. See next day. :
4 p.m. in SEO 427
Nov. 15, 2012
Burt Totaro :
4 p.m. in SEO 427
Abstract
Consider a smooth complex projective variety X. Hodge theory
shows that sections of exterior powers of the cotangent bundle
are related to the topology of X. What about symmetric powers
of the cotangent bundle? We discuss the relation
between the topology of X and its "symmetric differentials". One interest
of these results is that symmetric differentials give information
in the irection of "Kobayashi hyperbolicity"; for example, they limit
how many rational curves X can contain.
Nov. 21, 2012
No seminar :
4 p.m. in SEO 427
Nov. 28, 2012
Christian Schnell :
4 p.m. in SEO 427
Abstract
In the late 1980s, Green and Lazarsfeld studied the cohomology of topologically trivial line bundles on compact Kaehler manifolds. Among other things, they proved the "generic vanishing theorem": the cohomology of a generic such line bundle vanishes below a certain degree that only depends on the manifold. Recently, Mihnea Popa and I discovered that behind those results lies a certain class of D-modules on abelian varieties; in the talk, I am going to explain why.
Dec. 5, 2012
Jack Huizenga :
4 p.m. in SEO 427
Abstract
The Hilbert scheme of n points in the projective plane parameterizes zero-dimensional subschemes of length n. An interesting problem is to describe the birational geometry of this space, and give modular interpretations for its various birational models. A first step in this program is to determine the cone of effective divisors on the Hilbert scheme.
We show the sections of many stable vector bundles satisfy a natural interpolation condition, and
that these bundles always give rise to the edge of the effective cone. To do this, we
give a generalization of Gaeta's theorem on the resolution of the ideal sheaf of a general collection of n
points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that general ideal sheaves are always destabilized by exceptional bundles.
Dec. 11, 2012
Vasile Brinzanescu :
4 p.m. in SEO 427
Abstract
We study vector bundles and some of their moduli on non-Kaehler principal
elliptic bundles over compact complex manifolds of arbitrary dimension.
The main technical tools used are the twisted Fourier-Mukai transform and
a spectral cover construction.
Feb. 6, 2013
Sug Woo Shin :
4 p.m. in SEO 427
Abstract
A Newton stratum for Shimura varieties (certain moduli spaces of
abelian varieties with extra data in characteristic p) is the locus
where the p-divisible groups belong to a fixed isogeny class. In case
the moduli problem is defined by unitary or symplectic data (which are
unramified), Newton strata are known to be nonempty whenever expected
by Vasiu and Viehmann-Wedhorn based on earlier work by many, and also
recently by Kret. I will explain a different approach to prove the
result in the unramified case via Honda-Tate theory and Galois
cohomology.
Feb. 13, 2013
Mihnea Popa :
4 p.m. in SEO 427
Abstract
I will report on recent work with C. Schnell, in which we prove that every holomorphic one-form on a variety of general type must vanish
at some point (together with a suitable generalization to arbitrary Kodaira dimension). The proof makes use of generic vanishing theory
for Hodge $D$-modules on abelian varieties.
Feb. 19, 2013
Colleen Robles :
4 p.m. in SEO 1227
Abstract
I will characterize the Schubert varieties that arise as variations of Hodge structure (VHS). I will also discuss the central role that these Schubert VHS play in our study of arbitrary VHS. In particular: (i) infinitesimally their orbits under the isotropy action `span' the space of all VHS, yielding a complete description of the infinitesimal VHS; (ii) the cohomology classes dual to the Schubert VHS form an (integral) basis of the invariant characteristic cohomology associated to the system of PDE (Griffiths transversality) characterizing VHS.
Feb. 20, 2013
Melanie Wood :
4 p.m. in SEO 427
Abstract
We consider the "limiting behavior" of *discriminants* (or their
complements), by which we mean informally the closed locus in some
parameter space of some type of object where the objects have certain
singularities. We focus on the collection of unordered points on a
variety X, and linear systems on X. These are connected --- we use
the first to understand the second. We describe their classes in the
Grothendieck ring of varieties, as the number of points gets large, or
as the line bundle gets very positive. As applications, (i) we
show the motivic analogue of Poonen's point-counting result: the
motivic probability of a section of L being smooth (as L gets large)
is 1 / Z_X( \A^{-\dim X - 1} ) (where Z_X is the motivic zeta
functions), and (ii) show a priori unexpected structure in
configuration spaces of points on a variety, leading to many topological and
point-counting consequences and conjectures. This is joint work with
Ravi Vakil.
March 6, 2013
Michael Greenblatt :
4 p.m. in SEO 427
Abstract
In this talk, we describe the speaker's most recent local resolution
of singularities theorem. Motivating classical analysis problems will
be given, the properties of local resolution of singularities that are
needed will be explained, and the proof of the resolution of
singularities theorem will be described. The talk should be accessible
to non-analysts.
March 13, 2013
Dima Arinkin :
4 p.m. in SEO 427
Abstract
Let C be a (smooth projective algebraic) curve. It is well known that the Jacobian J of C is a principally polarized abelian variety. In other words, J is self-dual in the sense that J is identified with the space of topologically trivial line bundles on itself.
Suppose now that C is singular. The Jacobian J of C parametrizes topologically trivial line bundles on C; it is an algebraic group which is no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification J' of J.
In this talk, I consider (projective) curves C with planar singularities. The main result is that J' is self-dual: J' is identified with a space of torsion-free sheaves on itself. This autoduality naturally fits into the framework of the geometric Langlands conjecture; I hope to sketch this relation in my talk.
March 20, 2013
Ching-Jui Lai :
4 p.m. in SEO 427
Abstract
For the purpose of birational classification of projective
varieties, the minimal model program/conjecture (Mori's program) aims to
construct a good representative in the birational class of a given
variety. The MMP is established for varieties of general type by BCHM. For
the remaining cases, we show that the MMP (with abundance) can be reduced
to the case of varieties of Kodaira dimension zero and the Nonvanishing
conjecture.
April 3, 2013
Angela Gibney :
4 p.m. in SEO 427
Abstract
First Chern classes of globally generated vector bundles on a projective variety X are semi-ample divisor classes, which give rise to morphisms on X. In this talk I will introduce a class of globally generated vector bundles on the moduli space of stable pointed rational curves which come from the conformal field theory of Tsuchiya, Ueno and Yamada. Recently work of Fakhruddin has resulted in combinatorial methods for studying these divisor classes. I will explain the basic tools used to work with these divisors and some of their remarkable properties. I will also describe an open problem related to level rank duality of conformal blocks.
April 10, 2013
Martin Olsson :
4 p.m. in SEO 427
Abstract
I will give an overview of recent progress on several related questions on independence of $\ell $ for correspondences acting on $\ell $-adic sheaves.
The main focus will be on local terms whose rationality and independence of $\ell $ implies global independence of $\ell $ results using trace formulas.
In the case of constant coefficients the calculation of local terms is reduced to intersection theory calculations via the theory of Borel-Moore homology
and localized chern classes.
I will also discuss various results on independence of $\ell $ for correspondences acting on complexes of sheaves.
April 17, 2013
Artie Prendergast-Smith :
4 p.m. in SEO 427
Abstract
The Morrison--Kawamata cone conjecture predicts that, for a large class of "Calabi--Yau-like" varieties, certain cones of divisors are "finite up to automorphisms". I will start by explaining the conjecture and its geometric consequences. Then I will discuss how Fano manifolds of index n-1 give rise to a class of examples in which the conjecture can be verified.
This is joint work with Izzet Coskun.
April 24, 2013
Sam Grushevsky :
4 p.m. in SEO 427
Abstract
We use meromorphic differentials with real periods on Riemann surfaces to define local coordinates on the moduli space, similar to the period coordinates for the moduli of abelian differentials, and apply this construction to study the geometry of moduli spaces of curves, in particular focusing on complete subvarieties and homology, as well as cusps of plane curves. This talk will not assume advanced background in either moduli stacks or Teichmuller dynamics.
(joint work with Igor Krichever)
May 1, 2013
Jack Huizenga :
4 p.m. in SEO 427
Abstract
A fundamental problem in algebraic geometry is to determine when a zero-dimensional subscheme of a variety imposes independent conditions on sections of a line bundle. More generally, one can consider when a scheme imposes independent conditions on sections of a vector bundle. Studying such questions amounts to studying the birational geometry of Hilbert schemes of points, or base loci of theta divisors on moduli spaces of sheaves. We will discuss how Bridgeland stability gives a natural way of decomposing the ideal sheaf of a zero-scheme which helps solve the higher rank interpolation problem.
May 22, 2013
Lars Halvard Halle :
4 p.m. in SEO 427
Abstract
Let K be a complete discretely valued field with residue field k, and let X be a smooth K-variety
with trivial canonical sheaf. To such a variety one can associate an invariant known
as the "motivic zeta function" of X. This is a formal power series with coefficients
in the Grothendieck ring of k-varieties, which measures how the set of rational points of X
varies under ramified extension of K.
I will talk about joint work with J. Nicaise, where we investigate motivic zeta functions for
semi-abelian varieties. In particular, we prove that the motivic monodromy conjecture holds for these varieties.
Sept. 3, 2013
Rahul Pandharipande :
4 p.m. in SEO 1227
Abstract
The moduil space of curves carries tautological
cohomology classes. I will discuss the study of
relations amongst these classes starting with ideas
of Mumford in 1980s. The subject advanced in the
1990s with conjectures of Faber and Faber-Zagier.
I will explain the current state of affairs based
on Pixton's conjectures related to
cohomological field theories. The talk represents
joint work with A. Pixton and D. Zvonkine.
Sept. 11, 2013
Kevin Tucker :
4 p.m. in SEO 427
Abstract
The F-pure threshold is a positive characteristic invariant of singularities, and can be thought of as an analog
of the log canonical threshold in characteristic zero. These two invariants have numerous properties in
common, although showing them generally requires involves different methods. In this talk, I will describe
recent work with K. Schwede showing the rationality of the F-pure thresholds of ideals in power series rings.
Kevin Tucker :
4 p.m. in SEO 427
Sept. 18, 2013
Botong Wang :
4 p.m. in SEO 427
Abstract
Given a smooth manifold $X$, the set $\mathbf{R}(X, n)\stackrel{\textrm{def}}{=}Hom(\pi_1(X), Gl(n, \mathbb{C}))$ has naturally an algebraic variety structure.
Each element in $\mathbf{R}(X, n)$ corresponds to a local system on $X$. The local structure of $\mathbf{R}(X, n)$ at a point $\rho$ can be understood by studying
the deformation theory of the associated local system $L_\rho$. As a general principle, such deformation problem is governed by a differential graded Lie algebra (DGLA).
In this talk, we will discuss the local structure of some canonically defined subvarieties of $\mathbf{R}(X, n)$: $V^i_k(X, n)=\{\rho\in\mathbf{R}(X, n)|\dim H^i(X, L_\rho)\geq k\}$.
This is equivalent to studying deformation problem with cohomology constrains. We will introduce a new principle, that such deformation problem with cohomology constrains is governed by a DGLA together with a module of this DGLA.
Particularly nice results can be obtained, when $X$ is a compact K\"ahler manifold.
This is joint work with Nero Budur.
Oct. 2, 2013
Linquan Ma :
4 p.m. in SEO 427
Abstract
Let $(R,\mathfrak{m})$ be a local ring of positive characteristic. We show that when $H_{\mathfrak{m}}^i(R)$ has finite length for all $i<\dim R$, $R$ is F-injective if and only if every ideal generated by a system of parameters is Frobenius closed. As a corollary, we answer a question of Takagi that F-injective singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. Some partial results in characteristic 0 for Du Bois singularities will also be discussed.
Oct. 9, 2013
Jason Starr :
4 p.m. in SEO 427
Abstract
There are several classical results asserting existence of rational points of smooth, projective varieties defined over global function fields, e.g., $F_p(t)$: the Tsen-Lang theorem about points on low degree complete intersections in projective space, the Brauer - Hasse - Noether theorem that "period equals index", equivalent to existence of points on twists of Grassmannians, and the "split" case of Serre's "Conjecture II" (proved also in both the split and non-split case by Harder), equivalent to existence of points on twists of projective homogeneous varieties with Picard rank 1. In joint work with Chenyang Xu, and using work of Esnault and of de Jong - He - Starr in an essential way, we find a new, uniform proof of these results, as well as some extensions, by studying rational curves on a lift of the variety to characteristic 0. This will be a down-to-earth lecture with examples; no prior knowledge of "global function fields", "Brauer groups", "rational connectedness" or "rational simple connectedness" will be needed.
Oct. 16, 2013
Remke Kloosterman :
5 p.m. in SEO 427
Abstract
Cheltsov proved that a nodal hypersurface $X$ of degree $d$, which is
not Q-factorial, has at least $(d-1)^2$ nodes, and if equality holds
then $X$ contains a plane. We present a new proof for this result and
explain how one can generalize our methods to other cases such as
hypersurfaces with arbitrary semi-quasihomogeneous singularities, nodal
double solids and nodal complete intersection threefolds.
Orsola Tommasi :
4 p.m. in SEO 427
Abstract
It is well known that the cohomology of the moduli space A_g of
g-dimensional principally polarized abelian varieties stabilizes when
the degree is smaller than g. This is a classical result of Borel on the
stable cohomology of the symplectic group. By work of Charney and Lee,
also the stable cohomology of the minimal compactification of Ag, the
Satake compactification, is explicitly known.
In this talk, we consider the stable cohomology of toroidal
compactifications of A_g, concentrating on the perfect cone
compactification and the matroidal partial compactification. We prove
stability results for these compactifications and show that all stable
cohomology is algebraic. This is joint work with S. Grushevsky and K.
Hulek.
Oct. 23, 2013
Aaron Pixton :
4 p.m. in SEO 427
Abstract
The tautological ring of the moduli space of smooth curves of genus g is the subring of its Chow ring generated by the kappa classes. The Faber-Zagier relations are an explicit algebraic description of a large number of relations in this ring, possibly giving all the relations. I will discuss two families of tautological relations defined in a similar fashion to the Faber-Zagier relations.
Oct. 30, 2013
Claudiu Raicu :
4 p.m. in SEO 427
Abstract
The space $Mat(m,n)$ of $m\times n$ matrices admits a natural action of the group $\textrm{GL}_m \times \textrm{GL}_n$ via row and column operations on the matrix entries. The invariant closed subsets are the closures of the orbits of constant rank matrices. I will explain how to describe the local cohomology modules of the ring $S$ of polynomial functions on $Mat(m,n)$ with support in these orbit closures, and mention some consequences of the methods employed to computing minimal free resolutions of invariant ideals in $S$. These ideals correspond to nilpotent scheme structures on the orbit closures, and their study goes back to the work of De Concini, Eisenbud and Procesi in the 80s. Joint work with Jerzy Weyman.
Nov. 6, 2013
Dawei Chen :
4 p.m. in SEO 427
Abstract
The cone of effective divisors plays a central role in the birational geometry of a variety X. In this talk I will give an introduction to this subject and report some recent progress (joint with Izzet Coskun) when X is the moduli space of pointed genus one curves.
Nov. 7, 2013
Steven Sam :
noon in SEO 427
Abstract
I'll give a brief overview of some results in "Boij-Soderberg theory", which on the algebraic side is the study of cones of Betti tables of graded modules, and on the geometric side is the study of cones of cohomology tables of coherent sheaves. I'll state some problems and possible research directions (for example, related to noncommutative algebra and degeneracy loci formulae).
Nov. 13, 2013
John Lesieutre :
4 p.m. in SEO 427
Abstract
I will explain the failure of several "positivity"
properties of divisors: nefness is not an open condition in families;
the diminished base locus of a divisor is not always a closed set;
Zariski decompositions do not necessarily exist in dimension three;
and asymptotic multiplicity invariants are not always finite in the
relative setting.
Nov. 20, 2013
Morgan Brown :
4 p.m. in SEO 427
Abstract
The derived category of an algebraic variety is a categorical invariant which is coarser than the category of coherent sheaves. There are many interesting examples in geometry and representation theory of varieties or algebras with different categories of sheaves or modules but equivalent derived categories. For example, if $G$ is a finite subgroup of $SL(3, \mathbb{C})$, Bridgeland, King, and Reid showed there is a derived equivalence between $G$ equivariant sheaves on $\mathbb{C}^3$ and sheaves on a minimal resolution of the quotient. I will show how in many cases one can understand these equivalences by factoring them into simple equivalences called tilts.
Nov. 26, 2013
Karl Schwede :
noon in SEO 1227
Abstract
We prove a new inversion of adjunction statement for rational and Du Bois singularities. Roughly speaking, this says that if we have a family over a smooth base with Du Bois special fiber and rational generic fiber, then the total space also has rational singularities. Furthermore, we even generalize this result to the context of rational and Du Bois pairs as defined by Kollár and Kovács. Imprecisely, a pair
$(X, D)$ is Du Bois if the failure of $X$ to be Du Bois is equal to the failure of $D$ to be Du Bois. In order to accomplish our inversion of adjunction result we need to prove many recent results on Du Bois singularities for pairs, and I will describe some of these ideas. This is joint work with Sandor Kovács.
Jan. 15, 2014
Tiankai Liu :
4 p.m. in SEO 427
Abstract
The Coolidge-Nagata conjecture asserts that every rational curve in the complex projective plane that has only cusps (i.e., for which the normalization map is bijective) can be transformed into a line via a birational automorphism of the plane. We will discuss some progress towards this conjecture, and various techniques for studying cuspidal rational curves.
Jan. 29, 2014
Ilya Smirnov :
4 p.m. in SEO 427
Abstract
Hilbert-Kunz multiplicity is an invariant of a local ring of positive characteristic introduced by Paul Monsky. In the last 15 years, the Hilbert-Kunz theory became an active subject of research driven by its connection to tight closure, singularity theory and comparison to the classical theory of Hilbert-Samuel multiplicity.
In this talk, I will compare these multiplicity theories focusing on their use to study singularities.
Feb. 5, 2014
Wenbo Niu :
4 p.m. in SEO 427
Abstract
I will show first that on an algebraic curve (singular) a
MJ-multiplier ideal is essentially the same as an integrally closed ideal.
Secondly, I will show that on a curve (singular) by comparing
MJ-multiplier ideal with the conductor ideal, we can deduce a criterion
when the curve is locally a complete intersection.
Feb. 12, 2014
Valentino Tosatti :
4 p.m. in SEO 427
Abstract
A result of Nakamaye states that the augmented base locus of a
nef and big line bundle on a smooth projective variety over the complex
numbers equals its null locus, i.e. the union of all irreducible
subvarieties where the restriction of the bundle has volume zero. This was
later extended to R-divisors by Ein-Lazarsfeld-Mustata-Nakamaye-Popa, and
more recently there has been renewed interest in this theorem, especially
in positive characteristic.
I will discuss a different extension of this theorem, to all nef real
(1,1) classes on compact complex manifolds. The null locus of a (1,1)
class is defined in the same way as for a line bundle, but defining the
augmented base locus takes some work and was done by Boucksom (who called
it the non-Kahler locus). The main result I will discuss then says that on
any compact complex manifolds the null locus of any nef real (1,1) class
coincides with its non-Kahler locus. I will also mention some of the
consequences of this theorem. This is joint work with Tristan Collins.
Feb. 25, 2014
Roi Docampo :
4 p.m. in SEO 712
Feb. 26, 2014
Zhiyu Tian :
4 p.m. in SEO 427
Abstract
Given an algebraic variety $X$ over a field $F$ (e.g. number fields, function fields), a natural question is whether the set of rational points $X(F)$ is non-empty. And if it is non-empty, how many rational points are there? In particular, are they Zariski dense? Do they satisfy weak approximation? For cubic hypersurfaces defined over the function field of a complex curve, we know the existence of rational points by Tsen' s theorem or the Graber-Harris-Starr theorem. In this talk, I will discuss the weak approximation property of such hypersurfaces.
Feb. 27, 2014
Claude Sabbah :
2 p.m. in SEO 636
Abstract
Given a regular function f on a smooth quasi-projective variety U, the de Rham complex of U relative to the twisted differential d + df can be equipped canonically with a filtration (the irregular Hodge filtration) for which the associated hypercohomology spectral sequence degenerates at E_1. A logarithmic version of this de Rham complex (relative to a suitable compactification of U) has been introduced by M. Kontsevich, who showed the independence of the dimension of the corresponding cohomologies with respect to the differential ud +vdf, for u,v arbitrary complex numbers. This leads to bundles on the projective line of the (u:v) variable, on which we construct a natural connection for which the Harder-Narasimhan filtration satisfies the Griffiths transversality property and standard limiting properties at v=0. This is a joint work with Hélène Esnault (Berlin) and Jeng-Daw Yu (Taipei).
March 5, 2014
Colleen Robles :
3 p.m. in SEO 636
Abstract
I will describe a program to formulate and answer in a well-posed manner the question: what are the extremal degenerations of a smooth variety? That is, what is the `least singular' variety X can degenerate to? and what is the `most singular' variety X can degenerate to? These and related questions are being investigated, in various subsets of collaboration, by Mark Green, Phillip Griffiths, Matt Kerr, Greg Pearlstein and myself.
March 11, 2014
Shihoko Ishii :
3 p.m. in SEO 712
March 12, 2014
Yaim Cooper :
4 p.m. in SEO 427
Abstract
Stable quotient spaces provide an alternative to stable maps for compactifying spaces of maps. In this talk I will discuss spaces of stable quotients which compactify the space of degree $d$ maps of genus $g$ curves to $P^n$. I will describe what is known about the geometry of these spaces. I will also discuss the relationship between these spaces and the corresponding spaces of stable maps from the perspective of mirror symmetry and the perspective of the minimal model program.
March 13, 2014
Shihoko Ishii :
4 p.m. in SEO 712
April 2, 2014
Yi Zhu :
4 p.m. in SEO 427
Abstract
Iitaka's philosphy claims that whenever we have a theorem for proper varieties, we should have a counter-theorem for open varieties. In this talk, I will introduce this philosophy with several examples. Then I will report the recent progress on the theory of "log rational curves" on log pairs. As another piece of evidence of Iitaka's philosophy, this theory generalizes the classical theory of rational curves on proper varieties. This is a joint work with Qile Chen.
April 9, 2014
Chunyi Li :
4 p.m. in SEO 427
Abstract
The deformation of the Hilbert scheme of points on the projective plane is studied by Hitchin, Nevins and Stafford via different approaches. I will introduce these constructions, and talk about my recent results on the minimal model program of the deformation of Hilb P2.
April 16, 2014
Giulia Sacca :
4 p.m. in SEO 427
Abstract
We establish the semistablity of Lazersfeld-Mukai bundles for some
class of rank zero sheaves on a K3 surface, providing examples of
moduli spaces which, locally around a singular point, are isomorphic
to a quiver variety in the sense of Nakajima.
The singularities of these moduli spaces arise from the choice of a
specific polarization and admit natural symplectic resolutions
corresponding to the choice of a general polarization. We show that
these resolutions correspond, via the above isomorphism, to natural
symplectic resolutions of the quiver variety coming from variations of
GIT quotients. This is joint work with E. Arbarello.
April 18, 2014
Rahul Pandharipande :
3 p.m. in SEO 636
Abstract
I will explain our recent proof (with R. Thomas) of the KKV formula
governing higher genus curve counting in arbitrary classes on K3 surfaces.
The subject intertwines Gromov-Witten, Noether-Lefschetz, and Donaldson-Thomas
theories. A tour of these ideas will be included in the talk.
April 23, 2014
Noah Giansiracusa :
4 p.m. in SEO 427
Abstract
The moduli space $\bar{M}_{0,n}$, a compactification of the space of n distinct points on the Riemann sphere, has served as a fertile testing ground to explore many phenomena of moduli spaces in algebraic geometry. One tantalizing question is to describe the convex cone of effective divisor classes and Cox ring of these spaces. I'll discuss joint work with B. Doran and D. Jensen in which we provide a new perspective on this question in terms of simplicial complexes and show how this relates to recent exciting work of Castravet, Tevelev, and Opie.
July 24, 2014
Donghoon David Hyeon :
3 p.m. in SEO 427
Abstract
We study how state polytopes (from GIT, these tell you whether a given point is semistable or not) change according to the choice of the maximal torus. We define the notion of generic state polytope generalizing the notion of generic initial ideals, and prove that any point is stable with respect to a general maximal torus. This fundamental observation allows one to precisely formulate a conjecture of D. Bayer and I. Morrison about the geometry of the ideal and the computational complexity of its Groebner bases.
Sept. 3, 2014
Jack Huizenga :
4 p.m. in SEO 427
Abstract
Let $v$ be the set of numerical invariants of a sheaf on $\mathbb{P}^2$. The moduli space $M(v)$ parameterizes isomorphism classes of semistable sheaves with Chern character $v$. In this talk, I will discuss recent work with Izzet Coskun computing the cone of ample divisors on $M(v)$ for many choices of the character $v$. Our results in particular cover the case where the rank and first Chern class of $v$ are coprime and the discriminant of $v$ is sufficiently large.
Sept. 10, 2014
Yank\i\ Lekili :
4 p.m. in SEO 427
Abstract
Motivated by homological mirror symmetry, for each n, we define a certain
finite dimensional graded associative algebra $E_n$ and study the moduli
space of minimal $A_\infty$ structures on $E_n$. Surprisingly, we can identify
this moduli space with a modular compactification (due to Smyth) of the
moduli of curves of genus 1 with $n$ marked points. The corresponding moduli
stack, denoted by $\mathcal{M}_{1,n}(n-1)$, is a projective irreducible DM-stack. Our
realization of this space gives a description of these moduli spaces and
the universal curves over them by explicit equations. This enables us to
discover various geometric properties of $\mathcal{M}_{1,n}(n-1)$. For example, we prove
that they are normal and Gorenstein, show that their Picard groups have no
torsion and that they have rational singularities if and only if $n \leq 11$.
The case of $n=1$ and the algebra $E_1$ was studied earlier by the speaker and
Perutz, the current report is for $n>1$ on a joint work with A. Polishchuk.
Sept. 17, 2014
Matthew Woolf :
4 p.m. in SEO 427
Abstract
When can you write down the general solution to a polynomial equation in a way that gives you each solution only once? Algebraic geometers know that this condition corresponds to rationality of the corresponding hypersurface. It is is an easy fact that a smooth hypersurface of degree at least two more than its dimension cannot be uniruled, and in particular, cannot be rational. Improving on this result, János Kollár proved using reduction to positive characteristic that a very general hypersurface of degree greater than approximately two thirds its dimension is not rational. We will discuss recent work with Eric Riedl which extends this to hypersurfaces of degree greater than approximately half their dimension and certain singular hypersurfaces.
Sept. 24, 2014
Mihai Fulger :
4 p.m. in SEO 427
Abstract
If $\pi:X\to Y$ is a morphism of projective varieties over an algebraically closed field,
and $Z$ is an effective $k$-cycle on $X$, then $\pi_*Z=0$ iff $Z$ is a combination of subvarieties
of $X$ that are contracted by $\pi$.
When working not with cycles, but with cycle classes (modulo numerical equivalence),
it is natural to ask when can we expect a similar geometric conclusion given
the vanishing of a class $\pi_*\alpha$.
I will present progress on this question, in particular leading to new cases of two
conjectures essentially due to Debarre, Jiang, and Voisin. This is joint work with
B. Lehmann.
Oct. 1, 2014
Benjamin Antieau :
4 p.m. in SEO 427
Abstract
I will discuss recent joint work with Daniel Krashen and Matthew Ward on using techniques from dg categories to study derived categories of abelian fibrations. Time-permitting, I will talk about applications to elliptic Calabi-Yau threefolds and to a conjecture of Popa and Schnell.
Oct. 8, 2014
Jarek Bucynzki :
4 p.m. in SEO 427
Abstract
We fix a projective variety $X\subset \mathbb{P}^n$ and an integer $r$. We are interested in the defining equations of the $r$-th secant variety to the $d$-uple Veronese reembedding of $X$, and we assume $d$ is sufficiently large. One of the interesting cases is when $X= \mathbb{P}^n$. With these assumptions we prove that the $(r+1)$-minors of the catalecticant matrix with linear entries are sufficient to define the secant variety set-theoretically if and only if the Hilbert scheme parametrising $0$-dimensional Gorenstein subschemes of $X$ of length $r$ is irreducible. In particular, if $X$ is smooth and either $\dim X$ is at most $3$ or $r$ is at most $13$, then the minors are sufficient. If $\dim X$ is at least $4$ and $r$ is sufficiently large, then the locus defined by the minors has some additional components. These results motivate introducing cactus varieties, which generalise the secant varieties, and received a lot of attention since then.
The talk will be based on joint works with:
1) Adam Ginensky and Joseph Landsberg (JLMS 2013);
2) Weronika Buczynska (JAG 2014);
3) Joachim Jelisiejew (in preparation).
Oct. 15, 2014
Chris Skalit :
3 p.m. in SEO 636
Abstract
Suppose that $(A, \mathfrak{m})$ is a regular local ring whose $\mathfrak{m}$-adic completion is a power series over a discrete valuation ring. For properly-meeting, closed subschemes of complimentary dimension $Y, Z \subseteq \mathrm{Spec} (A)$, we show that the Serre intersection multiplicity $\chi^A(\mathcal{O}_Y,\mathcal{O}_Z) := \sum_{i=0}^{\infty}{(-1)^i \ell(\mathrm{Tor}_i^A(\mathcal{O}_Y,\mathcal{O}_Z))}$ is bounded below by the product of the Hilbert-Samuel multiplicities of $Y$ and $Z$. We also investigate the geometric significance of achieving this lower bound by examining the proper transforms of $Y$ and $Z$ under the blowup of $\mathrm{Spec} (A)$.
Oct. 21, 2014
Jun-Muk Hwang :
4 p.m. in SEO 636
Abstract
Let $X_1$ and $X_2$ with $\mathrm{dim} X_1 = \mathrm{dim} X_2$ be two projective manifolds of Picard number 1 in projective space.
Assume that both $X_1$ and $X_2$ are covered by lines. Let $\varphi: U_1 \to U_2$ be a biholomorphic map between two connected Euclidean
open subsets $U_1 \subset X_1$ and $U_2 \subset X_2$. Suppose that both $\varphi$ and $\varphi^{-1}$ send pieces of lines to pieces of lines.
We show that $\varphi$ can be extended to a biregular morphism $\Phi: X_1 \to X_2$. This was proved by Hwang-Mok in 2001
when the indices of $X_1$ and $X_2$ are bigger than 2 and the new result is when the indices are 2. In this case, the covering family
of lines form webs of rational curves. We exploit the monodromy of the webs of lines to extend the holomorphic map.
Oct. 22, 2014
Alex Kuronya :
4 p.m. in SEO 427
Abstract
Line bundles sharing some but not all the good properties of
ampleness have been investigated for quite some time, here we will focus on
the cohomological point of view. Building on earlier work of Sommese and
Demailly-Peternell-Schneider, Totaro came up with a very satisfactory theory
of line bundles with partially vanishing higher cohomology to which we will
refer as q-ample.
As it turns out, q-ampleness gives rise to interesting applications,
including a useful concept of ampleness for subvarieties, where among others,
one retains a Lefschetz hyperplane theorem (as showed by Ottem).
The main focus of this talk will be a generalization of Kodaira vanishing
to q-ample line bundles.
Oct. 29, 2014
Brooke Ullery :
4 p.m. in SEO 427
Abstract
If X is a smooth variety embedded in projective space, we can form a new variety by looking at the closure of the union of all the lines through 2 points on X. This is called the secant variety to X. Similarly, the Hilbert scheme of 2 points on X parametrizes all length 2 zero-dimensional subschemes. I will talk about how these two constructions are related. More specifically, I will show how we can use certain tautological vector bundles on the Hilbert scheme to help us understand the geometry of the secant variety, leading to a proof that for sufficiently positive embeddings of X, the secant variety is a normal variety.
Nov. 5, 2014
Daniel Litt :
4 p.m. in SEO 427
Abstract
Work of Lefschetz (in 1924) and Grothendieck (in SGA II) provides many relationships between properties of a smooth projective variety X and an ample divisor D in X. For example, the singular or l-adic cohomology of X agrees with that of D in low degree; X and D have the same Picard group if X has dimension at least 4; and X and D have the same fundamental group if X has dimension at least 3. I'll describe a general result which encompasses some of these Lefschetz hyperplane theorems and many new ones, comparing maps out of X to maps out of D. The case when the target of these maps is a moduli scheme or stack is of particular interest; for example, one may take the target to be Mg, and thus compare families of curves over X to families over D.
Nov. 12, 2014
Howard Nuer :
4 p.m. in SEO 427
Abstract
Since the work of Arcara, Bertram, Coskun, and Huizenga on the application of Bridgeland stability conditions to the study of the birational geometry of $\mathbb{P}^{2[n]}$, there has been much progress in applying similar ideas to a Hassett-Keel-type approach to the study of the birational geometry of more general moduli spaces of sheaves on other surfaces. In this talk, I will discuss previous and ongoing work on the application of Bridgeland stability techniques to running the MMP (minimal model program) on moduli spaces of stable sheaves on an Enriques surface. As an application of the tools I discuss, I will describe the nef cone of the Hilbert scheme of points on an Enriques surface explicitly in terms of the classical geometry of the Enriques surface as well as give a modular description of the first minimal model.
Nov. 14, 2014
Chenyang Xu :
1 p.m. in SEO 636
Abstract
(Joint with Xiaowei Wang) In the preface of the second version of the book Geometric Invariant Theory, the authors asked that whether the approach of using asymptotic chow stability to construct moduli space of canonically polarized manifolds could yield a natural compactification. By comparing different stability notions and the related invariants, we show that there exists families of canonically polarized manifolds, e.g., hypersurfaces in $\mathbb{P}^3$, which don't have asymptotical Chow semistable limits. This implies that unlike Giesker and Mumford's result in the curve case, in higher dimension, the method fails.
Nov. 19, 2014
Eric Riedl :
4 p.m. in SEO 427
Nov. 26, 2014
NO SEMINAR :
4 p.m. in SEO 427
Dec. 3, 2014
Emanuele Macri :
4 p.m. in SEO 427
Abstract
I will present a new proof and a generalization a result by Maciocia and Piyaratne on the existence of Bridgeland stability conditions on any abelian threefold.
As an application, we deduce the existence of Bridgeland stability conditions on a number of Calabi-Yau threefolds, namely Calabi-Yau threefolds of abelian type and Kummer threefolds.
As in the work of Maciocia and Piyaratne, the idea is to show a Bogomolov-Gieseker type inequality involving Chern classes of certain stable objects in the derived category; this was conjectured by Bayer, Toda, and myself.
Our approach uses the multiplication maps on abelian threefolds instead of Fourier-Mukai transforms.
This is joint work with Arend Bayer and Paolo Stellari.
Feb. 3, 2015
Dawei Chen :
3 p.m. in SEO 1227
Abstract
Consider strata of holomorphic one-forms on Riemann surfaces with prescribed number and multiplicity of zeros. They define flat structures realizing the underlying surfaces as plane polygons whose boundary edges are identified suitably. In this talk, I will report some work in progress on degenerations of holomorphic one-forms in the same stratum when the underlying Riemann surfaces become nodal, with a focus on the interplay between algebraic geometry and flat geometry.
Feb. 11, 2015
Evangelos Routis :
4 p.m. in SEO 427
Abstract
I will report on recent work on weighted compactifications of the configuration space of n labeled points on an arbitrary nonsingular variety. The construction that we will discuss provides a generalization of the Fulton- MacPherson compactification that is parallel to the generalization of the moduli space of n-pointed stable curves carried out by Hassett. As an application, I will give a presentation of the Chow ring of these weighted compactifications, which can be used to compute the Chow ring of Hassett's spaces in genus 0.
March 2, 2015
Robin Hartshorne :
4 p.m. in SEO 1227
Abstract
My goal is to introduce some of the basic ideas of deformation theory in algebraic geometry, with interesting examples. This will include first order deformations, the Hilbert scheme, smoothable singularities, higher order deformations and smoothness. Then as time permits, deformation of abstract varieties, rigid varieties, prorepresentable functors, fine and coarse moduli spaces, and examples from the moduli of curves.
The text for this course is my book "Deformation theory" Springer 2010, GTM 257.
Prerequisities are a basic knowledge of algebraic geometry such as in GTM 52.N
March 3, 2015
Robin Hartshorne :
4 p.m. in SEO 1227
Abstract
My goal is to introduce some of the basic ideas of deformation theory in algebraic geometry, with interesting examples. This will include first order deformations, the Hilbert scheme, smoothable singularities, higher order deformations and smoothness. Then as time permits, deformation of abstract varieties, rigid varieties, prorepresentable functors, fine and coarse moduli spaces, and examples from the moduli of curves.
The text for this course is my book "Deformation theory" Springer 2010, GTM 257.
Prerequisities are a basic knowledge of algebraic geometry such as in GTM 52.
March 4, 2015
Robin Hartshorne :
4 p.m. in SEO 427
Abstract
My goal is to introduce some of the basic ideas of deformation theory in algebraic geometry, with interesting examples. This will include first order deformations, the Hilbert scheme, smoothable singularities, higher order deformations and smoothness. Then as time permits, deformation of abstract varieties, rigid varieties, prorepresentable functors, fine and coarse moduli spaces, and examples from the moduli of curves.
The text for this course is my book "Deformation theory" Springer 2010, GTM 257.
Prerequisities are a basic knowledge of algebraic geometry such as in GTM 52.
March 6, 2015
Robin Hartshorne :
1 p.m. in SEO 427
Abstract
My goal is to introduce some of the basic ideas of deformation theory in algebraic geometry, with interesting examples. This will include first order deformations, the Hilbert scheme, smoothable singularities, higher order deformations and smoothness. Then as time permits, deformation of abstract varieties, rigid varieties, prorepresentable functors, fine and coarse moduli spaces, and examples from the moduli of curves.
The text for this course is my book "Deformation theory" Springer 2010, GTM 257.
Prerequisities are a basic knowledge of algebraic geometry such as in GTM 52.
March 11, 2015
Daniel Erman :
1 p.m. in SEO 1227
Abstract
The Kakeya Needle Problem has its origins in harmonic analysis, but it has led to a number of interesting related questions about algebra and geometry over finite fields. I’ll first give background on this famous problem. Then I’ll talk about recent work of myself and Jordan Ellenberg which uses degeneration techniques to make progress on some of the related algebraic questions.
March 18, 2015
John Lesieutre :
4 p.m. in SEO 427
Abstract
There are currently few known examples of automorphisms of smooth threefolds with positive entropy, i.e. for which the induced map on $N^1(X)$ has an eigenvalue larger than 1. I'll say a bit about why one might care, and what the situation is in dimension 2. Then I'll describe some constraints on smooth threefolds $X$ admitting such automorphisms. For example, I'll show that if $X$ is constructed as a blow-up of $\mathbb{P}^1 x \mathbb{P}^2$ or $\mathbb{P}^3$, any positive entropy automorphism admits an equivariant map to a surface. I'll also give a related example of a non-uniruled, terminal threefold with infinitely many $K_X$-negative extremal rays on the cone of curves.
April 1, 2015
Claire Voisin :
4 p.m. in SEO 427
Abstract
The Lüroth problem asks whether a unirational variety
is rational. It has a negative answer starting from dimension 3
and can be attacked by various geometric approaches. For the stable Lüroth
problem, where "rational" is replaced by "stably rational" , only
the Artin-Mumford approach had been used up to now
to solve the problem in dimension 3.
Using the notion of decomposition of the diagonal, we
exhibit many unirational threefolds which are not stably rational
while their Artin-Mumford invariant is trivial.
April 3, 2015
Rahul Pandharipande :
1 p.m. in SEO 427
April 8, 2015
Tom Nevins :
4 p.m. in SEO 427
Tom Nevins :
4 p.m. in SEO 427
Abstract
A D-module is a quasicoherent sheaf equipped with a flat connection. The existence of a flat connection strongly constrains a quasicoherent sheaf, thus providing some means of control over its space of global sections. I will explain a vanishing theorem for (twisted) D-modules in an equivariant setting, and sketch applications to geometric representation theory and topology.
April 15, 2015
S. K. Yang :
4 p.m. in SEO 427
Abstract
The smallest positive Euler number of a smooth surface of general type is $3$, which is
achieved by the $50$ pairs of fake projective planes and one pair of surfaces with first Betti number $2$, named
as Cartwright-Steger surfaces. We would introduce a few topics related to these surfaces, explain some joint work
with collaborators and answer some open problems in complex ball quotients and geometry of surfaces.
April 22, 2015
Bernd Ulrich :
4 p.m. in SEO 427
Abstract
Iterated socles of modules over a local ring are
obtained by repeated formations of socles. Iterated
socles are ubiquitous, and one of the goals of the
talk is to give explicit formulas for the generators
of iterated socles in terms of the matrices in a free
resolution of the original module. In addition, we
obtain bounds on iterated socles using the new notion
of distance in free resolutions, which serves as a
substitute for degree shifts in resolutions that are
not necessarily graded. This circle of ideas has
applications to integral closures of ideals, depths of
associated graded rings, and resolutions of multiplier
ideals.
April 29, 2015
David Yang :
4 p.m. in SEO 427
Abstract
It has recently been shown by Lev Borisov that the class of the affine line is a zero divisor in the Grothendieck ring of varieties. We place his construction in a more general context, and give a new proof using K3 surfaces rather than Calabi-Yau threefolds.
Aug. 26, 2015
---------- :
4 p.m. in SEO 427
Sept. 2, 2015
Benjamin Schmidt :
4 p.m. in SEO 427
Abstract
The theory of Bridgeland stability conditions has lead to deep results about the geometry of moduli spaces of sheaves on surfaces. One of the main obstacles to do the same on threefolds is the construction of stability conditions. Recent progress on this question in special cases raises the question whether the corresponding moduli spaces can be studied. I will present an approach that uses computations similar to those on surfaces. In the case of projective space, I will show examples of concrete wall-crossing behavior for some Hilbert schemes of curves.
Sept. 9, 2015
Alan Thompson :
4 p.m. in SEO 427
Abstract
I will describe recent joint work with C. Doran, A. Harder and A. Novoseltsev, in which we study the moduli spaces of certain Calabi-Yau threefolds with small Hodge number $h^{2,1}$. Many such Calabi-Yau threefolds admit fibrations by K3 surfaces that are polarized by lattices of high rank. In the case where the polarizing lattice has rank 19, the theory of such fibrations closely parallels the theory of elliptic surfaces: in particular, the coarse moduli space of the K3 surface fibres is a modular curve, and there are analogues of the functional and homological invariants which determine much of the geometry of the threefold total space. Using this structure, it is possible to explicitly map out the moduli spaces of Calabi-Yau threefolds fibred by such K3 surfaces. There is also a beautiful interpretation of mirror symmetry for these Calabi-Yau threefolds, related to (weak) Landau-Ginzburg models of Fano threefolds, which I will describe if time allows.
Sept. 16, 2015
Jinhyung Park :
4 p.m. in SEO 427
Abstract
The Okounkov body is a convex body associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this talk, I introduce two different ways to associate the Okounkov bodies with pseudoeffective divisors, and show that these convex bodies reflect some asymptotic invariants of given pseudoeffective divisors. This is joint work with Sung Rak Choi, Yoonsuk Hyun, and Joonyeong Won.
Sept. 23, 2015
---------- :
4 p.m. in SEO 427
Sept. 25, 2015
Chih-Chi Chou :
1 p.m. in SEO 427
Abstract
In this talk, I will talk about singularities of secant varieties
associated to smooth varieties embedded by sufficiently positive adjoint
line bundles. More specifically, I will show that they are always Du Bois and
talk about the condition of being rational singular.
If I have enough time, I will also talk about possible developments of the techniques
used in this talk.
(This work is joint with Lei Song)
Sept. 30, 2015
David Jensen :
4 p.m. in SEO 427
Abstract
The maximal rank conjecture, which has roots in the work of Noether and Severi in the late 19th and early 20th centuries, predicts the Hilbert function of the general embedding of a general curve. In recent joint work with Sam Payne, we show that this conjecture holds for the Hilbert function evaluated at $m=2$, meaning that such a curve is contained in the expected number of independent quadrics. From this we deduce that the general curve of genus $g$ and degree $d$ in projective space of dimension r is projectively normal if and only if $(r+2)(r+1)/2$ is at least $2d-g+1$. Our proof uses techniques from tropical and nonarchimedean geometry.
Oct. 6, 2015
Alexandru Dimca :
2 p.m. in SEO 427
Abstract
In this talk we recall the definition of a free projective hypersurface and discuss some geometric properties of free curves and
surfaces. This will lead us naturally to the definition of nearly free
hypersurfaces.
Oct. 7, 2015
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4 p.m. in SEO 427
Oct. 14, 2015
Nicola Tarasca :
4 p.m. in SEO 427
Abstract
In this talk, I will discuss loci of curves with subcanonical points inside moduli spaces of curves. For instance, the locus of curves of genus 3 with a marked subcanonical point has two components: the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. I will show how to compute the classes of the closures of these codimension-two loci in the moduli space of stable curves of genus 3 with a marked point. Similarly, I will present the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. Finally, I will discuss the geometric consequences of these computations. This is joint work with Dawei Chen.
Oct. 21, 2015
Matthew Kerr :
4 p.m. in SEO 427
Abstract
An algebraic cycle homologous to zero on a variety leads to an extension of Hodge-theoretic data, and in a variational context to a family of extensions called a normal function. These may be viewed as "horizontal" sections of a bundle of complex tori, and are used to detect cycles modulo algebraic (or rational) equivalence. Conversely, the existence of normal functions can be used to predict that interesting cycles are present...or absent: a famous theorem of Green and Voisin states that for projective hypersurfaces of large enough degree, there are no normal functions (into the intermediate Jacobian bundle associated to these hypersurfaces) over any etale neighborhood of the coarse moduli space.
Inspired by recent work of Friedman-Laza on Hermitian variations of Hodge structure and Oort's conjecture on special (i.e. Shimura) subvarieties in the Torelli locus, R. Keast and I wondered about the existence of normal functions over etale neighborhoods of Shimura varieties. Here the function is supposed to take values in a family of intermediate Jacobians associated to a representation of a reductive group. In this talk I will explain our classification of the cases where a Green-Voisin analogue does *not* hold and where one therefore expects interesting cycles to occur, and give some evidence that these predictions might be "sharp".
Nov. 4, 2015
David Swinarski :
4 p.m. in SEO 427
Abstract
A vector partition function is a function that counts the number of lattice points in a polytope defined by the function's arguments. It is conjectured that the ranks of vector bundles of conformal blocks on the moduli space of curves and the intersection numbers of their first Chern classes with F-curves are given by vector partition functions. I will discuss consequences of these conjectures and progress toward proving them.
Nov. 11, 2015
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4 p.m. in SEO 427
Nov. 12, 2015
Bangere Purnaprajna :
noon in SEO 427
Abstract
(joint work with R. V. Gurjar) I will talk about new results on fundamental groups for some classes of fibered algebraic surfaces with a finite group of automorphisms. The methods actually compute the fundamental groups of the surfaces under study upto finite index. The corollaries include an affirmative answer to Shafarevich conjecture on holomorphic convexity, Nori's well-known question on fundamental groups and free abelianness of second homotopy groups for these surfaces. We also prove a theorem that bounds the multiplicity of the multiple fibers of a fibration for any algebraic surface with a finite group of automorphisms $G$ in terms of the multiplicities of the induced fibration on $X/G$. If $X/G$ is a $\mathbb{P}^1$-fibration, we show
that the multiplicty actually divides $|G|$. This theorem on multiplicity, which is of independent interest, plays an
important role in our theorems.
Nov. 13, 2015
Bangere Purnaprajna :
1 p.m. in SEO 427
Abstract
(joint work with Jungkai Chen) Relations among fundamental invariants plays an important role in algebraic geometry. In this talk, we consider the relations between canonical volume and genus for varieties of general type. We prove an inequality for a $n$-dimensional minimal Gorenstein variety of general type and investigate the compelling extremal cae, when the inequality is an equality. These extremal varieties are natural higher dimensional analogue of Horikawa's surfaces whose invariants satisfy the equality in Noether's inequality. We prove that for extremal varieties of general type of arbitrary dimension, their canonical linear systems are base point free. We give a characterization of these varieties. Moreover, we show that the deformation of these varieties remain in the same type. It is also proved that these extremal varieties of general type are simply connected, and are pluri-regular (in the smooth case). Optimal results on projective normality of pluri-canonical linear systems will also be dealt in this talk. These results give a complete generalization of Horikawa's results in the Annals for all dimensions!
Nov. 18, 2015
Sijong Kwak :
4 p.m. in SEO 427
Abstract
For a projective variety (or scheme), the graded Betti numbers are defined from either
the minimal free resolution of the homogeneous coordinate ring or the Koszul complex.
These extrinsic numbers measure the complexity of the relations between the defining equations
and reflect the intrinsic and geometric information on a variety. In this talk, I'd like to introduce
the results of Castelnuovo and Fano on quadric equations and generalize them to the higher linear
syzygies in the first strand.
As a consequence, I'd like to characterize varieties of minimal degree and Del Pezzo varieties
with respect to linear syzygies. Main ideas are inner projections, mapping cone and partial
elimination ideals due to M. Green.
Nov. 25, 2015
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4 p.m. in SEO 427
Jan. 20, 2016
John Lesieutre :
4 p.m. in SEO 427
Abstract
Tba
Jan. 27, 2016
Chris Skalit :
4 p.m. in SEO 427
Abstract
Let $X$ be a smooth extension of a regular scheme $Y$. Given properly-meeting subschemes $V$ and $W$ of $X$, we show that the associated intersection multiplicities are positive whenever $\dim Y \leq 2$. When $Y$ is one-dimensional, we use these methods to investigate the extent to which the intersection multiplicity can detect transversality.
Feb. 3, 2016
Eric Riedl :
4 p.m. in SEO 427
Abstract
In joint work with David Yang, we prove that for d between (3n+1)/2 and 2n-3, a very general hypersurface in P^n will contain lines but no other rational curves. This partially resolves a conjecture of Voisin.
Feb. 10, 2016
Atanas Atanasov :
4 p.m. in SEO 427
Abstract
We aim to address the following: When is there a (smooth) curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$. Generalizations ask for the dimension of such curves, or replace the point incidence conditions with higher dimensional linear spaces. We will start by relating these statements to a property of the normal bundle of curves in projective space. Next, we will show how to address these questions for $r = 3$ and $d >= g + 3$. The demonstrated techniques generalize significantly and lead to an answer to our question for $d >= g + r$. This is joint work with E. Larson and D. Yang.
Feb. 15, 2016
Aaron Landesman :
2 p.m. in SEO 1227
Abstract
In this talk, we discuss interpolation of projective varieties through points.
It is well known that one can find
a rational normal curve in $\mathbb P^n$
through $n+3$ general points. More recently, it
was shown that one can always find nonspecial curves
through the expected number of general points.
We consider the generalization of this question
to varieties of all dimensions and explain why
rational normal scrolls satisfy interpolation.
We'll also discuss joint work
with Anand Patel on interpolation for del Pezzo surfaces
and present several interesting open interpolation problems.
We'll place particular emphasis on explaining the standard techniques used
to solve interpolation:
deformation theory, specialization, degeneration, and association.
Feb. 17, 2016
NO SEMINAR :
4 p.m. in SEO 427
Abstract
.
Feb. 24, 2016
Jason Starr :
4 p.m. in SEO 427
Abstract
A projective manifold is "Fano" if the expected dimension of the parameter space of rational curves of a given effective curve class increases with the multiple of that class. A conjecture of Cohen-Jones-Segal predicts the topology of these parameter spaces. I will focus on the simplest Fano manifolds, general low degree hypersurfaces in projective space. I will explain work of
Riedl-Yang on irreducibility of the parameter spaces, joint work with Zhiyu Tian on the Picard groups, and joint work with Coskun and Harris that gives the nef cones.
March 7, 2016
Patricia Hersh :
2 p.m. in SEO 1227
Abstract
Sergey Fomin and Michael Shapiro conjectured that certain topological spaces of totally nonnegative real matrices stratified according to which minors are positive and which are 0 are regular CW complexes homeomorphic to closed balls having the (lower) intervals of Bruhat order as their posets of closure relations. We will survey this area, including connections to Lusztig's theory of canonical bases, to electrical networks, and to cluster algebras, and we will discuss how combinatorics and topology were combined in somewhat non-standard ways to obtain a proof of this conjecture. This talk will not assume familiarity with these areas.
March 9, 2016
Roberto Svaldi :
4 p.m. in SEO 427
Abstract
The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.
April 6, 2016
Marton Hablicsek :
4 p.m. in SEO 427
Abstract
In a beautiful paper Deligne and Illusie proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. Later Kato generalized their results to logarithmic schemes. In the talk I give a geometric interpretation of Kato's result using derived intersections generalizing a result of Arinkin, Caldararu and myself.
April 13, 2016
Michael Groechenig :
4 p.m. in SEO 512
April 20, 2016
John Calabrese :
4 p.m. in SEO 427
Abstract
I will discuss a formula describing how some enumerative invariants for Calabi-Yau threefolds behave under birational transformations.
April 27, 2016
Karl Schwede :
4 p.m. in SEO 427
Abstract
We prove a new result relating the local cohomology of scheme-theoretic thickenings of Du Bois singularities to the local cohomology of the original Du Bois singularity. This yields a number of results. For instance, this allows us to generalize a result of Kollár-Kovács to non-projective families. It lets us answer a question of Eisenbud-Mustata-Stillman on the relation between Ext and local cohomology. Finally, it implies that singularities of dense F-injective type deform. This is joint work with Linquan Ma and Kazuma Shimomoto.
April 29, 2016
Asher Auel :
10:30 a.m. in SEO 512
Abstract
The notion of the universal triviality of the Chow group of 0-cycles has emerged as a powerful new invariant for obstructing the stable rationality of algebraic varieties. The degeneration method introduced by Voisin has led to incredible progress in the last few years. I will explain the geometric origins of universal cycles as well as the degeneration method and its application to the rationality problem for hypersurfaces, Fano threefolds, and quadric bundles.
Aug. 24, 2016
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4 p.m. in SEO 427
Aug. 31, 2016
Kevin Tucker :
4 p.m. in SEO 427
Abstract
The F-signature is a numerical invariant of singularities which measures the asymptotic number of splittings of iterates of Frobenius. The positivity of the F-signature characterizes F-regular singularities, which are closely related to KLT singularities in characteristic zero. After giving an overview, I will discuss new transformation rules for F-signature under finite maps. These transformation rules allow us to show finiteness of the etale local fundamental group for F-regular singularities, analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic zero. This is joint work with Javier Cravajal-Rojas and Karl Schwede.
Sept. 7, 2016
John Lesieutre :
4 p.m. in SEO 427
Abstract
Suppose that $X$ is a projective variety. Must the group $\textrm{Aut}(X)/\textrm{Aut}^0(X)$ be finitely generated?
Sept. 14, 2016
Henri Guenancia :
4 p.m. in SEO 427
Abstract
If X is a smooth projective variety (or compact Kähler manifold) with trivial first Chern class, then a famous result of Beauville and Bogomolov asserts that up to a finite étale cover, X is a product of varieties of three possible type: abelians varieties (or tori), Calabi-Yau's or Hyperkähler. These last two classes are defined using properties of the algebra of global holomorphic forms.
If X is singular though (say with torsion canonical bundle and klt singularities) this result is not known and presumably very difficult. In this talk, we will explain that if in addition X is assumed to be strongly stable (which is an infinitesimal version of irreducibility) then X falls into one of the singular analogues of the two categories above Calabi-Yau's and Hyperkähler.
This is ongoing joint work with Stefan Kebekus and Daniel Greb.
Sept. 28, 2016
Natalie Hobson :
4 p.m. in SEO 427
Abstract
Given a simple Lie algebra \g, a positive integer l and an n-tuple of dominant integral weights for \g at level l, one can define a vector bundle on the moduli space of curves known as a vector bundle of conformal blocks. These bundles are nef in the case that the genus is zero and so this family provides potentially an infinite number of elements in Nef(M_0,n\bar) to analyze.
It is natural to ask how this infinite family of conformal blocks divisors lives in Nef(M_0,n\bar). Is the subcone generated by conformal blocks divisors polyhedral? In this talk, we give several results to this question for specific cases of interest. To show our results, we use a correspondence of the ranks of these bundles with computations in the quantum cohomology of the Grassmannian.
Oct. 5, 2016
Matthew Woolf :
4 p.m. in SEO 427
Abstract
In this talk, I will discuss joint work with Eric Reidl showing that a general Calabi-Yau or general type complete intersection over a field of positive characteristic is not uniruled. I will also discuss applications of this work to deducing bounds on the dimension of complete intersections containing too many rational curves.
Oct. 12, 2016
Behrouz TAJI :
4 p.m. in SEO 427
Abstract
By proving Calabi's conjecture, Yau proved that the Chern classes of a compact manifold with
ample canonical bundle encode the symmetries of the Kahler-Einstein metric via a simple inequality
-- the so-called Miyaoka-Yau inequality. Furthermore it was shown that in the case of equality, the
universal cover is the ball. Later, Tsuji established the MY inequality for smooth minimal models of general
type by constructing singular Kahler-Einstein metrics. The singularity of these metrics are usually a major
obstacle towards uniformization; a problem that has not yet been resolved via analytic methods. In a joint
project with Greb, Kebekus and Peternell, we take a different approach, via Hermitian-Yang-Mills theory and
Simpson's groundbreaking work on complex variation of Hodge structures, and we prove the MY inequality
for minimal models of general type and establish a uniformization result for their canonical models.
Oct. 19, 2016
- :
4 p.m. in SEO 427
Oct. 26, 2016
Jian Xiao :
4 p.m. in SEO 427
Abstract
We present several (new) correspondences between convex bodies and the theory of holomorphic line bundles on smooth projective varieties or Kähler manifolds, thus extending the dictionary between convex geometry and complex geometry. An important ingredient is a refined structure of the movable cone of curves. This is joint work with Brian Lehmann.
Nov. 2, 2016
Giulia Sacca :
4 p.m. in SEO 427
Abstract
In recent years, there have been an increasing number of connections between
cubic 4folds and hyperkahler manifolds. The aim of the talk is to
give background in this area and then describe another instance of
this phenomenon, which is carried out in
joint work with R. Laza and C. Voisin:
Given a general cubic 4fold X, one may consider the universal family
Y_U \to U of smooth hyperplanes sections of X and the relative
Intermediate Jacobian fibration f: J_U \to U. In 1995 Donagi and
Markman constructed a holomorphic symplectic form on J_U, with respect
to which the fibration f is Lagrangian. Since then, there have been
many attempts to find a smooth hyperkahler compactification of J_U.
This was conjectured to exist and to be deformation equivalent to
O'Grady's 10--dimensional exceptional example. With Radu Laza and Claire
Voisin, we solve this conjecture by using relative compactified Prym
varieties.
Nov. 9, 2016
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4 p.m. in SEO 427
Nov. 15, 2016
Gabor Szekelyhidi :
11 a.m. in SEO 612
Abstract
We show that a polarized affine variety admits a Ricci flat
K\"ahler cone metric, if it is K-stable. This generalizes
Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to
K\"ahler cones, or equivalently, Sasakian manifolds. As an application
we show that the five-sphere admits infinitely many families of
Sasaki-Einstein metrics.
Nov. 16, 2016
Isabel Vogt :
4 p.m. in SEO 427
Abstract
In this talk we will discuss the following question: When does there exist a curve of degree d and genus g passing through n general points in P^r? We will focus on the case of space curves (r = 3).
Nov. 17, 2016
Alena Pirutka :
2 p.m. in SEO 636
Abstract
Let X be a projective algebraic variety, the set of solutions of a
system of homogeneous polynomial equations. Several classical notions
describe how ``unconstrained'' the solutions are, i.e., how close X is
to projective space: there are notions of rational, unirational and
stably rational varieties. Over the field of complex numbers, these
notions coincide in dimensions one and two, but diverge in higher
dimensions. In this talk I will discuss classical examples of
rational and nonrational varieties, as well as recent advances in this area.
Nov. 23, 2016
- :
4 p.m. in SEO 427
Nov. 30, 2016
Christian Schnell :
3 p.m. in SEO 1227
Abstract
In the past few years, people working on the analytic side of algebraic geometry have obtained two important new results: a version of the Ohsawa-Takegoshi extension theorem with sharp estimates (Blocki, Guan-Zhou), and the existence of canonical singular hermitian metrics on pushforwards of relative pluricanonical bundles (Berndtsson, Paun, Takayama, and others). In this talk, I will explore some consequences of this work for the study of morphisms to complex abelian varieties, including the recent proof of Iitaka's conjecture over abelian varieties (Cao-Paun). The talk will be understandable without any background in analysis.
Daniel Litt :
1 p.m. in SEO 427
Abstract
Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X.
As a sample application of our techniques, we show that if X is a smooth variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) satisfying the following: any irreducible p-adic representation of the fundamental group of X which arises from geometry is non-trivial mod p^N.
Jan. 23, 2017
Chenyang Xu :
4 p.m. in SEO 427
Abstract
In higher dimensional geometry, it has been known that from many perspectives a log terminal singularity is a local analogue of Fano varieties. Many statements of Fano varieties have a counterpart for log terminal singularities. One central topic on the geometry of a Fano variety is its stability which in particular reflects whether the Fano variety carries a canonical metric. In this talk, we will discuss a recent joint work with Chi Li (some part still in progress) in which we want to establish a local stability theory of a fixed log terminal singularity. Inspired by the study from differential geometry, (e.g. tangent cone, Sasakian-Einstein metric), for any log terminal singularity, we investigate the valuation which has the minimal normalized volume. Our goal is to prove various properties of this valuation which enable us to degenerate the singularity to a K-semistable T-singularity (with a torus action) in the Sasakian-Einstein sense.
Feb. 8, 2017
Chris Skalit :
4 p.m. in SEO 427
Abstract
A classical theorem of Auslander-Buchsbaum asserts that the divisor class group of a regular local ring $A$ is trivial. Several years later, R. Fossum conjectured that the same result ought to hold for all cycles of positive codimension on $\operatorname{Spec} A$. In this talk, we shall discuss the storied history of this problem, its connection with higher algebraic K-theory, and some recent progress.
Feb. 15, 2017
Eric Riedl :
4 p.m. in SEO 427
Abstract
Given a rational curve C in projective space, the normal bundle is an object that controls the deformations of C. Given a fixed vector bundle E, one can ask: What is the moduli space of rational curves with normal bundle E? Eisenbud and Van de Ven conjectured that these spaces are irreducible, but in joint work with Coskun, I show that this is not the case as soon as the dimension of projective space is at least 5.
Feb. 22, 2017
Christian Urech :
4 p.m. in SEO 427
Abstract
The Cremona group is the group of birational transformations of the projective space. While the plane Cremona group is well understood, many questions about Cremona groups in higher dimensions remain open. In this talk we will look at the question how the plane Cremona group can be embedded into Cremona groups in higher dimensions. In particular, I will give a classification of algebraic embeddings from the plane Cremona group to the group of birational transformations of a threefold and explain the geometry of some interesting examples in higher dimensions.
March 1, 2017
Javier Carvajal-Rojas :
4 p.m. in SEO 427
Abstract
In this talk I will start by summarizing recent work on the \'etale fundamental of $F$-regular singularities and schemes (joint work with subsets of B. Bhatt, P. Graf, K. Schwede and K. Tucker). After presenting some corollaries and a common limitation of these, I will discuss the necessity of considering a more general fundamental group, namely the Nori's fundamental group-scheme, which is well suited to the study of positive characteristic phenomena. As we will see, it gives a better understanding of the aforementioned corollaries. For example, we will see that the torsion, and not just the prime-to-$p$ torsion, of the Picard group of strongly $F$-regular singularities is bounded.
Howard Nuer :
5 p.m. in SEO 427
Abstract
A foundational tool in the study of families of Bridgeland semistable objects on a fixed variety is the notion of a constant sheaf of t-structures pioneered by Abramovich and Polishchuk. However, throughout algebraic geometry it is often useful to be able to deform the underlying variety to make the objects of study more tractable. In joint work with Lahoz, Macri, and Perry we develop a tool for studying how Bridgeland stability varies under such deformations to allow for the use of such techniques in Bridgeland stability. Although quite technical in full detail, we hope to share the general ideas involved in these so-called “relative Bridgeland stability conditions.” Instead of going into all of the details, we will cover in more detail two important applications of our tool. The first (which depends on some joint work with Bayer and Stellari as well as the above authors) is to proving the full version of Addington and Thomas’s equivalence between a cubic fourfold having an associated K3 surface in the hodge theoretic sense (due to Hassett) and having one in the derived category sense (due to Kuznetsov). The second is to the deformation invariance of Toda’s generalized DT invariants.
March 8, 2017
Rebecca Tramel :
4 p.m. in SEO 427
Abstract
In 2002, Bridgeland defined a notion of stability for objects in the derived category of a projective variety. This definition was meant to correspond to Douglas' definition of Pi-stability for D-branes in string theory. Since 2002, many connections have been made between stability conditions on a variety and its birational geometry. I will discuss some of these connections in the case of a smooth projective surface.
March 17, 2017
S. Takayama :
3 p.m. in SEO 427
Abstract
We consider degenerations of Calabi-Yau manifolds
over higher dimensional bases in general.
We then shall present a result on the
equivalence of a uniform
diameter bound as Ricci-flat Kaehler-Einstein manifolds
and that the limit varieties have canonical singularities at worst.
March 29, 2017
Samuel Grushevsky :
4 p.m. in SEO 427
Abstract
We describe a natural compactification of the moduli space of complex curves together with a meromorphic 1-form with prescribed multiplicities of zeroes and poles. Such a moduli space is the total space where the action of SL(2,R) is studied in Teichmuller dynamics, and is also the analog of the double ramification cycle on the moduli space of curves. Based on joint work with M. Bainbridge, D. Chen, Q. Gendron, M. Moeller.
April 5, 2017
Roya Beheshti :
4 p.m. in SEO 427
Abstract
I will talk about the geometry of moduli spaces of rational curves (and stable maps) on hypersurfaces and discuss some results concerning their dimension and birational geometry.
April 12, 2017
- :
4 p.m. in SEO 427
Abstract
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April 13, 2017
Rahul Pandharipande :
2 p.m. in SEO 427
Abstract
I will discuss kappa classes on the moduli space of quasi-polarized
K3 surfaces and relations obtained from the moduli spaces of stable
maps to the universal family. I will explain the proof of the generation of
the tautological ring by Noether-Lefschetz loci. There are a number of
open questions. Joint work with Qizheng Yin.
April 19, 2017
Kenta Sato :
4 p.m. in SEO 427
Abstract
Since the Bertini theorem for free linear series fails in positive characteristic,
it is not clear whether a general hyperplane section of a klt 3-fold in positive characteristic has only klt singularities or not.
We give an affirmative answer when the characteristic is larger than 5.
This talk is based on joint work with Professor Shunsuke Takagi.
April 24, 2017
Jean-Pierre DEMAILLY :
4 p.m. in SEO 427
Abstract
We describe a generalization of an L2 extension theorem due to Ohsawa-Takegoshi: the
holomorphic sections or cohomology classes defined on an algebraic
subscheme (or a non necessarily reduced analytic subvariety) can be extended
under a weak semipositivity assumption.
This even works with singular hermitian metrics, and the ambient subvariety need
only be Kaehler and holomorphically convex, the total space of a projective morphism
over an affine base being a typical situation.
April 26, 2017
Tommaso de Fernex :
4 p.m. in SEO 427
Abstract
The work of Greenberg, Nash, Kolchin, and Denef-Loeser has
set the basis for our understanding of the structure of arc spaces and
their connections to singularities and birational geometry. Most of
the focus in these studies is on the reduced structure of arc spaces
and their underlying topological spaces, and little is known about
their scheme structure. In joint work with Roi Docampo, we further
investigate the structure of arc spaces. Our main result gives a
description of the sheaves of Kahler differentials of the arc space.
The approach leads to new results on arc spaces as well as simpler and
more direct proofs of some of the theorems in the literature.
Jean-Pierre DEMAILLY :
11 a.m. in SEO 427
Abstract
On a projective variety of general type, one can prove the existence of sections of
certain jet bundles of sufficiently high order and degree, and even evaluate the growth of
their cohomology groups. New algebraic concepts of "strong general type"
and "jet algebraic hyperbolicity" can be derived from there, that imply
hyperbolicity properties for transcendental entire curves.
Related techniques have been used recently by Damian Brotbek to
confirm a version of the Kobayashi conjecture on the generic hyperbolicity of hypersurfaces of large degree.
Sept. 6, 2017
Howard Nuer :
4 p.m. in SEO 427
Abstract
We provide explicit descriptions of the generic members of Hassett’s divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and $d = 44$, thus giving unirationality of these $\mathcal C_d$.
We prove as a corollary that the moduli space $\mathcal N_d$ of polarized K3 surfaces of degree $d$ is unirational for $d = 14, 26, 38$. The case $d = 26$ is entirely new,
while the other two cases have been previously proven by Mukai. We also explain the construction of what we conjecture to be a new family of irreducible symplectic manifolds which are not birational to any moduli space of (twisted) sheaves on a K3 surface.
Time permitting, we explain how our results have been used by Russo and Stagliano to prove the rationality of the generic cubic fourfold in $\mathcal C_{38}$.
Sept. 13, 2017
Eric Riedl :
4 p.m. in SEO 427
Sept. 20, 2017
Chris Skalit :
4 p.m. in SEO 427
Sept. 27, 2017
Dan Abramovich :
4 p.m. in SEO 427
Abstract
A beginner can easily resolve toric singularities. While enormous progress was made on simplifying characteristic-0 resolution of singularities in general, wouldn't it be nice to have a straightforward way to transform any singularity to a toric singularity?
In joint work with Michael Temkin (Jerusalem) and Jarosław Włodarczyk (Purdue) we principalize an ideal, making it monomial on a variety which has only toroidal singularities, leading to such transformation, albeit using the language of stacks.
I'll try to show how this works in explicit examples.
Oct. 4, 2017
Chung Ching Lau :
4 p.m. in SEO 427
Abstract
TBA
Oct. 11, 2017
Feng Hao :
4 p.m. in SEO 427
Abstract
In this talk I will give a proof of the Weak Bounded Negativity Conjecture, which says that given any complex smooth projective surface, for any reduced curve $C$ in $X$ and integer $g$, assume that the geometric genus of each component of $C$ is bounded from above by $g$, then the self-intersection number $C^2$ is bounded from below. The Weak Bounded Negativity Conjecture is motivated by the old folklore Bounded Negativity conjecture, which says that given any complex smooth projective surface, the self-intersection number of any reduced curve is bounded from below. Also, the Bounded Negativity Conjecture has an interesting relation with the Nagata conjecture. I will introduce those background before the proof of the Weak Bounded Negativity Conjecture. Also, I will give some further thoughts towards the Bounded Negativity Conjecture.
Oct. 18, 2017
Eric Larson :
4 p.m. in SEO 427
Abstract
In this talk we give several results on the existence of a
curve of degree d and genus g passing through n general points in P^r.
We then discuss the application of these results to the determination
of the Hilbert function of a general curve.
Oct. 25, 2017
Daniel Litt :
4 p.m. in SEO 427
Abstract
In 1987, Deligne and Illusie famously gave an algebraic proof of the degeneration of the Hodge-to-de Rham spectral sequence and the Kodaira vanishing theorem. Their methods have been used since (by Arapura and others) to prove strong vanishing theorems. I'll discuss their methods, a conjecture that would strengthen them, and a proof of some important special cases of that conjecture. I'll also give some applications to toric varieties.
Nov. 1, 2017
Tristan COLLINS :
4 p.m. in SEO 427
Abstract
I will discuss the solvability of the J-equation,
which defines the critical point of Chen-Donaldson’s
J-functional. It is known that there do
not exist solutions to the J-equation in general – a
notion of algebro-geometric stability
has been proposed by Lejmi-Szekelyhidi which is
conjectured to be equivalent to the existence of solutions.
I will discuss a proof of this
conjecture on toric varieties, together with
some motivating connections with
mirror symmetry and Bridgeland stability.
This talk is based on joint
with G. Szekelyhidi, and A. Jacob and S.-T. Yau.
Nov. 8, 2017
Dhruv Ranganathan :
4 p.m. in SEO 427
Abstract
The Brill-Noether varieties of a curve C parameterize embeddings of C of prescribed degree into a projective space of prescribed dimension, i.e. equations for the curve. When C is general, these varieties are well understood: they are smooth, irreducible, and have the "expected" dimension. As one ventures deeper into the moduli space, past the general curve, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. Based on an analogous combinatorial problem on graphs, Pflueger conjectured a formula for the dimensions of the Brill-Noether varieties for general curves of a given gonality. I will present joint work with Dave Jensen, in which we prove Pflueger’s conjecture. The proof blends non-archimedean analytic techniques, ideas from logarithmic Gromov-Witten theory, and the geometry of scrolls.
Nov. 15, 2017
Remy van Dobben de Bruyn :
4 p.m. in SEO 427
Abstract
Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.
Nov. 16, 2017
Alex Küronya :
1 p.m. in SEO 427
Abstract
In a joint work with Victor Lozovanu we study syzygies of ample line bundles on abelian varieties, more specifically when property $(N_p)$ of Green and Lazarsfeld are satisfied. We give an equivalent characterization in dimension two, and look into what happens on abelian threefolds.
Nov. 22, 2017
No Seminar :
4 p.m. in SEO 427
Nov. 29, 2017
Gordon HEIER :
4 p.m. in SEO 427
Abstract
The interplay of various notions of hyperbolicity and the geometry and
structure of a projective manifold is an important topic in complex
geometry. In this spirit, we investigate a projective Kaehler manifold
$M$ of semi-negative holomorphic sectional curvature $H$. We will
begin with an overview of the recent progress on this topic. We will
then introduce a new differential geometric numerical rank invariant
which measures the number of linearly independent truly flat
directions of $H$ in the tangent spaces. This invariant turns out to
be bounded above by the nef dimension and bounded below by the
numerical Kodaira dimension of $M$. We will also discuss a splitting
theorem for $M$ in terms of the nef dimension and, under some
additional hypotheses, in terms of the new rank invariant. This is
joint work with S. Lu, B. Wong and F. Zheng.
Jan. 24, 2018
Stephane DRUEL :
4 p.m. in SEO 427
Abstract
The Beauville-Bogomolov decomposition theorem
asserts that any compact Kähler manifold with
numerically trivial canonical bundle admits an
étale cover that decomposes into a product of
a torus, an irreducible, simply-connected Calabi-Yau,
and holomorphic symplectic manifolds.
With the development of the minimal model program,
it became clear that singularities arise as an
inevitable part of higher dimensional life.
I will present recent works in
which a singular version of the decomposition theorem is established.
Feb. 7, 2018
Lawrence Ein :
4 p.m. in SEO 427
Feb. 14, 2018
Emre Sertoz :
4 p.m. in SEO 427
Abstract
Given a complex manifold X, the periods of X are complex numbers which describe the complex structure of X upon the underlying topological manifold.
The periods of a smooth algebraic variety reveal finer geometric data more readily than the defining equations alone. However, periods are typically very hard to compute. In the past 20 years, an algorithm for computing the periods existed only for plane curves. We will describe a different algorithm which can compute the periods of any smooth projective hypersurface.
As an application, we will demonstrate how to reliably guess the Picard rank of a quartic K3 surface from its periods computed up to numerical error.
Feb. 21, 2018
Kevin Tucker :
4 p.m. in SEO 427
Abstract
In characteristic zero, it is well known that multiplier ideals and log terminal singularities satisfy Bertini-type theorems for hyperplane sections. The analogous situation in characteristic p > 0 is more complicated. While F-regular singularities satisfy Bertini, the test ideal does not. In this talk, I will describe joint work with Karl Schwede and Javier Carvajal-Rojas showing that the F-signature -- a numerical invariant of singularities that detects F-regularity -- satisfies the relevant Bertini statements for hyperplane sections. In particular, one can view this as a generalization of the corresponding results for F-regularity.
Feb. 28, 2018
Izzet Coskun :
4 p.m. in SEO 427
Abstract
In this talk, I will discuss joint work with Jack Huizenga, on Brill-Noether Theorems for higher rank sheaves on rational surfaces. I will describe our classification of moduli spaces whose general member is globally generated on minimal rational surfaces. If time permits, I will discuss joint work with Howard Nuer and Kota Yoshioka on similar problems on K3 surfaces.
March 7, 2018
Wenbo Niu :
4 p.m. in SEO 427
Abstract
In this talk, we will discuss problems related to projective normality and higher syzygies for powers of line bundles on nonsingular projective varieties. We focus on two situations: powers of ample line bundles on Calabi-Yau varieties and pluricanonical divisors on varieties of general type. These two cases follow the same approach to consider how Arbarello-Sernesi module associated to the variety can be generated as a graded module, which can further be reduced to consider the surjectivity of multiplication maps of line bundles.
March 14, 2018
Simon Pepin Lehalleur :
4 p.m. in SEO 427
Abstract
Following Grothendieck's vision that many cohomolgical
invariants of of an algebraic
variety should be captured by a common motive, Voevodsky
introduced a triangulated category of mixed motives which partially realises
this idea. After describing this category, I will explain how to
define the motive of certain algebraic stacks in this context. I will
then report on joint work in progress with Victoria Hoskins, in which
we study the motive of the moduli stack of vector bundles on a smooth
projective curve and show that this motive can be described in terms of
the motive of this curve and its symmetric powers.
March 21, 2018
Julius Ross :
4 p.m. in SEO 427
April 4, 2018
Benson Farb :
4 p.m. in SEO 427
Abstract
The problem of understanding how the roots of a polynomial
depend on its coefficients goes back to the 16th century. In this talk I will explain a beautiful geometric point-of-view on this problem initiated by Klein and Hilbert, but which seems to be mostly forgotten. I will explain how these ideas are pertinent to Hilbert's 13th Problem and Sextic and Octic Conjectures, which are fundamental problems about formulas for roots of polynomials, and which we will relate to problems in enumerative geometry. This is ongoing work with Jesse Wolfson (UC Irvine) and Mark Kisin (Harvard).
April 18, 2018
Junyan CAO :
4 p.m. in SEO 427
Abstract
Let X be a simply connected projective manifold with nef anticanonical bundle.
We prove that X is a product of a rationally
connected manifold and a manifold with trivial canonical bundle.
As an application we describe the MRC fibration of any
projective manifold with nef anticanonical bundle.
It is a joint work with Andreas Höring
April 25, 2018
Benjamin Bakker :
5 p.m. in SEO 427
Abstract
Hodge structures on cohomology groups are fundamental invariants of algebraic varieties; they are parametrized by quotients $D/\Gamma$ of periods domains by arithmetic groups. Except for a few very special cases, such quotients are never algebraic varieties, and this leads to many difficulties in the general theory. We explain how to partially remedy this situation by equipping $D/\Gamma$ with an o-minimal structure, and show that period maps are "definable" with respect to this structure. As a consequence, we obtain an easy proof of a result of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci, a strong piece of evidence for the Hodge conjecture. The proof of the main theorem relies heavily on work of Schmid, Kashiwara, and Cattani--Kaplan--Schmid on the asymptotics of degenerations of Hodge structures. This is joint work with B. Klingler and J. Tsimerman.
Harold Blum :
4 p.m. in SEO 427
Abstract
In this talk, we will discuss two invariants that measure the singularities of anticanonical divisors on Fano varieties. The first is the global log canonical threshold, which is also known as Tian’s alpha invariant. The second is the stability threshold, an invariant recently introduced by Fujita and Odaka. Our approach to understanding these invariants involves valuations. Using results of Fujita and Li, we show that the K-semistability of a Fano variety is detected by the stability threshold. This talk is based on joint work with Mattias Jonsson.
May 2, 2018
Julien Keller :
4 p.m. in SEO 427
Abstract
We propose a new proof of algebraic nature of the correspondance. This is a joint work with Y. Hashimoto.
Aug. 29, 2018
Xudong Zheng :
4 p.m. in 427 SEO
Abstract
This is a report of work in progress on a conjectural identification of varieties in characteristic zero having globally F-regular type and Fano type. Aiming at showing any variety of globally F-regular in dimension three is of Fano type, I will discuss an approach using three dimensional minimal model program in positive characteristics.
Sept. 5, 2018
Charlie Stibitz :
4 p.m. in 427 SEO
Abstract
We will look at the relation between two concepts in the
geometry of Fano varieties: birational superrigidty, which comes from
the study of Mori fiber space structures on a Fano variety, and
K-stability, coming from the study of nice metrics of Fano manifolds.
We show that as long as the alpha invariant of a Fano variety is
greater than 1/2, any birationally superrigid Fano variety is K-stable.
Sept. 12, 2018
Dan Erman :
4 p.m. in 427 SEO
Abstract
I’ll consider limits of polynomial rings, as the number of variables goes to infinity. I’ll discuss the surprisingly simple structure of these limits and how this applies to some famous conjectures in algebraic geometry and commutative algebra, such as Stillman’s Conjecture on projective dimension and Hartshorne’s Conjecture on complete intersections. This is joint work with Steven Sam and Andrew Snowden.
Sept. 19, 2018
Yuri Tschinkel :
4 p.m. in 427 SEO
Abstract
A classical theme in algebraic geometry is to determine how far an algebraic variety is from projective space. Several notions have emerged in this context: rationality, stable rationality, unirationality, and rational connectedness. I will discuss new ideas and constructions that emerged in this area and that led to solutions of long-standing open problems.
Sept. 26, 2018
Rankeya Datta :
4 p.m. in 427 SEO
Abstract
We will prove a prime characteristic analogue of a result of Ein, Lazarsfeld and Smith on approximation of valuation ideals associated to real-valued Abhyankar (quasi-monomial) valuations.
Oct. 3, 2018
Matei Toma :
4 p.m. in 427 SEO
Abstract
For a complex projective manifold (X,\omega) the Kobayashi-Hitchin correspondence gives homeomorphisms between moduli spaces of irreducible Hermitian-Yang-Mills connections and moduli spaces of stable vector bundles on X. A by now classical paper of Jun Li from 1993 shows that when X is two-dimensional this correspondence can be extended as a homeomorphism between natural compactifications of these moduli spaces existing on the gauge theoretical and on the algebraic geometric side, respectively. As a consequence one gets a complex analytic structure on the Donaldson-Uhlenbeck compactification of the moduli space of Hermitian-Yang-Mills connections on a fixed hermitian vector bundle on X. In this talk we present joint recent work together with Daniel Greb, Benjamin Sibley and Richard Wentworth extending these results to the higher dimensional situation.
Oct. 10, 2018
Daniel Bragg :
4 p.m. in 427 SEO
Abstract
We will describe how the crystalline cohomology of a supersingular K3 surface gives rise to certain one-parameter families of K3 surfaces, which we call supersingular twistor spaces. Our construction relies on the special behavior of $p$-torsion classes in the Brauer group of a supersingular K3 surface, as well as techniques coming from the study of derived categories and Fourier-Mukai equivalences. As applications, we find new proofs of Ogus's crystalline Torelli theorem and Artin's conjecture on the unirationality of supersingular K3 surfaces. These results are new in small characteristic.
Oct. 24, 2018
Yuchen Liu :
4 p.m. in 427 SEO
Abstract
Motivated by work in differential geometry, Chi Li introduced the normalized volume of a klt singularity as the minimum normalized volume of all valuations centered at the singularity. This invariant carries some interesting geometric/topological information of the singularity. In this talk, we show that in a Q-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. As an application, we show that K-semistability is a very generic or empty property in a Q-Fano family. If time permits, I will discuss related results in positive characteristic. This talk is partly based on joint work with Harold Blum.
Oct. 31, 2018
Renzo Cavalieri :
4 p.m. in 427 SEO
Abstract
Kappa classes were introduced by Mumford, as a tool to explore the intersection theory of the moduli space of curves. Iterated use of the projection formula shows there is a close connection between the intersection theory of kappa classes on the moduli space of unpointed curves, and the intersection theory of psi classes on all moduli spaces. In terms of generating functions, we show that the potential for kappa classes is related to the Gromov-Witten potential of a point via a change of variables essentially given by complete symmetric polynomials, rediscovering a theorem of Manin and Zokgraf from '99. Surprisingly, the starting point of our story is a combinatorial formula that relates intersections of kappa classes and psi classes via a graph theoretic algorithm (the relevant graphs being dual graphs to stable curves). Further, this story is part of a large wall-crossing picture for the intersection theory of Hassett spaces, a family of birational models of the moduli space of curves.
This is joint work with Vance Blankers (arXiv:1810.11443) .
Nov. 7, 2018
Claudiu Raicu :
4 p.m. in 427 SEO
Abstract
Formulated in 1984, Green’s Conjecture predicts that one can recognize the intrinsic complexity of a smooth algebraic curve from the syzygies of its canonical embedding. In characteristic zero, Green's Conjecture for a general curve has been resolved using geometric methods in two landmark papers by Voisin in the early 00s. More direct approaches have been proposed over the years to solve Green's Conjecture for general curves, and one dates back at least to a paper of Eisenbud in the early 90s, and involves a connection with the syzygies of the tangent developable T to a rational normal curve. I will explain how the theory of Koszul modules allows for a complete characterization, in arbitrary characteristics, of the (non-)vanishing behavior of the syzygies of T, proving Green’s conjecture for general curves in almost all characteristics. Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman.
Nov. 14, 2018
Mircea Mustata :
4 p.m. in 427 SEO
Abstract
In this lecture I will introduce some invariants of singularities that come out of D-module theory. Given a nonzero element of the polynomial ring, the localization at this element is a module over the ring of differential operators, and Saito's theory of Mixed Hodge Modules endows it with a canonical filtration. I will explain how to relate the properties of this filtration to the singularities of the given polynomial and how to use this for geometric applications. This is based on joint work with Mihnea Popa.
Jan. 23, 2019
Izzet Coskun :
4 p.m. in 427 SEO
Abstract
Moduli spaces of Gieseker semistable sheaves on surfaces play a central role in mathematics and have many applications to cycles and linear systems on surfaces, Donaldson's 4-manifold invariants and mathematical physics. In this talk, I will describe a conjecture with Matthew Woolf on the cohomology of these moduli spaces. We conjecture that the Betti numbers of these moduli spaces stabilize as the discriminant tends to infinity and that the stable numbers are independent of the rank and the first Chern class. In particular, calculations of Gottsche determine the stable numbers. I will give some evidence for the conjecture. This is joint work with Matthew Woolf.
Feb. 6, 2019
Ronno Das :
4 p.m. in 427 SEO
Abstract
The Cayley-Salmon theorem states that every smooth cubic surface S in $\mathbb{C}\mathbb{P}^3$ has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a 'universal family' of cubic surfaces over M. One difficulty in understanding these spaces is that they are complements in affine space of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I'll then explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the typical smooth cubic cubic surface over a finite field $F_q$ contains 1 line and $q^2 + q + 1$ points.
Feb. 27, 2019
Giulia Sacca :
4 p.m. in 427 SEO
Abstract
A few years ago with Laza and Voisin we constructed a hyperkahler
compactification of the intermediate Jacobian fibration associated to
a *general* cubic fourfold. In this talk I will first show how a HK
compactification J(X) exists for *any* smooth cubic fourfold X and then
discuss how the birational geometry of the fibration is governed by
any extra algebraic cohomology classes on X.
March 6, 2019
Man-Wai (Mandy) Cheung :
4 p.m. in 427 SEO
Abstract
Cluster varieties are blow up of toric varieties. They come in pairs (A,X), with A and X built from dual tori. Compactifications of A, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties while the compactifications of X, studied by Fock and Goncharov, generalize the fan construction. The conjecture is that the A and the X cluster varieties are mirrors to each other. Together with Tim Magee, we have shown that there exists a positive polytope for the type A cluster varieties which give us a hint to the Batyrev-Borisov construction.
March 20, 2019
Eric Riedl :
4 p.m. in 427 SEO
Abstract
We discuss several related notions of hyperbolicity in projective space, focusing particularly on algebraic hyperbolicity and Brody hyperbolicity. We discuss what is known about these notions for very general hypersurfaces in projective space. In joint work with Coskun, we prove that quintic hypersurfaces in P^3 are algebraically hyperbolic, finally settling the last case of a conjecture of Demailly. In joint work with David Yang, we show that (a slightly stronger version of) the Green-Griffiths-Lang Conjecture implies the Kobayashi Conjecture.
April 3, 2019
Jay Kopper :
4 p.m. in 427 SEO
Abstract
Recent developments in the study of stable sheaves extend the notion of stability to the entire derived category. This broader perspective can be used to study the classical moduli space. In this talk I will discuss these ideas in the context of restriction theorems: situations in which a stable vector bundle remains stable when restricted to a subvariety. Derived category techniques can produce stronger restriction theorems than were previously available. I will discuss applications if time permits.
April 10, 2019
Alex Perry :
4 p.m. in 427 SEO
Abstract
I will discuss some surprising "homological" counterparts of constructions and results in classical projective geometry. This gives a powerful framework for producing varieties whose derived categories are equivalent, or more generally have a large subcategory in common. Besides being of intrinsic interest, such relations often have strong geometric consequences. This is joint work with Alexander Kuznetsov.
April 19, 2019
Rahul Pandharipande :
2 p.m. in 427 SEO
Abstract
A basic question in the theory of algebraic curves is whether a
divisor represents the zeros and poles of a rational function.
An explicit solution in terms of periods was given by the work of Abel
and Jacobi in the 19th century. In the past few years, a different
approach to the question has been pursued: what is the class
in the moduli of pointed curves of the locus of such divisors? The
answer in Gromov-Witten theory is given by Pixton's formula
for the double ramification cycle. I will discuss recent work
with F. Janda, A. Pixton, and D. Zvonkine which considers
double ramification cycles for target varieties X (where Pixton's
original question is viewed as the X=point case). I will also
discuss the associated relations studied by Y. Bae.
April 24, 2019
Dawei Chen :
4 p.m. in 427 SEO
Abstract
Computing volumes of moduli spaces has significance in many fields. For instance, the celebrated Witten's conjecture regarding intersection numbers on the Deligne-Mumford moduli space of stable curves has a fascinating connection to the Weil-Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. The initial two other proofs of Witten's conjecture by Kontsevich and by Okounkov-Pandharipande also used various ideas in ribbon graphs, Gromov-Witten theory, and Hurwitz theory. In this talk I will introduce an analogous formula of intersection numbers on moduli spaces of abelian differentials that computes the Masur-Veech volumes. This is joint work with Moeller, Sauvaget, and Zagier (arXiv:1901.01785).
May 1, 2019
Antoni Rangachev :
4 p.m. in 427 SEO
Abstract
In this talk I will introduce a class of singularities that generalizes the class of smoothable singularities: these are all singularities that admit deformations to singularities with deficient conormal spaces. I will discuss how this new class arises from problems in differential equisingularity and how it relates to the local volume of a line bundle.
Aug. 28, 2019
Jesse Wolfson :
4 p.m. in 427 SEO
Abstract
This seminar has been cancelled and will be rescheduled at a later date.
Sept. 4, 2019
Tyler Kelly (University of Birmingham, UK) :
4 p.m. in 1227 SEO
Abstract
Mirror Symmetry provides a link between symplectic and algebraic geometry through a duality in string theory. In particular, it asserts a link from the symplectic geometry of a space M to the algebraic geometry of its mirror space W. One way we see this is now known as classical mirror symmetry: the Gromov-Witten or enumerative theory of a symplectic space is encapsulated by the Hodge theory / periods of the mirror algebraic space. In the 90s this was articulated just for Calabi-Yau varieties, but it has expanded even further to Fano varieties; however, the mirror space is now not an algebraic variety but a mildly non-commutative object known as a Landau-Ginzburg model. Recently, this notion has been developed even to articulate mirror symmetry between Landau-Ginzburg models. In this talk, we will explain what non-commutative Hodge theory / periods look like for a Landau-Ginzburg model and how they predict phenomena in open enumerative theories for the mirror.
Sept. 9, 2019
Nadia Ott :
4 p.m. in 427 SEO
Sept. 16, 2019
Hannah Larson :
4 p.m. in 427 SEO
Abstract
The Brill-Noether theorem describes the maps of general curves to projective space. Recently, the Brill-Noether theory of general k-gonal curves C has gathered much interest: Coppens-Martens exhibited components of the Brill-Noether loci W^r_d(C) with different dimensions; work of Pflueger and Jensen-Ranganathan determined the dimension of the largest component. In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map C --> P^1, using the splitting type of push forwards of line bundles to P^1. In particular, studying this refinement determines the dimensions of all irreducible components of W^r_d(C) for general k-gonal C.
Sept. 23, 2019
Geoff Smith :
4 p.m. in 427 SEO
Abstract
The covering gonality of an irreducible projective variety over the complex numbers is the minimum gonality of a curve through a general point on the variety. This definition has two reasonable generalizations to positive characteristic, the covering gonality and the separable covering gonality. Of the two, separable covering gonalities are much easier to bound, and I'll give an easy lower bound for smooth hypersurfaces essentially due to Bastianelli-de Poi-Ein-Lazarsfeld-Ullery. I'll then give an analogous bound for the covering gonality of very general hypersurfaces, using a Chow-theoretic argument that extends work of Riedl-Woolf.
Oct. 7, 2019
Julius Ross :
4 p.m. in 427 SEO
Oct. 14, 2019
Gavril Farkas :
4 p.m. in 427 SEO
Abstract
Given two vector bundles E and F on a variety X and a
morphism from Sym^2(E) to F, we compute the cohomology class of the
locus in X where the kernel of this morphism contains a quadric of
prescribed rank. Our formulas have many applications to moduli theory:
(i) a simple proof of Borcherds' result that the Hodge class on the
moduli space of polarized K3 surfaces of fixed genus is of
Noether-Lefschetz type, (ii) an explicit canonical divisor on the
Hurwitz space parametrizing degree k covers of the projective line
from curves of genus 2k-1, (iii) a closed formula for the Petri
divisor on the moduli space of curves consisting of canonical curves
which lie on a rank 3 quadric and (iv) myriads of effective divisors
of small slope on M_g. Joint work with Rimanyi.
Oct. 21, 2019
Mingyi Zhang :
4 p.m. in 427 SEO
Abstract
In this talk, I will present my work on studying a sequence of invariants, called Hodge ideals, which detect singularities of a hypersurface on a smooth complex variety and measures the Hodge theory on the complement of the hypersurface. These Hodge ideals arise naturally from Saito’s theory on the Hodge filtration of Hodge modules associated to the localization along a hypersurface and give a good generalization of multiplier ideals. I will give a general introduction to Hodge ideals for Q-divisors and show some applications in singularity theory. In particular, I will give explicit formulas of these ideals in some special cases and develop computational results of various invariants of singularities, e.g., generating level of Hodge filtration, roots of Bernstein-Sato polynomials and Hodge ideal spectrum.
Oct. 28, 2019
Martino Fassina :
4 p.m. in 427 SEO
Abstract
In 1979 Kohn introduced a procedure to prove subelliptic estimates for the Cauchy-Riemann equations. Over the years, many people have studied algebraic aspects of this algorithm, and in particular the question of its effectiveness. I will show how, in the polynomial case, the problem can be tackled by applying effectiveness results of Kollár and Jelonek from commutative algebra. Simple examples show that the Kohn algorithm is not effective in general. I will prove that every modified effective algorithm in the holomorphic case yields an effective procedure to prove subellipticity on a wide class of domains with real analytic boundary satisfying a condition slightly stronger than pseudoconvexity.
Nov. 4, 2019
Howard Nuer :
4 p.m. in 427 SEO
Nov. 18, 2019
Carl Lian :
4 p.m. in 427 SEO
Abstract
We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.
Feb. 10, 2020
Kenny Ascher :
4 p.m. in 427 SEO
Abstract
K-stability has become a central tool in the study of compact moduli of Fano varieties. In this talk I will discuss K-stability compactifications of the moduli space of log Fano pairs (P2, aC), where C is a plane curve of degree at least 4 and a is a rational number. We establish a wall-crossing framework to study the behavior of these moduli spaces as the weight a varies. We show that when a is small, the K-moduli compactification is isomorphic to the GIT moduli space, and that the first wall crossing is a weighted blowup of Kirwan type. We describe all wall-crossings for degree 4, 5 and 6 and relate the final K-moduli spaces to Hacking's moduli space and some compact moduli of K3 surfaces. This is joint work with K. DeVleming and Y. Liu.
Feb. 17, 2020
Wenliang Zhang :
4 p.m. in 427 SEO
Feb. 24, 2020
Kristin DeVleming :
4 p.m. in 427 SEO
Abstract
I will discuss compactifications of the moduli space of (d,d) curves on P1xP1, focusing in particular on the case d = 4. We regard such a curve as a log Fano pair (P1xP1, aC), where a is a rational number, and study the compactifications coming from K stability and establish a wall crossing framework as a varies. In the case d = 4, Laza and O'Grady show that one can interpolate between the GIT moduli space of (4,4) curves and a Baily-Borel compactification of degree 4 K3 surfaces with a series of explicit VGIT wall crossings. We show that these VGIT walls coincide exactly with the K moduli walls described above. This is joint work with Kenneth Ascher and Yuchen Liu.
March 2, 2020
Michael Kemeny :
4 p.m. in 427 SEO
Abstract
The classical theorems of Noether--Petri on the ideals of canonically embedded curves are central in the theory of curves. In the 80s, Mark Green realized that these results should extend to a far broader statement about the entire resolution of the ideal. No major progress was made until Voisin resolved this conjecture for generic curves in 02 and 05. Voisin's proof was extremely sophisticated and used in a deep way the geometry of the situation. We will give a very short proof of her result, using little more than the basic yoga developed by Green, Ein and Lazarsfeld in the 80s. For even genus, our proof also resolves a deeper (and previously open) conjecture, describing in depth the structure of the extremal syzygy space.
March 9, 2020
Dennis Tseng :
4 p.m. in 427 SEO
Abstract
In a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2-plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel.
March 16, 2020
Nick Addington :
4 p.m. in 427 SEO
March 30, 2020
Eduardo Esteves :
4 p.m. in 427 SEO
April 3, 2020
Jesse Wolfson :
10 a.m. in 427 SEO
April 13, 2020
Mihnea Popa :
4 p.m. in 427 SEO
Sept. 14, 2020
John Kopper :
4:30 p.m. in Zoom
Abstract
A general stable vector bundle on a smooth curve is globally generated as soon as its Euler characteristic is greater than its rank. The complement of the locus of globally generated stable bundles thus has positive codimension and describing its geometry is a topic of interest in the higher rank Brill-Noether theory of the curve. In this talk, I will discuss some new results about the dimension and irreducibility of this non-globally generated locus. We are able to compute its dimension in all cases and show that it is irreducible under certain numerical hypotheses. This is joint work with Sayanta Mandal.
Sept. 21, 2020
Jack Huizenga :
4 p.m. in Zoom
Abstract
Using recent advances in the Minimal Model Program for moduli spaces of
sheaves on the projective plane, we compute the cohomology of the tensor
product of general semistable bundles on the projective plane. More
precisely, let V and W be two general stable bundles, and suppose the
numerical invariants of W are sufficiently divisible. We fully compute
the cohomology of the tensor product of V and W. In particular, we show
that if W is exceptional, then the tensor product of V and W has at most
one nonzero cohomology group determined by the slope and the Euler
characteristic, generalizing foundational results of Drézet, Göttsche
and Hirschowitz. We also characterize when the tensor product of V and W
is globally generated. Crucially, our computation is canonical given the
birational geometry of the moduli space, providing a roadmap for
tackling analogous problems on other surfaces. This is joint work with
Izzet Coskun and John Kopper.
Sept. 28, 2020
Naoki Koseki :
4 p.m. in Zoom
Abstract
Tilt-stability is a certain stability notion for objects in the derived categories of coherent sheaves. It has been applied to several classical problems in algebraic geometry. In this talk, I will explain about some of these applications, including stronger Bogomolov-Gieseker inequalities and the construction of Bridgeland stability on some Calabi-Yau threefolds.
Oct. 5, 2020
Samuel Grushevsky :
4:15 p.m. in Zoom
Abstract
The moduli space of cubic threefolds can be thought of as a GIT quotient of the projective space of all cubic polynomials, studied via the period map to a ball quotient, or via the intermediate Jacobians. We describe the relations between various compactifications of the moduli space of cubic threefolds that arise in these ways, and compute their cohomology. Based on joint works with S. Casalaina-Martin, K. Hulek, R. Laza.
Oct. 12, 2020
David Stapleton :
4 p.m. in Zoom
Abstract
The degree of irrationality measures how far a variety is from being rational. In the case of curves the degree of irrationality coincides with the gonality, which is controlled by the positivity of the canonical bundle. In higher dimensions, the positivity of the canonical bundle plays an important role in controlling the degree of irrationality but it is interesting to ask what can be said when the canonical bundle is antiample. In this talk we discuss joint work with Nathan Chen where we show that Fano hypersurfaces can have arbitrarily large degrees of irrationality. We follow a degeneration to characteristic p argument of Kollár, where specializations of hypersurfaces can admit many holomorphic forms.
Oct. 26, 2020
Madeline Brandt :
4 p.m. in Zoom
Abstract
I will discuss an active project in computing the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of $A_g$. Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.
Nov. 2, 2020
Rohini Ramadas :
4 p.m. in Zoom
Abstract
A degree $d>1$ self-map $f$ of $\mathbb{P}^n$ is called post critically finite (PCF) if its critical hypersurface $C_f$ is pre-periodic for $f$, that is, if there exist integers $r \geq 0$ and $k>0$ such that $f^{r+k}(C_f)$ is contained in $f^{r}(C_f)$.
I will discuss the question: what does the locus of PCF maps look like as a subset of the moduli space of degree $d$ self-maps on $\mathbb{P}^n$? I’ll give a survey of many known results and some conjectures in dimension 1 (i.e. for $n=1$). I’ll then present a result, joint with Joseph H. Silverman and Patrick Ingram, that suggests that in dimensions two or greater, PCF maps are comparatively scarce in the moduli space of all self-maps.
Nov. 23, 2020
No seminar :
4 p.m. in Zoom
Jan. 25, 2021
Nicholas Addington :
3 p.m. in Zoom
Abstract
It often happens that if M is a moduli space of vector bundles on a curve
C, then C is also a moduli space of vector bundles on M, where the bundles
on M come from taking "wrong-way slices" of the the universal bundle on M
x C. This story starts in the '70s and is due to Narasimhan and Ramanan,
Newstead, and others. Reede and Zhang recently observed that a similar
result holds for Hilbert schemes of points on surfaces, and for certain
moduli spaces of rank-0 sheaves on K3 surfaces. I will discuss joint work
with my student Andrew Wray, showing that it holds for moduli spaces of
high-rank sheaves on K3 surfaces. Techniques include the Quillen metric
on determinant line bundles and twistor families of hyperkaehler
manifolds.
Feb. 1, 2021
Justin Sawon :
3 p.m. in Zoom
Abstract
Lagrangian fibrations on holomorphic symplectic manifolds and orbifolds are higher-dimensional generalizations of elliptic K3 surfaces. They are fibrations whose general fibres are abelian varieties that are Lagrangian with respect to the symplectic form. Markushevich and Tikhomirov described the first example whose fibres are Prym varieties, and their construction was further developed by Arbarello, Ferretti, and Sacca and by Matteini to yield more examples. In this talk we describe the general framework, and consider a new example. We describe its singularities and show that it is a ‘primitive’ symplectic variety. We also construct the dual fibration, using ideas of Menet. This is joint work with Chen Shen.
Feb. 8, 2021
Paul Kruse :
3 p.m. in Zoom
Abstract
The study of certain moduli spaces of sheaves on smooth projective K3 surfaces has been closely related to the study of Hilbert Schemes of Points on K3 surfaces. Recently, the use of Bridgeland Stability has taken advantage of this connection to produce spaces birational to these Hilbert Schemes. In this talk, we present some wall crossings and associated birational modifications to the moduli spaces of Bridgeland Stable objects on K3 surfaces. In particular, we focus our attention to objects with Chern characters equal to those of ideal sheaves of points.
Feb. 15, 2021
Fatemeh Rezaee :
3 p.m. in Zoom
Abstract
I will describe a new wall-crossing phenomenon of (Bridgeland) stable objects on the projective 3-space that induces non-Q-factorial singularities; hence it cannot be detected as an operation in the Minimal Model Program of the moduli space, unlike the case for many surfaces.
Feb. 22, 2021
Eric Larson :
3 p.m. in Zoom
March 1, 2021
Erik Carlsson :
3 p.m. in Zoom
Abstract
I'll present a new result with A. Mellit, which gives a combinatorial formula for a remarkable diagonalizing operator for the modified Macdonald polynomials, known as the nabla operator. This formula was discovered by finding a Schubert-type basis of a certain explicit module from Haiman's polygraph theory, which is conjecturally identified with the equivariant homology of the unramified affine Springer fiber studied by Goresky, Kottwitz, and Macpherson.
March 8, 2021
Libby Taylor :
3 p.m. in Zoom
Abstract
There is a great deal of interest in studying the question of when two varieties have equivalent derived categories. In low dimensions, this is mostly understood, but in higher dimensions, many fewer examples are known. In this talk, we will produce families of derived equivalent fourfolds using the theory of moduli spaces of sheaves on a K3 surface.
March 15, 2021
Ravi Vakil :
3 p.m. in Zoom
Abstract
A recurring theme in geometry and topology is that moduli spaces become better and better behaved "in the limit".
(i) Stabilization of the Grothendieck ring is one algebro-geometric analogue of stabilization in topology. After briefly introducing stabilization in the Grothendieck ring (joint with Wood), I will describe how it applies to low-degree Hurwitz spaces (in analogy with Bhargavology), which is made simpler thanks to powerful ideas of Bilu and Howe. (This is joint with Landesman and Wood.)
(ii) H. Larson recently completely described (integrally) the "characteristic classes" of vector bundles on $\mathbf{P}^1$-bundles, in the Chow ring. Bott periodicity relates vector bundles on a topological space $X$ to vector bundles on $X \times S^2$: the "moduli space" $BU$ of complex vector bundles is "basically the same as" the "moduli space" maps of a sphere to $BU$. I will try to explain an algebro-geometric incarnation of Bott periodicity. (This is work in progress with H. Larson.)
March 29, 2021
Yilong Zhang :
3 p.m. in Zoom
Abstract
For a smooth cubic threefold Y, its Hilbert scheme with Hilbert polynomial 2n+2 has two irreducible components H and H'. The general member for H is a pair of skew lines and a general member for H' is a conic union an isolated point. We will show that the component H is smooth and is isomorphic to the blow-up of the 2nd symmetric product of Fano surface of lines on Y along the diagonal. This work is based on the work on Hilbert schemes of skew lines on projective spaces by Chen, Coskun and Nollet in 2011. Moreover, I'll also explain the relation of the component H to the stable moduli space considered by Altavilla-Petkovic-Rota and the compactification of locus of vanishing cycles on hyperplane sections.
April 5, 2021
Sarah Peluse :
3 p.m. in Zoom
Abstract
In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $p$ the density of entries in the character table of $S_n$ that are divisible by $p$ goes to $1$ as $n$ goes to infinity. I’ll describe a proof of this conjecture, which is joint work with K. Soundararajan. I will also discuss the (still open) problem of determining the asymptotic density of zeros in the character table of $S_n$, where it is not even clear from computational data what one should expect.
April 12, 2021
Hannah Larson :
3 p.m. in Zoom
Abstract
Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss joint work with Eric Larson and Isabel Vogt that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting.
April 19, 2021
Jordan Ellenberg :
3 p.m. in Zoom
Abstract
I will talk about a program, joint with Matt Satriano and David Zureick-Brown, for formulating a notion of height for points on stacks over global fields, which opens up many new questions about arithmetic distribution of rational points on stacks; in particular, we’ll talk about a common generalization of the Malle conjecture for counting number fields and the Batyrev-Manin conjecture for counting rational points on Fano varieties and describe what is known about it so far.
April 26, 2021
Samir Canning :
3 p.m. in Zoom
Abstract
The rational Chow ring of the moduli space of smooth curves is known when the genus is at most 6 by work of Mumford (g=2), Faber (g=3,4), Izadi (g=5), and Penev-Vakil (g=6). In each case, it is generated by the tautological classes. On the other hand, van Zelm has shown that the bielliptic locus is not tautological when g=12. In recent joint work with Hannah Larson, we show that the Chow rings of M_7, M_8, and M_9 are generated by tautological classes, which determines the Chow rings by work of Faber. I will explain an overview of the proof with an emphasis on the special geometry of curves of low genus and low gonality.
Aug. 30, 2021
Joaquin Moraga :
3 p.m. in Zoom
Abstract
In this talk, I will discuss some recent progress on toroidalization principles for klt singularities.
These toroidalizations allow us to prove theorems about the topology of klt singularities and about their minimal log discrepancies.
If time permits, I will also explain the relationship between these toroidalization principles and the termination of flips.
Sept. 13, 2021
Geoffrey Smith :
3 p.m. in Zoom
Abstract
I will present a result allowing us to control the normal bundle of a rational curve in certain complete intersections in a variety X. In particular, given a rational curve C in X, under certain hypotheses this control allows us to find complete intersections in X such that the normal bundle to C in Y is "as general as possible." By using this tool, I will present some new examples of separably rational connected Fano varieties in arbitrary characteristic. For instance, a general Fano complete intersection of hypersurfaces of degree at least 3 in a Grassmannian is separably rationally connected in any characteristic. This talk is based on joint work with Izzet Coskun.
Sept. 20, 2021
Tim Ryan :
3 p.m. in Zoom
Abstract
Recent work on the birational geometry of moduli spaces has largely worked along two lines; either it has used the machinery of Bridgeland stability conditions or it has solved the interpolation problems for vector bundles. In this talk, I will discuss recent work with Manuel Leal and Cesar Lozano Huerta in which we connect these approaches to the minimal free resolutions of sheaves. In particular, I will show that the base locus of (primary) extremal chamber of the effective cone of a moduli space of sheaves on the projective plane can be characterized in terms of the map in the Gaeta minimal free resolution. Time permitting, I will discuss a conjecture for the exact relationship between the minimal free resolution, Bridgeland destabilizing objects, and the stable base locus decomposition.
Sept. 27, 2021
Ritvik Ramkumar :
3 p.m. in Zoom
Abstract
For a smooth surface S the Hilbert scheme of points S^(n) is a well studied smooth parameter space. In this talk I will consider a natural generalization, the nested Hilbert scheme of points S^(n,m) which parameterizes pairs of subschemes X \supseteq Y of S with deg(X) = n and deg(Y) = m. In contrast to the usual Hilbert scheme of points, S^(n,m) is almost always singular and it is known that S(n,1) has rational singularities. I will discuss some general techniques to study S^(n,m) and apply them to show that S^(n,2) also has rational singularities. This relies on a connection between S^(n,2) and a certain variety of matrices, and involves square-free Gröbner degenerations as well as the Kempf-Weyman geometric technique. This is joint work with Alessio Sammartano.
Oct. 4, 2021
Janet Page :
3 p.m. in Zoom
Abstract
What is the most singular possible (reduced) hypersurface in positive characteristic? One answer to this question comes from finding a lower bound on an invariant called the F-pure threshold of a polynomial in terms of its degree. In this talk, I'll introduce a new class of hypersurfaces which obtain a minimal F-pure threshold and discuss some of their surprising algebraic and geometric properties. They are cut out by polynomials that we call Frobenius forms, which have a rich algebraic structure coming from the fact that they have a matrix factorization mirroring the theory of quadratic forms. In the surface case, we'll see that they share some geometric properties with cubic surfaces. This is based on joint work with Zhibek Kadyrsizova, Jennifer Kenkel, Jyoti Singh, Karen E Smith, Adela Vraciu, and Emily E Witt, as well as more recent joint work with Anna Brosowsky, Tim Ryan, and Karen Smith.
Oct. 11, 2021
John Kopper :
3 p.m. in Zoom
Abstract
Ample bundles are among the most important "positive" vector bundles in algebraic geometry. Unfortunately, they cannot be classified by their Chern classes alone. An approach to this problem was suggested by Le Potier, who asks for a classification of those Chern characters for which there exists a stable ample bundle. When the moduli space of stable bundles is irreducible, this is equivalent to asking for the general stable bundle to be ample. I will discuss some recent progress on this problem for (minimal) rational surfaces. This is joint work with Jack Huizenga.
Oct. 18, 2021
Karl Schwede :
3 p.m. in Zoom
Abstract
Building on breakthrough results of Andr\'e, Bhatt, Gabber and
others, Ma and the speaker introduced a theory of mixed characteristic
test ideals / multiplier ideals. There was a gap in this theory, it was
defined only for complete local rings and the formation of these ideals
did not seem to commute with localization. By utilizing ideas from
Bhatt-Ma-Patakfalvi-Tucker-Waldron-Witsazek and the author (also see
Takamatsu-Yoshikawa), we introduce a notion of multiplier / test ideals
for normal schemes finite type over a complete local ring (in particular,
our notion commutes with localization). We use our theory to study the
non-nef locus and so obtain mixed characteristic versions of results on
the non-nef locus for varieties over fields due to
Ein-Lazarsfeld-Mustata-Nakamaye-Popa, Mustata, and Nakayama. This is joint
work with Christopher Hacon and Alicia Lamarche.
Oct. 25, 2021
Takumi Murayama :
3 p.m. in Zoom
Abstract
In 1953, Kodaira proved what is now called the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations due to Grauert–Riemenschneider, Kawamata–Viehweg, Kollár, and others have become indispensable tools in algebraic geometry over fields of characteristic zero, in particular in birational geometry and the minimal model program. Even in this context, however, it is often necessary to work with schemes that are not of finite type over fields, and a fundamental problem in this more general context has been the lack of Kodaira-type vanishing theorems. We prove generalizations of Kodaira's vanishing theorem for proper morphisms of schemes of equal characteristic zero in arbitrary dimension, answering questions of Boutot, Kollár, and Kawakita. These results are optimal given known counterexamples to these vanishing theorems in positive and mixed characteristic.
Nov. 1, 2021
Greg Taylor :
3 p.m. in Zoom
Abstract
In this talk, we will discuss the asymptotic behavior of the minimal free resolution of the secant variety of a smooth curve. In particular, we will cover the asymptotic purity of the Boij-Soederberg decomposition, some of its corollaries, and directions for further inquiry.
Nov. 8, 2021
Hang (Amy) Huang :
3 p.m. in Zoom
Abstract
Tensors are just multi-dimensional arrays. Notions of ranks and border rank abound in the literature. Tensor decompositions also have a lot of application in data analysis, physics, and other areas of science. I will try to give a colloquium-style talk surveying my recent two results about tensor ranks and their application to matrix multiplication complexity. The first result relates different notion of tensor ranks to polynomials of vanishing Hessian. The second one computes the border rank of 3 X 3 permanent. I will also briefly discuss the newest technique we used to achieve our results: border apolarity. This talk assumes little background in geometry or algebra.
Nov. 15, 2021
Ziquan Zhuang :
3 p.m. in Zoom
Abstract
K-stability is an algebraic condition that characterizes the existence of K\"ahler-Einstein metrics on Fano varieties. Recently there has been a lot of work on the construction of the K-moduli space, i.e. a good moduli space parametrizing K-polystable Fano varieties. Motivated by results in differential geometry, it is conjectured that this K-moduli space is proper and projective. In this talk, I'll discuss some recent progress in birational geometry that leads to a full solution of this conjecture. Based on joint work with Yuchen Liu and Chenyang Xu.
Nov. 29, 2021
Christian Liedtke :
2 p.m. in Zoom
Abstract
We study isolated quotient singularities by finite group schemes in positive characteristic. We compute invariants, study the uniqueness of the quotient presentation, and compute some deformation spaces. A special emphasis is laid on the dichotomy between quotient singularities by linearly reductive group schemes and by group schemes that are not linearly reductive. We essentially classify the linearly reductive ones, give applications, and make some conjectures. This is joint work with Gebhard Martin (Bonn) and Yuya Matsumoto (Tokyo).
Feb. 28, 2022
Nolan Shock :
3 p.m. in Zoom
Abstract
The Grothendieck-Knudsen compactification of the moduli space of n-pointed rational curves satisfies a number of remarkable properties: it has a modular interpretation (by construction), it is the log canonical compactification (roughly, the smallest compactification with reasonable boundary singularities), and its Chow ring is the same as its cohomology ring and looks like the Chow ring/cohomology ring of a toric variety. I will discuss how these results can be partially generalized to compactifications of moduli of higher-dimensional varieties (namely, moduli of hyperplane arrangements and marked del Pezzo surfaces) by using some simple ideas in tropical geometry.
March 7, 2022
Hang (Amy) Huang :
3 p.m. in Zoom
Abstract
Tensors are just multi-dimensional arrays. Notions of ranks and border rank abound in the literature. Tensor decompositions also have a lot of application in data analysis, physics, and other areas of science. I will try to give a colloquium-style talk surveying my recent two results about tensor ranks and their application to matrix multiplication complexity. The first result relates different notion of tensor ranks to polynomials of vanishing Hessian. The second one computes the border rank of 3 X 3 permanent. I will also briefly discuss the newest technique we used to achieve our results: border apolarity. This talk assumes little background in geometry or algebra.
March 14, 2022
Isabel Vogt :
3 p.m. in Zoom
Abstract
We will prove that the normal bundle of a general canonical curve of genus g not 4 or 6 is semistable. This is joint work with Izzet Coskun and Eric Larson.
March 28, 2022
Anand Patel :
3 p.m. in Zoom
Abstract
In this talk I will provide an overview of progress (collectively joint with Anand Deopurkar, Mitchell Lee, Hunter Spink, and Dennis Tseng) on a basic problem in enumerative geometry: counting hypersurfaces.
In a nutshell, the problem asks to determine universal formulas which count the number of times a particular hypersurface (up to PGL-equivalence) arises in any family. For a prototype: The symmetric Thom-Porteous formula of Harris and Tu, determining the class of the locus where a symmetric map of vector bundles has a particular rank, comprises the "quadric hypersurface case" of the problem.
In particular, I will report on the cases of hyperplane arrangements, quartic plane curves, and cubic surfaces.
April 4, 2022
Wern Yeen Yeong :
3 p.m. in Zoom
Abstract
A complex algebraic variety is said to be hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. Demailly introduced algebraic hyperbolicity as an algebraic version of this property, and it has since been well-studied as a means for understanding Kobayashi’s conjecture, which says that a generic hypersurface in projective space is hyperbolic whenever its degree is large enough. In this talk, we study the algebraic hyperbolicity of very general hypersurfaces of high bi-degrees in Pm x Pn and completely classify them by their bi-degrees, except for a few cases in P3 x P1. We present three techniques to do that, which build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As another application of these techniques, we improve the known result that very general hypersurfaces in Pn of degree at least 2n − 2 are algebraically hyperbolic when n is at least 6 to when n is at least 5, leaving n = 4 as the only open case.
April 11, 2022
John Lesieutre :
3 p.m. in Zoom
Abstract
If L is a line bundle on a variety X, then it is a basic result that h^0(mL)
grows roughly polynomially in m. In birational geometry, it is frequently
useful to instead fix an ample divisor A and consider the growth of h0(mL+A)
as m increases. I will show that the behavior of this growth can be quite
strange.
April 18, 2022
Ian Cavey :
3 p.m. in Zoom
Abstract
The Hilbert scheme of n points in the (affine) plane parametrizes finite, length n subschemes of C^2. In this talk I will explain how to compute the “Newton-Okounkov bodies” of these Hilbert schemes. These Newton-Okounkov bodies are (unbounded) polyhedra which encode geometric information about the Hilbert schemes. I will also discuss some partial results and conjectures for Hilbert schemes of points on complete toric surfaces.
April 25, 2022
Eric Jovinelly :
3 p.m. in Zoom
Abstract
The cone of effective divisors controls the rational maps from a variety. We study this important object for M_{0,n}, the moduli space of stable rational curves with n markings. Fulton once conjectured the effective cones for each n would follow a certain combinatorial pattern. However, this pattern holds true only for n < 6. Despite many subsequent attempts to describe the effective cones for all n, we still lack even a conjectural description. We study the simplest open case, n=7, and identify the first known difference between characteristic 0 and characteristic p. Although a full description of the effective cone for n=7 remains open, our methods allowed us to compute the entire effective cones of spaces associated with other stability conditions.
June 8, 2022
Isabel Vogt and Eric Larson :
11 a.m. in 636 SEO
Sept. 12, 2022
Nolan Schock :
3 p.m. in 636 SEO
Abstract
I will discuss approaches to understanding interesting (e.g. stable pair) compactifications of moduli of del Pezzo surfaces using tropical geometry and combinatorics of the root system $E_n$. Such an approach was first considered by Hacking, Keel, and Tevelev, dating back to works of Naruki and Sekiguchi; however, new ideas may lead to a more complete understanding of these compactifications. Similar approaches for other root systems may yield other interesting compactifications of moduli spaces.
Sept. 19, 2022
Miguel Moreira :
3 p.m. in Zoom
Abstract
Virasoro constraints for Gromov-Witten invariants have a rich history tied to the very beginning of the subject. Recently, Virasoro constraints for moduli spaces of stable pairs on 3-folds were found using the Gromov-Witten/Stable pairs correspondence. This discovery led to a new study of such constraints for integrals of descendents in different moduli of sheaves. This talk will be based on joint work with A. Bojko and W. Lim, where we propose a general conjecture and fit the Virasoro operators in the vertex algebra Joyce recently introduced to study wall-crossing. We then use this framework to show compatibility between the constraints and wall-crossing. As an application, we prove that Virasoro holds for moduli of stable sheaves on curves and surfaces with h^{0,1}=h^{0,2}=0.
Sept. 26, 2022
Colleen Robichaux :
3 p.m. in Zoom
Abstract
We give an explicit formula for the degree of a vexillary Grothendieck polynomial. This generalizes a previous result of J. Rajchgot-Y. Ren-C. Robichaux-A. St. Dizier-A. Weigandt for degrees of symmetric Grothendieck polynomials. We apply our work to compute the Castelnuovo-Mumford regularity of certain matrix Schubert varieties. We also derive formulas for the regularities of Kazhdan-Lusztig varieties coming from open patches of Grassmannians as well as the regularities of certain ladder determinantal ideals. This is joint work with Jenna Rajchgot and Anna Weigandt.
Oct. 3, 2022
No Seminar :
3 p.m. in 636 SEO
Oct. 10, 2022
Takumi Murayama :
3 p.m. in 636 SEO
Abstract
The nef cone of a projective variety X is a fundamental object that controls morphisms from X to other projective varieties. Computing this cone for specific varieties is a notoriously difficult problem, even for the product CxC of a curve C with itself. We construct new nef classes on self-products of very general complex projective curves of genus g > 2, which are the first non-trivial examples on the boundary of the nef cone that exist for all g > 2. This is joint work with Mihai Fulger.
Oct. 17, 2022
Geoffrey Smith :
3 p.m. in 636 SEO
Abstract
While vector bundles on $\mathbb{P}^2$ are well-studied, much less is known about vector bundles on higher-dimensional projective spaces. In this talk, I will partially address this problem by discussing properties of vector bundles on projective space that can be described as the kernel of a general map between relatively simple bundles (e.g., direct sums of line bundles and (co)tangent bundles). More precisely, I will describe circumstances under which these bundles have cohomology that is as simple as possible, are stable, and are ample. This talk is based on joint work with Izzet Coskun and Jack Huizenga.
Oct. 24, 2022
Jarod Alper :
3 p.m. in 636 SEO
Abstract
Grothendieck's Existence Theorem asserts that a coherent sheaf on a scheme proper over a complete local noetherian ring is the same as a compatible system of coherent sheaves on the thickenings of its central fiber. This is a fundamental result with important applications to moduli theory. We will discuss generalizations of this result to algebraic stacks beginning with a review of the characteristic 0 situation where a satisfactory answer is known: any quotient stack [Spec A/G] whose invariant ring A^G is a complete local k-algebra is coherently complete along its unique closed point. We will report on partial progress in joint work with Hall and Lim on extending this result to positive characteristic.
Oct. 31, 2022
Jack Huizenga :
3 p.m. in 636 SEO
Abstract
We investigate certain moduli spaces of rank 2 vector bundles on blowups
of the projective plane at 10 or more very general points. Assuming the
SHGH conjecture, we show that as the polarization varies these spaces
can have arbitrarily many
components of arbitrarily high dimensions. In the case of 10 points,
the components correspond to continued fractions of the square root of
10. This is joint work with Izzet Coskun.
Nov. 7, 2022
Joe Waldron :
3 p.m. in 636 SEO
Abstract
Given a field K of characteristic p, a classical result of Jacobson provides a Galois correspondence between finite purely inseparable subfields of K of exponent one (i.e. those which contain K^p), and sub-restricted Lie algebras of Der(K). I will discuss joint work with Lukas Brantner in which we extend this Galois correspondence to subfields of arbitrary exponent using methods from derived algebraic geometry.
Nov. 14, 2022
Junliang Shen :
3 p.m. in 636 SEO
Abstract
In 2010, de Cataldo-Hausel-Migliorini proposed a conjecture connecting topology of the Hitchin system and Hodge theory of the corresponding character variety via the non-abelian Hodge theory. This conjecture is now referred to as the P=W conjecture. The purpose of this talk is to explain a recent proof of this conjecture (for GL_n) in joint work with Davesh Maulik for any rank and genus, where we combine tools from algebraic geometry and representation theory.
Nov. 21, 2022
No Seminar :
3 p.m. in 636 SEO
Nov. 28, 2022
Ritvik Ramkumar :
1:30 p.m. in 636 SEO
Abstract
It is a classical, but also difficult, problem to determine the defining equations of the blowup of P^n along a given subvariety X. In this talk, I will focus on the cases where X is defined by the ideal of maximal minors of a 2xn matrix of linear forms. I will explain how to determine the defining equations and the singularities of their blowup algebras. This relies on a stratification of the Hilbert scheme of determinantal ideals and the combinatorics of square-free degenerations. This is joint work with Alessio Sammartano.
March 6, 2023
Gwyneth Moreland :
noon in 636 SEO
Abstract
We compute some higher (co)dimension nef and effective cones of the Hilbert scheme of 3 points in $\mathbb{P}^3$. This involves studying the orbits of the PGL action on the Hilbert scheme, as well as extending Mallavibarrena and Sols' bases for the Chow groups of Hilbert schemes of points on $\mathbb{P}^2$ to the case of the Hilbert scheme of 3 points in $\mathbb{P}^3$. This work builds on results of Ryan and Stathis.
March 13, 2023
Jinhyung Park :
noon in 636 SEO
Abstract
Recently, Aprodu-Farkas-Papadima-Raicu-Weyman and Raicu-Sam obtained new proofs of generic Green's conjecture by studying syzygies of tangent developable surfaces of rational normal curves and K3 carpets. Using secant varieties of rational normal curves, we give simple geometric proofs of their results. As a consequence, we obtain a quick proof of generic Green's conjecture. We also discuss the syzygies of tangent developable surfaces of arbitrary smooth projective curves.
March 27, 2023
Fernando Figueroa :
noon in 427 SEO
Abstract
We will start this talk by discussing general results about the fundamental group of the link of a singularity.
We will continue by studying the singularities of the Minimal Model Program. We will start by discussing the state of the art for log terminal singularities. Lastly, we will study restrictions for fundamental groups of log canonical singularities in low dimensions, and construct log canonical singularities with certain prescribed fundamental groups.
This is based on joint work with Joaquín Moraga
April 10, 2023
Siddarth Kannan :
noon in 636 SEO
Abstract
Given integers g, m, and n, the heavy/light moduli space Mbar_{g, m|n} is a compactification of the moduli space of smooth (m+n)-marked curves of genus g. These spaces are particular examples of Hassett’s moduli spaces of weighted stable curves. Their rational cohomology gives a rich family of representations of products of symmetric groups. I’ll discuss recent work on the structure of this family of representations, and how they relate to the S_n-representations determined by the cohomology of Deligne-Mumford compactifications. This talk is based on joint work with Stefano Serpente and Claudia Yun.
April 17, 2023
Ben Bakker :
noon in 636 SEO
Abstract
Compact hyperkahler manifolds are higher-dimensional generalizations of K3 surfaces; their geometry is tightly constrained by the existence of a holomorphic symplectic form. For example, a result of Matsushita says the only nontrivial fibration structures they admit are fibrations by Lagrangian tori. In this talk I'll explain how to prove a conjecture of Matsushita that such fibrations are either isotrivial or vary maximally in moduli. I will also discuss some other features of the topology of Lagrangian fibrations and a result about the density of torsion points in sections.
April 24, 2023
Frederik Benirschke :
3 p.m. in 636 SEO
Abstract
One of the easiest ways of producing classes in the cohomology or Chow ring of the moduli space of curves is by taking the fundamental class of the hyperelliptic locus (or other loci of curves with rational functions of specified ramification).
While we know that these classes are computable in theory by results of Faber-Pandharipande, no closed formulas are known in general.
We explain a new approach using differential forms instead of rational functions. This allows to use the recent constructions of compactifications of moduli spaces of differentials by Bainbridge-Chen-Gendron-Grushevsky-Moeller.
It turns out that, lifted to the moduli space of differentials, the classes are simply products of divisors.
Sept. 1, 2023
Linquan Ma :
11 a.m. in 636 SEO
Abstract
We introduced a mixed characteristic test ideal using the p-adic Riemann-Hilbert correspondence of Bhatt-Lurie. We show that, under mild finiteness assumptions, this version of test ideal commutes with localization and can be computed by a single alteration up to small perturbation. This is based on joint work in progress with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.
Sept. 11, 2023
Yuchen Liu :
3 p.m. in 636 SEO
Abstract
K-stability provides a powerful tool for constructing compact moduli spaces, known as K-moduli spaces, for Fano varieties. However, determining the K-moduli space for specific Fano varieties, such as Fano hypersurfaces, can be a challenging problem. Previously, K-moduli space for cubic hypersurfaces was shown to be the same as GIT up to dimension 4. In this talk, I’ll discuss some recent progress on the K-moduli space of quartic threefolds where K-moduli and GIT differ significantly. We find a new codimension 3 locus in the K-moduli space that parametrizes certain weighted complete intersections. Moreover, we show that this locus is closed by relating the K-stability of such complete intersections to certain del Pezzo surface pairs. This is joint work with Hamid Abban, Ivan Cheltsov, Alexander Kasprzyk, and Andrea Petracci.
Sept. 18, 2023
Emanuela Marangone :
3 p.m. in 636 SEO
Abstract
An Artinian Algebra $A$ has the Weak Lefschetz Property (WLP) if there
is a linear form, $\ell$, such that the multiplication map $\times \ell$ from $A_i$ to $A_{i+1}$ has
maximal rank for each integer $i$. We want to study the set of linear forms
for which maximal rank fails, this is called the non-Lefschetz locus and has
a natural scheme structure.
An important result by Boij–Migliore–Miro-Roig–Nagel states that for a
general Artinian complete intersection of height 3, the non-Lefschetz locus
has the expected codimension and the expected degree.
In this talk, we will define in a similar way the non-Lefschetz locus for
conics. We say that $C$, a homogeneous polynomial of degree 2, is a Lefschetz
conic for $A$ if the multiplication map $\times C$ from $A_i$ to $A_{i+2}$ has maximal rank
for each integer $i$. We will show that for a general complete intersection of
height 3, the non-Lefschetz locus of conics has the expected codimension as
a subscheme of $\mathbb{P}^5$, and that the same does not hold for certain monomial complete intersections.
The study of the non-Lefschetz locus for Artinian complete intersections
can be generalized to modules $M = H^1_{∗}(\mathbb{P}^2,E)$ where $E$ is a vector bundle of
rank 2. The non-Lefschetz locus, in this case, is exactly the set of jumping
lines of $E$, and the expected codimension is achieved under the assumption
that $E$ is general.
In the case of conics, the same is not true. The non-Lefschetz locus of
conics is a subset of the jumping conics, but it is a proper subset when $E$ is
semistable with first Chern class even.
Sept. 25, 2023
No Seminar :
3 p.m. in 636 SEO
Oct. 2, 2023
Junyan Zhao :
3 p.m. in 636 SEO
Abstract
The K-moduli theory provides us with an approach to study moduli of curves. In this talk, I will introduce the K-moduli of certain log Fano pairs and how it relates to moduli of curves. We will see that the K-moduli spaces interpolate between different compactifications of moduli of curves. In particular, the K-moduli gives the last several Hassett-Keel models of moduli of curves of genus six.
Oct. 9, 2023
No Seminar :
3 p.m. in 636 SEO
Oct. 13, 2023
Nathan Chen :
11 a.m. in 636 SEO
Abstract
A result of Popa and Schnell shows that any holomorphic one-form on a smooth complex projective variety of general type vanishes somewhere. In this talk, we will explore a complementary question: given a variety of intermediate Kodaira dimension, what can we say about the geometry of the variety if it carries a set of g pointwise linearly independent holomorphic one-forms? Our main result is a classification of all such varieties if the Kodaira ``codimension" is equal to g. This is joint work with Ben Church and Feng Hao.
Oct. 16, 2023
Yeqin Liu :
3 p.m. in 636 SEO
Abstract
Vector bundles E with Hom*(E, E)=C are called exceptional, and they play an important role in the study of derived categories and stable sheaves. Unlike P^1 and P^2, classifying exceptional bundles on P^3 is a challenging problem. In this talk we introduce new techniques to approach this problem, by studying stable spherical bundles on quartic surfaces. We show the first nonexistence results: there is no exceptional bundle on P^3 with degree d and maximal possible rank 2d^2+1 when |d|>3. We will also discuss many future developments of this program.
Oct. 23, 2023
Lucas Mioranci :
3 p.m. in 636 SEO
Abstract
A complex projective variety $X$ is algebraically hyperbolic if there exists an ample divisor $H$ and a real number $\epsilon > 0$ such that the geometric genus $g(C)$ and the degree of any integral curve $C\subset X$ satisfy the inequality
\[ 2g(C) - 2\ge \epsilon \deg_H (C). \]
The algebraic hyperbolicity is an important property to characterize varieties of general type, and it is connected to famous conjectures such as the Lang Conjectures and Green-Griffiths Conjecture.
By building on recent work, I classify algebraic hyperbolic hypersurfaces in homogeneous varieties, thus obtaining explicit bounds for the hyperbolicity in plenty of open cases, including Grassmannians, flag varieties, and their products.
Oct. 30, 2023
Eamon Quinlan-Gallego :
3 p.m. in 636 SEO
Abstract
The theory of F-modules, pioneered by Lyubeznik, is a powerful machinery that allows us to prove finiteness results about local cohomology of regular rings in positive characteristic. In this talk I will explain how this theory can be extended to rings with mild singularities (namely: rings with finite F-representation type). If time allows I will then show how one can recover and extend some results on local cohomology for these rings.
Nov. 1, 2023
Swaraj Pande :
11 a.m. in 427 SEO
Abstract
The Alpha invariant of a complex Fano manifold was introduced by Tian to detect its K-stability, an algebraic condition that implies the existence of a Kähler–Einstein metric. Demailly later reinterpreted the Alpha invariant algebraically in terms of a singularity invariant called the log canonical threshold. In this talk, we will present an analog of the Alpha invariant for Fano varieties in positive characteristics, called the Frobenius-Alpha invariant. This analog is obtained by replacing “log canonical threshold” with “F-pure threshold”, a singularity invariant defined using the Frobenius map. We will review the definition of these invariants and the relations between them. The main theorem proves some interesting properties of the Frobenius-Alpha invariant; namely, we will show that its value is always at most 1/2 and make connections to a version of local volume called the F-signature. Time permitting, we will also discuss the semicontinuity properties of the Frobenius-Alpha invariant.
Nov. 13, 2023
Felix Janda :
3 p.m. in 636 SEO
Abstract
I will present joint work with Tony Yue Yu, which introduces a new type
of curve counting invariants, which count, in a naive way, curves in a
smooth projective variety that pass through prescribed subvarieties
with tangencies. Examples of enumerative and non-enumerative
invariants, and a connection to mirror symmetry will be provided.
Nov. 20, 2023
No Seminar :
3 p.m. in 636 SEO
Jan. 8, 2024
Jihao Liu :
3 p.m. in 636 SEO
Abstract
I will report the establishment of the minimal model program for algebraically integrable foliations on klt varieties and it applications, such as the minimal model program for generalized pairs and the canonical bundle formula. If time permits, I will discuss some related open problems and their connections to moduli theory. This talk is partially based on a series of joint works of myself with Guodu Chen, Jingjun Han, Fanjun Meng, and Lingyao Xie.
Jan. 19, 2024
Eric Larson :
11 a.m. in 636 SEO
Abstract
By Bezout's theorem, a space curve of degree d intersects a quadric Q in 2d points. The dimensions of the Hilbert schemes of Brill--Noether space curves of degree d and genus g, and of 2d points on Q, are both of dimension 4d. It is therefore natural to expect that intersecting with a quadric induces an etale map between these Hilbert schemes. In characteristic zero, this is the case with exactly 6 exceptions. This talk will focus on understanding what happens in characteristic 2, and in particular, why the analogous statement fails in a dramatic way.
Jan. 22, 2024
No Seminar :
3 p.m. in 427 SEO
Jan. 29, 2024
Ethan Reed :
3 p.m. in 636 SEO
Abstract
In characteristic 0, the sheaf cohomology groups for line bundles on the full flag variety are given by the Borel-Weil-Bott Theorem. However, in positive characteristic a full description is not known. I will discuss some progress in positive characteristic including recent stabilization results of Raicu and Vandebogert. Further, Raicu and Vandebogert computed special cases of these stable cohomology groups using certain arithmetic complexes (arithmetic in the sense that they are defined over the integers). I will then discuss joint work with Luca Fiorindo, Shahriyar Roshan-Zamir, and Hongmiao Yu in which we prove an isomorphism of generalizations of these complexes defined over the ring of integer valued polynomials as conjectured by Gao, Raicu, and Vandebogert. In particular, this gives a more conceptual proof of an identification between the stable sheaf cohomology groups of hook and two column partition Schur functors applied to the cotangent sheaf of projective space.
Feb. 5, 2024
Keller VandeBogert :
3 p.m. in 636 SEO
Abstract
The Borel-Weil-Bott (BWB) theorem is a fundamental result that gives a (relatively simple) method of computing the cohomology of line bundles on flag varieties over a field of characteristic 0. The analogue of BWB in positive characteristic is a wide-open problem despite many important results over the decades, and it remains out of reach even from a computational perspective. In this talk, I'll speak on joint work with Claudiu Raicu that shows that, despite the chaos, there is a notion of stability for the cohomology of line bundles on flags in arbitrary characteristic. Moreover, there are many cases where we can compute this stable sheaf cohomology explicitly, and these computations yield sharp, characteristic-free vanishing results for finite-length Koszul modules.
Feb. 12, 2024
Junyan Zhao :
3 p.m. in 636 SEO
Abstract
As the last step to the Calabi Problem, we are asked to find all the Kaehler-Einstein limits of each deformation family of Fano varieties. In this talk, I will illustrate the application of the moduli continuity method in conjunction with approaches like wall-crossing and moduli of K3 surfaces to explicitly describe the K-moduli space of a specific deformation family of Fano threefold. This is a recent work joint with Yuchen Liu.
Feb. 19, 2024
Lena Ji :
3 p.m. in 636 SEO
Abstract
In this talk, we study the 6-dimensional moduli space of a family of Fano threefolds, and we construct a compactification using K-stability. These threefolds admit a conic bundle structure---we relate the K-moduli space of the threefolds to the GIT moduli space of the discriminant curves, and we study the behavior of the conic bundle structure on the boundary. The technique we use is wall-crossings in K-moduli for certain log Fano pairs (X, cD) as the coefficient c varies. Our work is the first to systematically study these K-moduli spaces when D is not proportional to the anticanonical divisor of X, and we find surprising wall-crossing behavior in this setting. This work is joint with Kristin DeVleming, Patrick Kennedy-Hunt, and Ming Hao Quek.
Feb. 26, 2024
Mircea Mustaţă :
3 p.m. in 636 SEO
Abstract
I will give an introduction to higher-order versions of the classical notions of Du Bois and rational singularities and I will discuss an invariant that governs these notions for local complete intersections. This is based on joint work with Qianyu Chen, Bradley Dirks, Sebastian Olano,and Mihnea Popa.
March 1, 2024
Hannah Larson :
11 a.m. in 636 SEO
Abstract
The moduli space M_g of genus g curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of M_g is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of M_g called the tautological ring. The definition of the tautological ring was later extended to the compactification M_g-bar and the moduli spaces with marked points M_{g,n}-bar. While the full cohomology ring of M_{g,n}-bar is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I'll ask the question: which cohomology groups H^k(M_{g,n}-bar) are tautological? And when they are not, how can we better understand them? This is joint work with Samir Canning and Sam Payne.
March 4, 2024
Dawei Chen :
3 p.m. in 636 SEO
Abstract
We investigate the count of meromorphic differentials on the Riemann sphere possessing a single zero, multiple poles with prescribed orders, and fixed residues at each pole. Gendron and Tahar previously examined this problem with respect to general residues using flat geometry, while Sugiyama approached it from the perspective of fixed-point multipliers of polynomial maps in the case of simple poles. In our study, we employ intersection theory on compactified moduli spaces of differentials, enabling us to handle arbitrary residue conditions and provide a complete solution to this problem. This is joint work with Miguel Prado.
March 11, 2024
Ziquan Zhuang :
11 a.m. in 636 SEO
Abstract
Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when the weights drop. I will discuss some joint work with Fanjun Meng that constructs similar morphisms between moduli of stable pairs in higher dimensions.
March 29, 2024
Brian Lehmann :
11 a.m. in 636 SEO
Abstract
Let X be a smooth projective variety and let E be a vector bundle on X. A common way to analyze E is to fix a family of curves C on X and to study the restrictions of E to C. In this talk I will give several qualitative statements describing the behavior of these restrictions. This is joint work with Eric Riedl and Sho Tanimoto.
April 15, 2024
Bruno De Oliveira :
3 p.m. in 636 SEO
Abstract
We investigate the components determining bigness of the cotangent bundle $\Omega^1_X$ of smooth models $X$ in the birational class $\mathcal {Y}$ of an orbifold surface of general type $Y$, with a focus on the contribution given by the singularities of $Y$. A criterion for bigness of $\Omega_X^1$ is given involving only topological and singularity data on $Y$. We single out a special case, the Canonical Model Singularities (CMS) criterion, when $Y$ is the canonical model of $\mathcal Y$. We study the singularity invariants appearing in the criterion and determine them for $A_n$ singularities. Knowledge of these invariants for $A_n$ singularities allows one to evaluate the $(c_2,c^2_1)-$geographical range of the CMS criterion and compare it to other criteria. We obtain new examples of surfaces with big cotangent bundle. (Joint work with Y. Asega and M.Weiss)
April 19, 2024
Sebastian Casalaina-Martin :
11 a.m. in 636 SEO
Abstract
In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus of the talk will be on some results regarding non-isomorphic smooth compactifications of the moduli space of cubic surfaces, showing that two natural desingularizations of the moduli space have the same cohomology, and are both blow-ups of the moduli space at the same point, but are nevertheless, not isomorphic, and in fact, not even K-equivalent. I will also discuss a related moduli space, the moduli space of cubic surfaces with a marked line.
April 22, 2024
Eric Riedl :
3 p.m. in 636 SEO
Abstract
Given a smooth Fano variety and a smooth curve B, let
Hom(B,X) be the moduli space of maps from B to X. Let M be a component
of Hom(B,X). If a general curve parameterized by M is free, it means M
has the expected dimension and good deformation behavior. Components M
consisting entirely of curves that are not free are more mysterious.
We give a geometric characterization of which curves can be nonfree,
explaining that roughly they come from fibrations. We show how this
result can be seen as the analogue of Manin's Conjecture, which
predicts the number of rational on a variety of bounded height. This
is joint work with Brian Lehmann and Sho Tanimoto.
Sept. 9, 2024
Philip Engel :
3 p.m. in 636 SEO
Abstract
Deligne proved in 1987 that only finitely many Z-local systems of a fixed
rank underlie a polarized variation of Hodge structure, over a fixed
quasiprojective variety. He conjectured that this finiteness also holds
in families of quasiprojective varieties. In the 1990’s, Simpson’s refined
this conjecture in the following form: the nonabelian Hodge locus is
algebraic. I will discuss joint work with Salim Tayou proving these
conjectures when the algebraic monodromy group is cocompact.
Sept. 16, 2024
Maya Banks :
3 p.m. in 636 SEO
Abstract
A differential module is a module equipped with a square-zero endomorphism and is a natural generalization of a chain complex. We use deformation theoretic techniques to give a geometric description of the set of differential modules with homology isomorphic to a given module.
Sept. 23, 2024
Aaron Landesman :
3 p.m. in 636 SEO
Abstract
What is the smallest genus h of a non-isotrivial curve over the generic genus g curve?
In joint work with Daniel Litt, we show h is more than $\sqrt{g}$ by proving a
more general result about variations of Hodge structure on sufficiently general curves.
As a consequence, we show that local systems on a sufficiently general curve of geometric origin are not Zariski dense in the character variety
parameterizing such local systems. This gives counterexamples to conjectures of Esnault-Kerz and Budur-Wang.
Sept. 30, 2024
Martin Bishop :
3 p.m. in 636 SEO
Abstract
We will discuss the importance of Chow rings in algebraic geometry, as well as one of the central issues in their computation: can one find the Chow ring of a space given the Chow ring of an open and its complement? This is the so called patching problem, and we will discuss multiple ways of solving it. Our main examples will be the moduli stack of curves, as well as root gerbes and root stacks.
Oct. 7, 2024
Jenia Tevelev :
3 p.m. in 636 SEO
Abstract
Given a Fano manifold M with extremal contractions to Fano manifolds A and B, it is expected that the derived category of M contains two semi-orthogonal decompositions, related by the action of the braid group, which refine the semi-orthogonal decompositions of the derived categories of A and B. I will discuss strategies for proving this expectation when the Fano manifolds have moduli interpretations. As one application, we construct a semi-orthogonal decomposition of the derived category of the moduli space of stable rank 2 vector bundles on a smooth projective curve, as conjectured by Narasimhan and by Belmans, Galkin, and Mukhopadhyay.
Oct. 14, 2024
Ian Cavey :
3 p.m. in 636 SEO
Abstract
The Hilbert schemes of points on a smooth algebraic surface are smooth varieties that parametrize finite closed subschemes of the surface of a fixed length. When the underlying surface is toric, global sections of line bundles on the surface correspond to integer points in an associated polygon. In this talk, I will explain how in certain cases the corresponding problem on the Hilbert scheme can be interpreted as a packing problem for integer points in the same polygon satisfying a certain separation condition. Such an interpretation is known for all ample line bundles on Hilbert schemes of points on Hirzebruch surfaces (for example $\mathbb{P}^1\times\mathbb{P^1})$ and is expected to hold more generally. Based on this counting interpretation for sections of ample line bundles, I will also give formula for the Euler characteristic of any line bundle on the Hilbert schemes of points on $\mathbb{P}^1\times\mathbb{P^1}$. The latter formula has applications to the Verlinde series introduced by Ellingsrud, Göttsche, and Lehn.
Oct. 21, 2024
Salim Tayou :
3 p.m. in 636 SEO
Abstract
The Ceresa cycle is a homologically trivial cycle that
lives on the Jacobian of any smooth proper curve of genus g. Its
image under the Abel-Jacobi map defines a normal function on M_g
and Ceresa famously proved that this normal function is generically
non-torsion. In this talk, I will explain a joint recent work with
Matt Kerr where we prove that the positive-dimensional part of
the torsion locus of the Ceresa normal function in M_g is not Zariski
dense when g>2. Moreover, it has only finitely many components with
generic Mumford-Tate group equal to GSp_2g, these components are
defined over the algebraic closure of Q and their union is closed
under the action of the absolute Galois group of Q. This result
follow from a general study of the distribution of the torsion
locus of arbitrary admissible normal functions.
Oct. 28, 2024
Mark Walker :
3 p.m. in 636 SEO
Abstract
An "Ulrich module" for a local ring is a non-zero maximal Cohen-Macaulay module of minimal multiplicity. An "Ulrich sheaf" for a projective scheme is a non-zero coherent sheaf whose cohomology table looks like the cohomology table of a direct sum of copies of the structure sheaf on projective space. The mere existence of an Ulrich module or an Ulrich sheaf implies a collection of desirable results. For instance, if a local ring R admits an Ulrich module, then Lech's conjecture holds for faithfully flat extensions of R. It has been asked if every Cohen-Macaulay ring admits an Ulrich module. In this talk, I'll explain the connection between Ulrich modules and Ulrich sheaves, and use it prove there exist complete local complete intersection rings of dimension two that do not have any Ulrich modules. This result is joint work with Srikanth Iyengar, Linquan Ma, and Ziquan Zhuang.
Nov. 4, 2024
Patricia Klein :
3 p.m. in 636 SEO
Abstract
Roughly speaking, enumerative geometry is a field whose goal is to count the "typical" number of solutions to certain types of families of polynomial equations, particularly when that number is finite. With a great deal of effort, especially in the wake of the work of Hermann Schubert around the turn of the 20th century, mathematicians made rigorous the notion of a "typical" answer and also made rigorous certain simplifying strategies Schubert had suggested. Indeed, making Schubert's arguments precise was the topic of Hilbert's 15th problem, and the field born from this study is now called Schubert calculus. The simplifications Schubert had suggested entail sliding or deforming the geometric objects to be studied while preserving the total number of whatever it is one wants to count. These strategies are what are now called degeneration techniques. In this talk, we will describe some modern questions in Schubert calculus and explain how these questions are studied via Gröbner degeneration in particular.
Nov. 11, 2024
Francois Greer :
3 p.m. in 636 SEO
Abstract
Elliptic surfaces are a fairly well understood class of complex
projective surfaces. They come with two discrete invariants,
$g$ and $d$, both nonnegative integers. I will discuss some
new results (joint with P. Engel, A. Ward, and Y. Zhang) about
the moduli space and Hodge theory of elliptic surfaces with
$(g,d)=(1,1)$. While they have Kodaira dimension one, they behave
like K3 surfaces in many respects, and they provide an interesting
test case for the Hodge Conjecture in dimension 4.
Nov. 18, 2024
Noah Giansiracusa :
3 p.m. in 636 SEO
Abstract
I'll discuss joint work with Alessio Caminata, Luca Schaffler, and Han-Bom Moon in which we study equations defining (the closure of) the locus of n points in projective space that lie on a rational normal curve and apply these equations to resolve a question of Lior Pachter and David Speyer from 2004 on the tropical geometry of the space of phylogenetic trees.
Nov. 25, 2024
NO SEMINAR :
3 p.m. in 636 SEO
Jan. 13, 2025
Roberto Svaldi :
3 p.m. in 636 SEO
Abstract
In recent years there has been considerable progress in extending the
ideas and techniques of the Minimal Model Program beyond the realm of
algebraic varieties to the study of foliations. For the case of
foliations on surfaces, McQuillan, Brunella and Mendes have obtained a
detailed classification — analogous to the Enriques-Kodaira
classification. In this seminar, I will explain how, using the
birational classification of foliations on surfaces and MMP techniques,
we can start constructing moduli spaces for minimal foliations that
have maximum Kodaira dimension on surfaces. While there are many
similarities between the birational theory and the theory of foliations,
some new important phenomena appear in the latter.
In the seminar, I will try to explain what these new phenomena are and
what new difficulties they introduce into the identification of a good
functor of moduli for the aforementioned foliations, in comparison to
the case of KSBA moduli spaces.
The talk will feature joint work with C. Spicer, and joint work in
progress with M. McQuillan, C. Spicer, and S. Velazquez.
Jan. 27, 2025
Kevin Tucker :
3 p.m. in 636 SEO
Abstract
The log canonical threshold (lct) is an important numerical invariant of singularities in complex algebraic geometry, with analytic origins. Via standard reduction to characteristic $p>0$ techniques, it is closely related to the $F$-pure threshold in positive characteristic defined in terms of the Frobenius endomorphism. These equal characteristic thresholds admit an analogue in the
developing theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this talk, I will review these notions and discuss a computation of the plus-pure thresholds of some mixed characteristic cusp-like singularities (such as $p^2 + x^3 \in \mathbb{Z}_p[[ x ]]$). This talk is based on joint work with Hanlin Cai, Suchitra Pande, Eamon Quinlan-Gallego, and Karl Schwede.
Feb. 3, 2025
Philip Engel :
3 p.m. in 636 SEO
Abstract
I will discuss forthcoming joint work with S. Filipazzi,
F. Greer, M. Mauri, and R. Svaldi that (1) irreducible Calabi-Yau
varieties admitting an abelian fibration, in any fixed dimension,
form a birationally bounded class and (2) symplectic varieties
admitting a Lagrangian fibration, in any fixed dimension,
form an analytically bounded class.
Feb. 10, 2025
Izzet Coskun :
3 p.m. in 636 SEO
Feb. 17, 2025
Eric Jovinelly :
3 p.m. in 636 SEO
Abstract
Rational curves play a critical role in understanding the birational geometry of varieties. Free curves are the easiest to work with, but on Fano varieties that are even mildly singular, it remains an open question whether these free rational curves exist. In this talk, we discuss free curves of higher genus. Using some ideas on stability of vector bundles, we show that any klt Fano variety has higher-genus free curves. We then use the existence of these free curves to get some applications: we prove the existence of free rational curves in terminal Fano threefolds, obtain an optimal upper bound for any klt pair on the length of extreme rays of its Mori cone of curves, and study the fundamental group of the smooth locus of a klt Fano variety. This is joint work with Brian Lehmann and Eric Riedl.
Feb. 24, 2025
Desmond Coles :
3 p.m. in 636 SEO
Abstract
The study of algebraic varieties by constructing an associated 'combinatorial shadow', has proven to be a valuable toolkit in many areas of algebraic geometry including the study of moduli spaces, enumerative geometry, and more. The starting point for constructing these 'shadows' has commonly been the tropicalization of toric varieties. In this talk I will explain how tropicalization of toric varieties can be extended to a larger class of varieties, spherical varieties. I will review the literature on this subject, and discuss some recent work of mine on a balancing condition for spherical tropical varieties.
Feb. 26, 2025
David Anderson :
3 p.m. in 1227 SEO
Abstract
Given a matrix of homogeneous polynomials, one often wants to specify rank bounds on various submatrices – the result is called a degeneracy locus. An old problem, considered by many 19th century mathematicians, asks for a formula for the degree of such a variety. It turns out that the universal such formulas – the Schubert polynomials – have an incredibly rich algebra and combinatorics in their own right.
I’ll describe recent work on Schubert polynomials, including developments by Lam, Lee, and Shimozono, as well as joint work with William Fulton. These polynomials possess a striking and subtle positivity property. As I’ll explain, this positivity is an artifact of a new Kleiman-Bertini-type transversality theorem, applied to subvarieties of flag varieties.
March 10, 2025
Lisa Marquand :
3 p.m. in 636 SEO
Abstract
Cubic fourfolds have been classically studied up to birational
equivalence, with an eye towards rationality problems. On the
other hand, the Fano variety of lines F(X) on a cubic fourfold X
is a hyperkahler manifold, and the rationality/irrationality of
X is conjecturely reflected in the geometry of the Fano variety
of lines. We give examples of conjecturally irrational cubic
fourfolds with birationally equivalent Fano varieties of lines.
Two of our examples, which are special families in C_12, provide
new examples of pairs of cubic fourfolds with equivalent Kuznetsov
components. Further, we show the cubic fourfolds themselves are
birational. Our examples were discovered by studying the group of
birational transformations of the Fano varieties of lines of these
cubic fourfolds. This is joint work with Corey Brooke and Sarah Frei,
building on our previous work with Xuqiang Qin.
March 17, 2025
Laure Flapan :
3 p.m. in 636 SEO
Abstract
We study the cone of pseudoeffective divisors on moduli spaces of K3 surfaces. We give numerical criteria for when (the irreducible components of) a Noether-Lefschetz divisor on these moduli spaces is an extremal ray of the pseudoeffective cone and use this to exhibit many new extremal divisors. We also discuss the question of whether the pseudoeffective cone is generated by Noether-Lefschetz divisors.
March 31, 2025
Samuel Grushevsky :
3 p.m. in 712 SEO
Abstract
A stratum of differentials is the moduli space of curves together with a meromorphic form with prescribed multiplicities of zeroes and poles. The strata are phase spaces of an action of SL(2,R) and thus the central object of study in Teichmueller dynamics. On the other hand, they give natural high codimension subvarieties of the moduli of curves with marked points. The strata are non-compact, and we determine the number of their ends, and discuss a viewpoint towards further homology computations. This uses an algebraic compactification of the strata. Based on a joint work with Ben Dozier.
April 7, 2025
Nikon Kurnosov :
3 p.m. in 636 SEO
Abstract
In this pop-up talk I'll speak on some progress on study of
Riemann-Roch polynomials of hyperkahler manifolds. In particular,
I will try to gently introduce main objects needed for this, some
interesting conjectures around, and conditions whenever the second
Chern class belongs to Verbitsky component of LLV decomposition.
April 14, 2025
Tim Ryan :
3 p.m. in 636 SEO
Abstract
While the general surface of degree at least 4 in projective 3-space contains no lines, the maximum possible number of lines on any surface of degree at least 4 over a field k is a classical question dating back to at least Clebsch's work in 1861. When the degree is less than the (positive) characteristic and always in characteristic 0, the number of lines has an upper bound which is quadratic in the degree when the degree is at least 4. In contrast, once the degree is at least one more than the characteristic, it has long been known that there are surfaces with vastly more lines. In this talk, we answer this classical question over an arbitrary field. In particular, we prove that the maximum number of lines on any smooth surface of degree d over any field k is $d^4-3d^3+3d^2$ and show that, up to projective equivalence, a unique surface obtains this sharp upper bound in the infinitely many degrees and characteristics where it is obtained.
April 16, 2025
Ben Tighe :
3 p.m. in 636 SEO
Abstract
I will discuss some vanishing results for symplectic varieties which generalize the rational and Du Bois properties for singularities. These vanishing theorems arise from Hodge-theoretic symmetries on the Du Bois and intersection cohomology complex which generalize the extra symmetries you see on the Hodge diamond of a compact hyperkahler manifold. Along the way, I will discuss why these vanishing theorems are related to the existence of symplectic resolutions of singularities, new perspectives in deformation theory, and the LLV algebra.
April 21, 2025
Morgan Opie :
3 p.m. in 636 SEO
Abstract
Given a variety X over a field, it is generally difficult to understand
the structure of vector bundles on X. As a first approximation, we might
try to understand vector bundles only up to isomorphism. Classical
isomorphism invariants of algebraic vector bundles include Chern classes
and Euler classes, so we can study the extent to which these invariants
determine a vector bundle. The analogous question in topology is a finite
one: given a finite-dimensional manifold M, there are only finitely many
isomorphism classes of complex rank r topological vector bundles on M
with given topological Chern classes. However, such a finiteness result
is not, in general, known in algebraic geometry. In this talk, I will
discuss conditions under which algebraic characteristic classes
determine algebraic rank 2 vector bundles on a given smooth affine
fourfold up to finite choices. As a consequence, I will deduce complete
isomorphism classification results for certain examples. This is joint
work with Thomas Brazelton and Tariq Syed.
April 28, 2025
Yu-Shen Lin :
3 p.m. in 636 SEO
Abstract
The Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds
admit special Lagrangian fibrations and provides a recipe for the
mirror construction. In this talk, I will explain the existence of
special Lagrangian fibrations in certain log Calabi-Yau surfaces.
Moreover, with the suitable mirror map the special Lagrangian
fibrations on the mirror pairs are dual to each other. The study of
these special Lagrangian fibrations also accidentally proved the
Torelli theorems of certain types of gravitational instantons.
Moreover, these setup the foundation for the equivalence of certain
open Gromov-Witten invariants and log Gromov-Witten invariants.
Sept. 8, 2025
Maya Banks :
3 p.m. in 636 SEO
Abstract
Rational normal scrolls are ubiquitous objects in algebraic geometry. They constitute almost all of ``minimal degree varieties" in $\mathbb{P}^n$ and also have among the simplest minimal free resolutions. We'll discuss analogs of rational normal scrolls in weighted projective space and use them to explore the relationship between degree and syzygies for weighted projective varieties.
Sept. 15, 2025
Jack Huizenga :
3 p.m. in 636 SEO
Abstract
The Brill-Noether theory of curves plays a fundamental role in the
theory of curves and their moduli and has been intensively studied since
the 19th century. In contrast, Brill-Noether theory for vector bundles
and higher dimensional varieties is less understood. It is hard to
determine when Brill-Noether loci are nonempty and these loci can be
reducible and of larger than the expected dimension.
In this talk, we will study Brill-Noether loci for vector bundles on the
projective plane in the case where the number of sections is close to
the largest possible number. When the number of sections is very large,
Brill-Noether problems are all "trivial"--the Brill-Noether loci are
either empty or the entire moduli space. As the number of sections
decreases, we find that there is a "first" nontrivial Brill-Noether
locus, and we discuss its geometry.
Sept. 22, 2025
Philip Engel :
3 p.m. in 636 SEO
Abstract
We will discuss a proof that the integral Hodge conjecture is false
for a very general abelian variety of dimension ≥ 4. Associated to
any regular matroid is a degeneration of principally polarized abelian
varieties. We introduce a new combinatorial invariant of regular
matroids, which obstructs the algebraicity of the minimal curve class,
on the very general fiber of the associated degeneration. In concert
with a result of Voisin, one deduces (via the intermediate Jacobian)
the stable irrationality of a very general cubic threefold. This is
joint work with Olivier de Gaay Fortman, and Stefan Schreieder.
Sept. 29, 2025
Izzet Coskun :
3 p.m. in 636 SEO
Abstract
Given a class in the cohomology of a projective manifold, one can ask whether the class can be represented by an irreducible subvariety. If the class is represented by an irreducible subvariety, we say that the class is realizable. One can further ask whether the subvariety can be taken to satisfy additional properties such as smooth, nondegenerate, rational, etc. These questions are closely related to central problems in algebraic geometry such as the Hodge Conjecture or the Hartshorne Conjecture. Recently, June Huh and collaborators have made significant progress in understanding realizable classes in products of projective spaces. In this talk, I will give a survey of this circle of ideas and discuss recent joint work with Julius Ross on realizable classes in Grassmannians.
Oct. 6, 2025
Zhijia Zhang :
3 p.m. in 636 SEO
Abstract
The notion of G-varieties was introduced by Manin when he studied rationality problems of surfaces. Broadly speaking, a G-variety is a variety X carrying an action of a group G. The group can act via automorphisms of X or via Galois actions if the base field is non-closed. There are close connections, as well as drastic differences between these two types of actions from the perspective of birational geometry. In this talk, I will explore these similarities and differences with a focus on equivariant unirationality of Fano threefolds. This is joint work with Yuri Tschinkel and Ivan Cheltsov.
Oct. 13, 2025
Jakub Witaszek :
3 p.m. in 636 SEO
Abstract
I will review recent developments in the study of higher Du Bois and rational singularities in characteristic zero. Then I will discuss new results on inversion of adjunction for higher rational singularities joint with T. Kawakami.
Oct. 20, 2025
Jefferson Baudin :
3 p.m. in 636 SEO
Abstract
A useful vanishing theorem for understanding characteristic zero singularities is Grauert-Riemenschneider vanishing, which asserts that if f: Y -> X is a projective birational morphism and Y is smooth, then higher pushfowards of \omega_Y vanish. A remarkable consequence of this result is that characteristic zero klt singularities are rational. As one could expect, this vanishing theorem fails in positive characteristic. In this talk, we will explain how to prove a Witt vector version of Grauert-Riemenchneider vanishing, and consequences on the Witt-rationality of certain singularities in positive characteristic.
Oct. 27, 2025
Nathan Chen :
3 p.m. in 636 SEO
Abstract
In this talk, we will explore several invariants for curves on (very) general hypersurfaces
and complete intersections, which will have applications to measures of irrationality. This
is joint work with Ben Church and Junyan Zhao, and separately with David Yang.
Nov. 3, 2025
Eric Riedl :
3 p.m. in 636 SEO
Abstract
In a series of results dating back to Morin, it is shown that smooth hypersurfaces in a large number of variables are unirational. The basic technique shows an important relationship between the spaces of k-planes in these hypersurfaces and their unirationality. We investigate these questions using the notion of strength coming from commutative algebra. In particular, we prove that hypersurfaces having high secondary strength are also unirational, providing a new source of examples of (singular) unirational hypersurfaces. Along the way, we see that notions of strength allow for a very short proof of a weak form of the de Jong-Debarre conjecture. This is joint with Daniel Erman.
Nov. 10, 2025
Erik Paemurru :
3 p.m. in 636 SEO
Abstract
We generalize an intersection-theoretic local inequality
of Fulton–Lazarsfeld to weighted blowups. Using this together with the
classification of 3-dimensional divisorial contractions, we prove
nonrationality of many families of terminal Fano 3-folds. This is a
joint work with Igor Krylov and Takuzo Okada.
Nov. 17, 2025
Joel Castillo :
3 p.m. in 636 SEO
Abstract
The Watanabe–Yoshida conjecture states that the Hilbert–Kunz multiplicity attains its minimal value across singularities exactly at quadric hypersurfaces. It further claims that these are characterised by this property, but this part of the conjecture remained largely unaddressed in the literature.
We present an affirmative answer to this problem for complete intersections in every positive characteristic, improving a theorem by Enescu and Shimomoto, thus settling the conjecture for this family of singularities. The proof relies on advanced characteristic-dependent applications of a technique developed by Han and Monsky, and critically includes a explicit calculation needed to fill the gaps for the often-overlooked characteristic 2 case.
Jan. 12, 2026
Daniil Serebrennikov :
3 p.m. in 636 SEO
Abstract
The Kawamata–Morrison cone conjecture is a long-standing problem in
birational geometry. Totaro generalized the conjecture and proved it
for klt Calabi–Yau pairs in dimension two. The conjecture predicts
that such a pair has only finitely many birational contractions modulo
its automorphism group. I will explain that the finiteness of the
targets of these contractions follows once they admit polarizations
of bounded degree. In dimension two, this provides a new proof of the
generalized Kawamata–Matsuki conjecture on the finiteness (up to log
isomorphism) of weak log canonical models within a birational class.
Jan. 26, 2026
Feliks Raczka :
3 p.m. in 636 SEO
Abstract
The talk will be devoted to D-affinity of smooth projective varieties over fields of positive characteristic. First, I will recall the notion of a D-affine variety and justify its importance. Then, I will explain how in positive characteristic this notion relates to other properties defined in terms of the Frobenius morphism: the tilting property of Frobenius pushforwards of the structure sheaf, GFFRT, etc. In the last part of the talk I will present the results from my recent preprint https://arxiv.org/abs/2601.13340.
Feb. 2, 2026
Josh Pollitz :
3 p.m. in 636 SEO
Abstract
A semi-classical question of Avramov asks whether embedded deformations of a local ring correspond exactly to central elements in the homotopy Lie algebra of the ring. In this talk, I will explain the question and some recent insights. The latter is based on joint work with Briggs, Grifo, and Walker.
Feb. 9, 2026
Matthew Hase-Liu :
3 p.m. in 636 SEO
Abstract
The space of rational curves on a Fano variety X serves as a powerful tool for probing the geometry of X. Even for hypersurfaces, characterizing these spaces is difficult; however, work by Riedl–Yang established they are irreducible and have the expected dimension. In this talk, I will discuss another aspect, namely the singular locus. Specifically, I will show the singular locus of the moduli space of smooth degree e curves on a general low-degree hypersurface is small, i.e. has codimension growing linearly with e. This turns out to use a weird combination of 1. Lehmann–Riedl–Tanimoto's recent work on geometric Manin’s conjecture and 2. Sawin's work on Waring's problem from analytic number theory.
Feb. 16, 2026
Suchitra Pande :
3 p.m. in 636 SEO
Abstract
The F-signature is a numerical invariant of singularities in positive characteristic that measures asymptotic properties of the Frobenius map. While initially studied as an algebraic invariant of local rings, there has been recent interest in the geometric and global aspects of this theory. In previous work with Seungsu Lee, we studied the F-signature of a projective variety as a continuous function on the ample cone. In this talk, I will discuss continuation of our work where we extend the F-signature function to the big cone. The results include existence of the F-signature for big divisors, continuity and positivity of the F-signature on the big cone and transformation rules under birational contractions for big and semi-ample divisors. As a key tool, we also study similar properties for the Frobenius-alpha invariant. The geometric aspects of our techniques will be presented and emphasized.
Feb. 23, 2026
Younghan Bae :
3 p.m. in 636 SEO
Abstract
By Beauville, and Deninger-Murre, cycles on abelian schemes have a
multiplicative weight decomposition. Recent developments surrounding the
moduli space of Higgs bundles suggest that analogous properties may hold
for abelian fibrations with singular fibers.
In this talk, I will study the cohomological and Chow theoretic study of
fine compactified Jacobians. I will first show that there exist two fine
compactified Jacobians whose rational cohomology rings are not
isomorphic. To address this issue, we degenerate the ring structure via
the perverse filtration, and prove that the resulting ring is independent
of the choice of stability condition. This intrinsic ring structure
further lifts to the level of algebraic cycles. Finally I will present
explicit calculations using the Fourier transform and logarithmic
Abel-Jacobi theory.
This is a joint work with D. Maulik, J. Shen, Q. Yin; A. Pixton;
and S. Molcho and A. Pixton.
Feb. 27, 2026
Brad Dirks :
3 p.m. in 636 SEO
Abstract
Local cohomology is a fundamental tool in commutative algebra and algebraic geometry. Over the complex numbers, the local cohomology of a smooth variety along a subvariety admits an action by differential operators and has an associated Hodge and weight filtration (due to M. Saito). These filtrations contain important singularity information about the subvariety (as evidenced by Mustațǎ-Popa's Hodge ideals). Mustațǎ-Popa also showed that local cohomology can detect Du Bois and rational singularities. I will explain how this point of view gives a new perspective on inversion of adjunction for such singularities, as well as their higher analogues in the LCI setting, based on joint work with Qianyu Chen and Sebastián Olano.
March 2, 2026
Juliette Bruce :
3 p.m. in 636 SEO
Abstract
I will discuss how the asymptotic Castelnuovo-Mumford regularity for powers of ideals sheaves on a smooth projective toric variety is closely related to a convex region, called the Seshadri region, that measures the positivity along subvarieties.
March 9, 2026
Ben Church :
3 p.m. in 636 SEO
Abstract
A variety is "unirational" if it admits a dominant rational map from projective space. For moduli spaces this amounts to an explicit “recipe” for writing down a general member of the universal family. In characteristic zero, tensor forms obstruct unirationality -- famously employed by Harris--Mumford (1982) to prove that M_g is not unirational for g > 22. In positive characteristic, unirationality behaves much wilder due to the existence of inseparable maps. Consequently, we know the (non)-unirationality of few moduli spaces in positive characteristic. I will exhibit new techniques to obstruct unirationality in positive characteristic inspired by methods used to prove hyperbolicity in complex geometry. As applications, I will present a counterexample to a 1977 conjecture of Shioda regarding the unirationality of general type surfaces and prove that many Hilbert modular varieties over positive characteristics are not unirational.
March 16, 2026
Ruijie Yang :
3 p.m. in 636 SEO
Abstract
Given a polynomial, the Strong Monodromy Conjecture predicts a mysterious relationship between its p-adic zeta function and Bernstein-Sato polynomial. While the conjecture remains widely open in general, progress has been made for specific classes of polynomials. In 2009, Budur-Mustațǎ-Teitler introduced the n/d conjecture and showed that it would imply the Strong Monodromy Conjecture for all hyperplane arrangements.
In this talk, I will present a solution of the n/d conjecture, based on our new theory of multivariate V-filtration and a wall crossing theory for mixed Hodge modules. The latter is inspired by the recent breakthrough on the unitary dual problem of real Lie groups, by Davis-Vilonen. The talk is based on the upcoming work, joint with Dougal Davis.
March 18, 2026
Christian Schnell :
3 p.m. in 712 SEO
Abstract
My talk is about group actions on projective varieties.
Every algebraic group G over the complex numbers is an extension of an
abelian variety A by a linear algebraic group L (Chevalley's theorem).
The main result is that if G acts algebraically on a projective
variety X, then the cohomology of X is "free" over the cohomology of
A. The precise statement involves Hopf algebras and comodules. This is
joint work with Mark de Cataldo and Yoonjoo Kim.
March 30, 2026
Jack Jeffries :
3 p.m. in 612 SEO
Abstract
In this talk, we will discuss a general result about the regularity of the associated graded ring of localizations at the prime ideals of a fixed ring. We will then apply this result to give a simple proof of a result of Ballard, Iyengar, Lank, Mukhopadhyay, and Pollitz on generation of the derived category in positive characteristic. This is based on joint work with De Stefani, KC, and Núñez-Betancourt.
April 6, 2026
Ritvik Ramkumar :
3 p.m. in 636 SEO
Ritvik Ramkumar :
3 p.m. in 612 SEO
Abstract
The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will focus on the case where X is a threefold, as there are several open questions regarding its singularities. I will describe the structure of the smooth points of this Hilbert scheme and, time permitting, discuss the structure of the mildly singular points. This is all joint (ongoing) work with Joachim Jelisiejew and Alessio Sammartano.
April 13, 2026
Daebeom Choi :
3 p.m. in 636 SEO
Abstract
In this talk, we discuss the existence and nonexistence of certain birational contractions of \(\overline{\mathrm{M}}_{g,n}\). Somewhat surprisingly, this depends on the characteristic of the base field: many such contractions exist only in positive characteristic. We present a precise form of this phenomenon and discuss two examples that highlight the difference between characteristic zero and positive characteristic. The first is a simple and explicit contraction that exists only in positive characteristic, and the second is a modular interpretation of the morphisms associated with psi classes on \(\overline{\mathrm{M}}_{1,n}\). We also offer some speculation on why such characteristic-dependent phenomena arise.
April 27, 2026
Raluca Vlad :
3 p.m. in 636 SEO
Abstract
A locally symmetric variety is a non-compact complex algebraic variety obtained as the quotient of a Hermitian symmetric domain by the action of an arithmetic group. I will start by reviewing the theory of toroidal compactifications of these varieties, originally due to Ash-Mumford-Rapoport-Tai. Building on this construction, we define the tropicalization of a locally symmetric variety to be a combinatorial (polyhedral) object encoding the boundary strata of a toroidal compactification of the variety. I will discuss applications of this theory to the cohomology of moduli spaces and arithmetic groups, with an emphasis on the case of moduli of abelian varieties and general linear groups. Based on joint work with Assaf, Brandt, Bruce, and Chan.
April 29, 2026
Austyn Simpson :
3 p.m. in 1227 SEO
Abstract
Given a local ring R of prime characteristic and a nonzero divisor x such that R/xR is F-injective, it is a longstanding open problem to determine whether R must itself be F-injective. There are partial affirmative results that rely on R/xR being cohomologically full, while examples of F-injective rings which lack this property are very sparse. In this talk I will describe a family of such examples which are geometrically normal over an F-finite field, and discuss potential implications for the deformation problem. I will also highlight a feature of these rings which distinguishes them from Du Bois singularities in characteristic zero. Joint with A. De Stefani and T. Polstra.