Graduate Algebraic Geometry Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Sept. 3, 2008
Benjamin Antieau :
3 p.m. in SEO 1227
Sept. 10, 2008
Tuan Pham :
3 p.m. in SEO 1227
Sept. 17, 2008
Matt Wechter :
3 p.m. in SEO 1227
Sept. 24, 2008
Luigi Lombardi :
3 p.m. in SEO 1227
Oct. 1, 2008
Wenbo Niu :
3 p.m. in SEO 1227
Oct. 8, 2008
Rong Du :
3 p.m. in SEO 1227
Oct. 15, 2008
Richard Abdelkerim :
3 p.m. in SEO 1227
Nov. 5, 2008
Roi Docampo Alvarez :
3 p.m. in SEO 1227
Nov. 12, 2008
Fei Ye :
3 p.m. in SEO 1227
Nov. 19, 2008
Tuan Pham :
3 p.m. in SEO 1227
Dec. 3, 2008
Benjamin Antieau :
3 p.m. in SEO 1227
Jan. 20, 2009
Wenbo Niu :
3:30 p.m. in SEO 636
Jan. 27, 2009
Paul Reschke :
3:30 p.m. in SEO 636
Feb. 3, 2009
Matthew Wechter :
3:30 p.m. in SEO 636
Feb. 10, 2009
Wenbo Niu :
3:30 p.m. in SEO 636
Feb. 17, 2009
Chih-Chi Chou :
3:30 p.m. in SEO 636
Feb. 24, 2009
Robert Krzyzanowski :
3:30 p.m. in SEO 636
March 3, 2009
Tuan Pham :
3:30 p.m. in SEO 636
March 10, 2009
Richard Abdelkerim :
3:30 p.m. in SEO 636
March 17, 2009
Tuan Pham :
3:30 p.m. in SEO 636
March 31, 2009
Luigi Lombardi :
3:30 p.m. in SEO 636
April 7, 2009
Tuan Pham :
3:30 p.m. in SEO 636
April 14, 2009
Fei Ye :
3:30 p.m. in SEO 636
April 21, 2009
Robert Krzyzanowski :
3:30 p.m. in SEO 636
April 28, 2009
Robert Krzyzanowski and Fei Ye :
3:30 p.m. in SEO 636
July 9, 2009
Fei YE :
2 p.m. in SEO 712
July 16, 2009
Chih-Chi Chou :
2 p.m. in SEO 712
July 23, 2009
Luigi Lombardi :
2 p.m. in SEO 712
Aug. 6, 2009
Tuan Pham :
2 p.m. in SEO 712
Aug. 31, 2009
Fei YE :
4 p.m. in SEO 427
Sept. 14, 2009
David Benjamin Antieau :
4 p.m. in SEO 427
Sept. 16, 2009
Manuel Gonzalez :
4:30 p.m. in SEO 427
Sept. 21, 2009
Tuan Pham :
4 p.m. in SEO 427
Sept. 23, 2009
Manuel Gonzalez :
4:30 p.m. in SEO 712
Sept. 28, 2009
Richard Abdelkerim :
4 p.m. in SEO 427
Sept. 30, 2009
Manuel Gonzalez :
4:30 p.m. in SEO 712
Oct. 5, 2009
Chih-Chi Chou :
4 p.m. in SEO 427
Oct. 7, 2009
Manuel Gonzalez :
4:30 p.m. in SEO 712
Oct. 12, 2009
Luigi Lombardi :
4 p.m. in SEO 427
Oct. 14, 2009
Manuel Gonzalez :
4:30 p.m. in SEO 712
Oct. 19, 2009
Matthew Wechter :
4 p.m. in SEO 427
Oct. 26, 2009
Wenbo Niu :
4 p.m. in SEO 427
Nov. 2, 2009
Wenbo NIU :
4 p.m. in SEO 427
Nov. 4, 2009
Manuel Gonzalez :
4:30 p.m. in SEO 712
Nov. 9, 2009
Luigi Lombardi :
4 p.m. in SEO 427
Nov. 16, 2009
Fei YE :
4 p.m. in SEO 427
Nov. 23, 2009
Richard Abdelkerim :
4 p.m. in SEO 427
Nov. 30, 2009
Fei YE :
4 p.m. in SEO 427
Jan. 26, 2010
Luigi Lombardi :
4 p.m. in SEO 712
Feb. 2, 2010
Luigi Lombardi :
4 p.m. in SEO 712
Feb. 9, 2010
Matthew Wechter :
4 p.m. in SEO 712
Feb. 16, 2010
Matthew Wechter :
4 p.m. in SEO 712
Feb. 23, 2010
Richard Abdelkerim :
4:30 p.m. in SEO 712
March 2, 2010
Robert Krzyzanowski :
4 p.m. in SEO 712
March 9, 2010
Chih-Chi Chou :
4 p.m. in SEO 712
March 16, 2010
Wenbo Niu :
4 p.m. in SEO 712
March 30, 2010
Fei Ye :
4 p.m. in SEO 712
April 6, 2010
Fei Ye :
4 p.m. in SEO 712
April 13, 2010
Tuan Pham :
4 p.m. in SEO 712
April 20, 2010
Tuan Pham :
4 p.m. in SEO 712
April 27, 2010
Wenbo Niu :
4 p.m. in SEO 712
July 6, 2010
Richard Abdelkerim :
1 p.m. in SEO 612
Abstract
We will discuss the formal functorial definitions of moduli spaces, including when a moduli space is fine or coarse, and provide examples.
July 13, 2010
Lei Song :
1 p.m. in SEO 612
Abstract
We will give the construction of the Grassmannian and do some preliminary calculations with its cohomology classes.
July 20, 2010
Wenbo Niu :
1 p.m. in SEO 612
Abstract
We will discuss some important results and formulas relating to the multiplication of cohomology classes of the Grassmannian.
July 27, 2010
Matt Wechter :
1 p.m. in SEO 612
Abstract
We will construct tangent spaces to Grassmannians and relate the construction of the Grassmannian to the formal definition of moduli space.
Aug. 3, 2010
Matt Wechter :
1 p.m. in SEO 612
Abstract
We will construct tangent spaces to Grassmannians and relate the construction of the Grassmannian to the formal definition of moduli space.
Aug. 10, 2010
Richard Abdelkerim :
1 p.m. in SEO 612
Abstract
We will give an algorithm using Young Tableaux to apply the Littlewood-Richardson rule, which allows one to calculate the product of any two Schubert classes in the cohomology ring of the Grassmannian.
Aug. 24, 2010
Richard Abdelkerim :
3 p.m. in SEO 636
Aug. 31, 2010
Cesar Lozano :
3 p.m. in SEO 1227
Sept. 8, 2010
Tuan Pham :
4 p.m. in SEO 712
Sept. 15, 2010
Cesar Lozano :
4 p.m. in SEO 712
Sept. 22, 2010
Xudong Zheng :
4 p.m. in SEO 712
Sept. 29, 2010
Fei Ye :
4 p.m. in SEO 712
Oct. 6, 2010
Lei Song :
4 p.m. in SEO 712
Oct. 13, 2010
Chih-Chi Chou :
4 p.m. in SEO 712
Oct. 20, 2010
---- :
4 p.m. in SEO 712
Oct. 27, 2010
Matt Wechter :
4 p.m. in SEO 712
Nov. 3, 2010
Luigi Lombardi :
4 p.m. in SEO 712
Nov. 10, 2010
Wenbo Niu :
4 p.m. in SEO 712
Nov. 17, 2010
Luigi Lombardi :
4 p.m. in SEO 712
Nov. 24, 2010
Cesar Lozano Huerta :
4 p.m. in SEO 712
Dec. 1, 2010
Fei Ye :
4 p.m. in SEO 712
Jan. 27, 2011
Cesar Lozano Huerta :
3:30 p.m. in SEO 712
Abstract
We'll review some constructions of toric varieties using Geometric Invariant Theory (GIT) and then we will look at the construction of the moduli space of Riemann surfaces. This will all be done paying close attention to the concept of "linearization".
Feb. 3, 2011
Cesar Lozano Huerta :
3:30 p.m. in SEO 712
Abstract
We will start constructing moduli spaces as GIT quotients. However, instead of going straight to the general case of the moduli space of Riemann Surfaces of genus g $M_g$, we will take a closer look at genus 2, 3 and 4. First, in constructing these spaces, by performing a "naive" GIT quotient, we will get a "wrong" space, it wont be $M_g$ itself, but which turns out to be a birrational model for $M_g$.
This gives us an example of the birrational geometry of $M_g$ and points out a subtle ingredient of Mumford's construction of $M_g$ as a GIT quotient.
Feb. 10, 2011
Cesar Lozano Huerta :
3:30 p.m. in SEO 712
Abstract
We will start constructing moduli spaces as GIT quotients. However, instead of going straight to the general case of the moduli space of Riemann Surfaces of genus g $M_g$, we will take a closer look at genus 2, 3 and 4. First, in constructing these spaces, by performing a "naive" GIT quotient, we will get a "wrong" space, it wont be $M_g$ itself, but which turns out to be a birrational model for $M_g$.
This gives us an example of the birrational geometry of $M_g$ and points out a subtle ingredient of Mumford's construction of $M_g$ as a GIT quotient.
Feb. 17, 2011
Rita Jimenez Rolland :
3:30 p.m. in SEO 712
Abstract
In this talk we will explore the relation between the structure of the mapping class group Mod_g (i.e. the group of homotopy classes of orientation preserving diffeomorphisms of a compact orientable surface of genus g) and the topology of M_g, the moduli space of smooth projective curves of genus g. We will describe the the moduli space M_g as a quotient of Teichmuller space by a natural action of Mod_g, and how this gives us a "rational" classifying space for the discrete group Mod_g.
Feb. 24, 2011
Luigi Lombardi :
3:30 p.m. in SEO 712
March 3, 2011
Chih-Chi Chou :
3:30 p.m. in SEO 712
March 17, 2011
Lei Song :
3:30 p.m. in SEO 712
March 31, 2011
Matt Wechter :
3:30 p.m. in SEO 712
April 7, 2011
Cesar Lozano Huerta :
3:30 p.m. in SEO 712
Abstract
We will count rational curves on algebraic surfaces (4 manifolds) using elementary methods.
Then, we'll contrast these numbers with those one can get through the formalism of quantum cohomology,
This talk will be very concrete, for instance we'll be computing numbers of the following type: there are 620 rational quartics on $CP^2$ passing through 11 fixed points.
April 21, 2011
Luigi Lombardi :
3:30 p.m. in SEO 712
Aug. 23, 2011
:
4 p.m. in SEO 712
Sept. 1, 2011
Robert Krzyzanowski :
4 p.m. in SEO 712
Abstract
We will introduce toric varieties, normal varieties that contain an algebraic torus as a dense open subset such that the action of the torus on itself extends naturally to the full variety. We will start with the standard construction given by
piecing together strongly convex rational polyhedral cones.
Sept. 22, 2011
Joel Sievert :
3 p.m. in SEO 712
Oct. 6, 2011
Cesar Lozano Huerta :
3:30 p.m. in SEO 712
Oct. 13, 2011
Cesar Lozano Huerta :
3:30 p.m. in SEO 712
Abstract
We will explore the the torus action and the quotient space it induces.
Oct. 20, 2011
Matthew Wechter :
3:30 p.m. in SEO 712
Nov. 10, 2011
Xudong Zheng :
3:30 p.m. in SEO 712
Dec. 1, 2011
Matt Wechter :
3:30 p.m. in SEO 712
Jan. 24, 2012
Xudong Zheng :
3 p.m. in SEO 712
Abstract
In this talk I will first characterize nef divisors of a toric variety whose fan has convex support of full dimension. The toric cone theorem is presented.
Feb. 14, 2012
Joel Sievert :
3 p.m. in SEO 712
Abstract
We will look at the conditions under which a cone gives rise to a nonsingular affine toric variety, and then describe how one can subdivide a given fan into such cones, giving rise to a nonsingular toric variety with a proper birational map to the variety given by the original fan.
Feb. 28, 2012
Matt Wechter :
3 p.m. in SEO 712
Abstract
I will attempt to do an explicit example or two of a resolution of singularities for a toric variety and then discuss the theory of polytopes for toric varieties.
Sept. 27, 2012
Xudong Zheng :
3 p.m. in SEO 712
Abstract
Du Val singularities have been studied since antiquity, and there are many ways of characterizing them. They appear throughout the classification of surfaces, and in many other areas of algebraic geometry, singularity theory and group theory. I will give an introduction continuing Joel's talk last week.
Oct. 4, 2012
Lei Song :
3 p.m. in SEO 712
Abstract
I will define elliptic singularities for a normal surface, and discuss a classification theorem on it, due to Laufer and Reid in 1970s. Cohomological techniques will be explored.
Oct. 11, 2012
César Lozano Huerta :
3 p.m. in SEO 612
Abstract
This talk will be divided into two parts: 20min devoted to background and 30 minutes of discussion on families curves.
Oct. 18, 2012
César Lozano Huerta :
3 p.m. in SEO 712
Abstract
I will continue with explicit examples of curves on various surfaces. Towards the end I will compute the dimension of the families of the curves we encountered and this will in particular give us a proof that the Hilbert Scheme of genus 10 and degree 9 curves in $\mathbb{P}^3$ is reducible.
Dec. 6, 2012
Lei Song :
3 p.m. in SEO 712
Abstract
I will follow the titled Kollár's paper. Given a log canonical pair $(X, \Delta)$, we can define the notion of log canonical center (LC center).
Kawamata's subadjunction says by restricting the pair $(X, \Delta+\epsilon H)$, for small $\epsilon>0$ and ample $H$, to an exceptional LC center $W$, we can get a klt pair $(W, \Delta_W)$. Kodaira-type canonical bundle formula plays an essential role in the proof. The talk will explain how to choose moduli and boundary divisors in the kind of formula in a natural way.
Jan. 17, 2013
Matt Wechter :
3 p.m. in SEO 512
March 21, 2013
Cesar Lozano Huerta :
4 p.m. in SEO 712
Abstract
In the talk I'll go over particular examples of contractions and flips that arise in the study of the Hilbert scheme of curves in P3.
Aug. 29, 2013
N/A :
3:30 p.m. in SEO 712
Abstract
This will be an organizational meeting to decide what topics we would like to discuss as well as what time we would like to meet.
Sept. 5, 2013
Xudong Zheng :
3:30 p.m. in SEO 712
Abstract
A Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general
projective scheme). The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. I will
formulate the construction problem of the Hilbert scheme and outline the idea of the construction following D. Mumford's "<i>
Lectures on Curves on an Algebraic Surface</i>".
Sept. 19, 2013
Tim Ryan :
3:30 p.m. in SEO 712
Abstract
Taking a step back from the abstract construction of the Hilbert scheme and regularity, we will discuss a concrete example. That is the Hilbert scheme of twisted cubics in projective space.
Sept. 26, 2013
Lei Song :
3:30 p.m. in SEO 712
Abstract
Deformation theory is an important tool for the study of moduli spaces. I will talk about the notions of first order deformation, higher-order deformations and obstruction groups involved. As an application, I will show that the Hilbert scheme of n-points on a smooth surface is smooth.
Oct. 3, 2013
Joe Berner :
3:30 p.m. in SEO 712
Abstract
We will give an overview of the construction of the Picard scheme.
Oct. 10, 2013
Alexander Stathis :
3:30 p.m. in SEO 712
Abstract
Let $X$ be a smooth projective surface, and fix a positive integer $n$. In this talk, we will follow the work of
Fogarty in "Algebraic Families on an Algebraic Surface" to show that ${Hilb}^n(X) := X^{[n]}$, the Hilbert scheme
parameterizing zero dimensional subschemes $Z \subset X$ of length $n$, is connected and smooth, and that the
Hilbert-Chow morphism $h : X^{[n]} \to X^{(n)}$, where $X^{(n)}$ is the $n$-th symmetric product of $X$, is birational.
Oct. 17, 2013
Alexander Stathis :
3:30 p.m. in SEO 712
Abstract
Let $X$ be a smooth projective surface, and fix a positive integer $n$. In this talk, we will follow the work of Fogarty in "Algebraic Families on an Algebraic Surface" to
show that ${Hilb}^n(X) := X^{[n]}$, the Hilbert scheme parameterizing zero dimensional subschemes $Z \subset X$ of length $n$, is connected and smooth, and that
the Hilbert-Chow morphism $h : X^{[n]} \to X^{(n)}$, where $X^{(n)}$ is the $n$-th symmetric product of $X$, is birational.
We will begin this discussion where we left off last time.
Nov. 14, 2013
Yajnaseni Dutta :
3:30 p.m. in SEO 712
Abstract
We will prove the existence of moduli space of elliptic curves and show that it is not a fine moduli. Assuming the existence of
the moduli space of curves $\mathcal{M}_g$ ($g\geq 2$) we will calculate it's dimension using first order deformation and check it
for lower genus curves.
Jan. 27, 2014
Tabes Bridges :
4 p.m. in SEO 712
Abstract
This week we begin our study of positivity with a review of the classical notions of divisors and linear series and the problem of embedding varieties in projective space, as well as the more modern language of line bundles. After discussing intersection products and some basic facts about numerical equivalence of divisors, we will state asymptotic Riemann-Roch for divisors and coherent sheaves. If time remains, I may say a few first words about ampleness.
Feb. 24, 2014
Xudong Zheng :
4 p.m. in SEO 712
Abstract
This talk continues on Section 1.4 of <i>Positivity, vol I</i>. Specifically, we shall discuss Kleiman's numerical criterion of ampleness of $\mathbb{R}$-divisors in terms of cones in the Néron-Severi space and its dual space of numerical classes of one cycles of the variety.
In the second part, we introduce Fujita's vanishing theorem and some applications. In particular, we show that the numerically trivial line bundles form a bounded family for a projective variety.
April 14, 2014
Xudong Zheng :
4 p.m. in SEO 712
Abstract
This talk will cover the section in <i>Positivity</i> on several inequalities on an irreducible complete variety of Hodge type, following the work of Teissier, Khovanskii-Matsusaka-Teissier, Demailly, Beltrametti-Sommese, Ein-Fulton. The global version generalizes the Hodge index theorem, and the local version applies to mixed multiplicities.
Aug. 27, 2014
Alex Stathis :
2 p.m. in SEO 427
Abstract
We'll begin by recalling the basic properties of the Hilbert scheme of points in the projective plane, and then discuss its Picard group and its generators. We'll then define Brill-Noether divisors and use these to compute some top intersection numbers of the generators. We'll also briefly discuss volumes of divisors and their relation to top intersection numbers. If we have time, we'll discuss some of interesting geometric observations we can make using these numbers.
Sept. 3, 2014
Xudong Zheng :
2 p.m. in SEO 427
Abstract
Let $X$ be a smooth projective surface and $L$ a big and nef line bundle. The pluricanonical $|nK_X|$ and adjoint linear systems $|K_X + L|$ have been well studied. The aim of this talk is to introduce the circle of ideas of Bombieri, Mumford, Van de Ven, Sommese, Reider, among others on the positivity and global generation of $|K_X + L|$ in the case that the base field has characteristic zero.
Sept. 10, 2014
Xudong Zheng :
2 p.m. in SEO 712
Abstract
This is the second part of the talk on adjoint linear systems on surfaces, in characteristic $p > 0$. In circumventing the classical vanishing theorems and Bogomolov's inequality on rank 2 vector bundles in characteristic zero, we briefly introduce the construction of Ekedahl and Shepherd-Barron on purely inseparable degree $p$ coverings.
Sept. 17, 2014
Chris Perez :
2 p.m. in SEO 712
Abstract
Chris will give an introduction to tropical geometry and tropical curves.
Sept. 24, 2014
Seckin Adali :
2 p.m. in SEO 712
Abstract
Schubert varieties have well understood singularities, in particular an explicit description can be given using a number of different approaches. In this talk I will give a description of the singular loci of Schubert varieties in Grassmannians using two different arguments: One uses the Bott-Samelson resolution, the other gives a description of their tangent spaces.
Oct. 1, 2014
Joe Berner :
2 p.m. in SEO 712
Abstract
A brief and incomplete history of topological methods in algebraic geometry will be presented, and then we will build up some of the basic definitions of the étale site leading to loosely defining the étale homotopy type of a scheme. If time permits we'll give some examples of étale homotopy types an application.
Oct. 8, 2014
Joe Berner :
2 p.m. in SEO 712
Abstract
We'll take a more detailed look at the etale homotopy type and examine some basic examples, theorems, and applications.
Oct. 15, 2014
Tabes Bridges :
2 p.m. in SEO 712
Abstract
The moduli space of smooth algebraic curves (a.k.a. compact Riemann surfaces) $M_g$ and its variants are objects of central interest in algebraic geometry, topology, combinatorics, and mathematical physics. I will describe the basic topological structure (in particular, Riemann's famous parameter count), state some of what is known (e.g. concerning the birational geometry of such spaces), and attempt to give a brief taste of the divisor class theory on $\overline{M}_{g,n}$, the moduli space of stable genus-$g$ curves with $n$ marked points.
Oct. 22, 2014
Daniel McLaury :
2 p.m. in SEO 712
Abstract
I will present a degeneration-based proof of the Noether-Lefschetz theorem by Griffiths and Harris.
Oct. 29, 2014
Yang Shuhang :
2 p.m. in SEO 712
Abstract
I will talk about basics of elliptic surfaces. An elliptic surface is, by definition, an elliptic fibration over an algebraic curve. We can regard it as an elliptic curve over the function field of the curve. I will talk about the singular fibres, especially Kodaira's classification of singular fibres. Also we will see that algebraic equivalence and numerical equivalence are the same for elliptic surfaces with a section.
Nov. 5, 2014
Sam Shideler :
2 p.m. in SEO 712
Abstract
I will begin with an introduction to Frobenius splitting in the local algebraic spirit of Hochster and Roberts. That will include introductions to F-regularity and the F-signature. I will continue with an introduction to the global properties of F-splitting in the spirit of Mehta and Ramanathan.
Nov. 12, 2014
Tim Ryan :
2 p.m. in SEO 712
Abstract
I will give a very basic introduction to elements of birational geometry. All grad students are welcome; a basic knowledge of divisors on varieties is the only prerequisite(come talk to me if that is an issue).
I will define various cones in the Neron-Severi space including why we care about these cones. Next I will describe some current methods of computing these cones. I will finish with a few examples of these methods, focusing on the Hilbert schemes of points on $\mathbb {P}^2$ and, time permitting, $\mathbb {P}^1 \times \mathbb {P}^1$.
Dec. 3, 2014
Janet Page :
2 p.m. in SEO 712
Abstract
We'll start by introducing Betti diagrams for finitely generated graded modules over a polynomial ring, and talking about their relation to the Hilbert function. In the process, we'll define Castelnuovo-Mumford regularity. We'll conclude our discussion by showing that the Betti diagram can give even more geometry than the hilbert function, by working out the example of 7 points in $\mathbb{P}^3$.
Jan. 21, 2015
Alex Stathis :
3 p.m. in SEO 712
Abstract
The volume of a line bundle on a smooth projective variety is an asymptotic birational invariant which is closely related to classical birational invariants. We’ll define and discuss the volume following the exposition of Ein, Lazarsfeld, Mustata, Nakayame, and Popa in “Asymptotic Invariants of Line Bundles” (ELMNP). I’ll attempt to fill in the details left out by ELMNP and work some examples fully. An alternative reference is Lazarsfeld’s “Positivity in Algebraic Geometry: Volume 1”.
Jan. 28, 2015
Alex Stathis :
3 p.m. in SEO 712
Abstract
We will recall the definitions from last time, and then compute the effective and nef cones of the blowup of the projective plane at a point as well as the volume function on this space. We'll then define a Mori Dream Space and informally discuss its relation to the volume function.
Feb. 4, 2015
Xudong Zheng :
3 p.m. in SEO 712
Abstract
A singularity in characteristic zero is said to be of <i>dense F-pure type </i> if its modulo $p$ reduction is locally Frobenius split for infinitely many $p$. In this talk I will present that if $x \in X$ is an isolated log canonical singularity with certain additional condition, then it is of dense $F$-pure type.
Feb. 11, 2015
Tim Ryan :
3 p.m. in SEO 712
Abstract
I will define deformations in increasingly general settings starting with deformations of closed subschemes following Hartshornes's book on deformation theory. In order to do so, I will define the $T^i$ functors. I will finish with a discussion of the infinitesimal lifting property and what the vanishing of the $T^i$ tells us.
Feb. 18, 2015
Sam Shideler :
3 p.m. in SEO 712
Abstract
I will define local cohomology modules and discuss some basic properties.
Feb. 25, 2015
Janet Page :
3 p.m. in SEO 712
Abstract
This week we'll continue our discussion of local cohomology. In particular, we'll look at its connection to sheaf cohomology and give an alternate charactization as cohomology with supports. Towards the end of the lecture, we'll look at some examples and applications.
March 11, 2015
Daniel McLaury :
3 p.m. in SEO 712
March 18, 2015
Tabes Bridges :
3 p.m. in SEO 712
Abstract
I will begin by briefly recalling the basic objects associated with the moduli spaces of curves, then present an overview of some things that are currently known about the birational geometry and the topology of these spaces. To give a feel for the study of these spaces, I will conclude with some intersection-theoretic calculations on $\overline{M}_g$.
April 1, 2015
Lei Wu :
3 p.m. in SEO 712
Abstract
Morihiko Saito's Hodge module is a beautiful theory generalizing hodge
structure and variation of hodge structure. We will start with recalling
terminologies of D-modules involved in Hodge modules. Then I will give a
very brief introduction of definitions of pure Hodge modules and mixed
Hodge modules. At last, I will give some applications of Hodge modules
like Kodaira-Saito's vanishing and Kollar's conjecture involving the
higher direct images of dualizing sheaves.
April 15, 2015
Shuhang Yang :
3 p.m. in SEO 712
Abstract
I will talk about the classification of singular fibres of elliptic fibration over an algebraic curve, then introduce the Weierstrass fibration. We will define the fundamental line bundle of a Weierstrass fibration. We will see that Weierstrass fibration can be regarded as a divisor of a rank 2 vector bundle over the base curve, and a double cover of a ruled surface with branch locus over which singular fibres appears. We will also calculate the invariants and Hodge numbers of Weiestrass fibration.
April 22, 2015
Seckin Adali :
3 p.m. in SEO 712
Abstract
This will be an introductory talk aimed at explaining the combinatorial characterization of which Schubert varieties are Gorenstein given in the article with the same title by A. Woo and A. Yong. We will look at the preliminary definitions needed to understand the characterization. The proof is based on an approach using the global geometry of Schubert varieties instead of local ring structure; we will sketch the main ideas omitting most of the details.
Aug. 26, 2015
Janet Page/Sam Shideler :
3 p.m. in SEO 712
Sept. 2, 2015
Xudong Zheng :
3 p.m. in SEO 712
Abstract
In this talk, I will present some constructions of algebraic surfaces in characteristic $p$ which violate the Kodaira vanishing theorem. These surfaces also provide counterexamples to Fujita's positivity theorem for fibered surfaces. Further pathologies on these examples include the existence of non-closed global differential 1-forms and global vector fields, which are both known to be impossible in characteristic zero. I will survey the comparison of characteristic zero with characteristic $p$, sketch the construction, and mention how I encountered these results by noticing errors in a recent published paper.
Sept. 9, 2015
Sam Shideler :
3 p.m. in SEO 712
Abstract
I will discuss two characteristic $p$ invariants of a ring $R$ - the Hilbert-Kunz multiplicity $e_{HK}(R)$ and the F-signature $s(R)$. Both are invariants defined using Frobenius that give information about the singularities of $R$.
I will prove basic facts about these invariants, discuss their computations, and give a broad overview of known results and further questions.
The talk will primarily follow the survey article by Huneke with the same title.
Sept. 16, 2015
Tabes Bridges :
3 p.m. in SEO 712
Abstract
I will review the basics of the geometry of the moduli space of stable curves $\overline{M}_g$, in particular its Picard group and its relation to the Picard group of the moduli stack $\overline{\mathcal{M}}_g$. I will then briefly discuss some classical results on the birational geometry for small $g$ before sketching the proof of the Eisenbud-Harris-Mumford Thereom that $M_g$ is of general type for $g\geq 23$. Along the way we will explore techniques for protecting oneself when "general" arguments become too much for some people.
Sept. 23, 2015
Alexander Stathis :
3 p.m. in SEO 712
Abstract
The Grassmannian $G(k,n)$ of $k$-planes in an $n$-dimensional vector space $V$ has a basis for the
Chow ring given by the classes of the closures of "Schubert cells". These Schubert cells are indexed and defined
by Young tableaux with no more than $k$ rows and $n-k$ boxes per row. Furthermore, the intersection product
can be computed using a set of combinatorial identities involving the Young tableaux. This is known as
Schubert calculus. I'll talk about my recent work in attempting to provide an analog of Schubert calculus
for the Hilbert scheme of $n$ points in the projective plane $\mathbb{P}^2$. I'll describe the combinatorial
basis and show a method for computing intersection products of cycles with complementary codimension.
Sept. 30, 2015
Tim Ryan :
3 p.m. in SEO 712
Abstract
We will introduce moduli spaces of sheaves starting with the history of their construction. We will then specialize to the case of moduli spaces of sheaves on $\mathbb{P}^1 \times \mathbb{P}^1$. In this setting, we will discuss properties of the moduli spaces including nonemptiness. Furthermore, we will consider results showing they are $\mathbb{Q}$-factorial and Mori dream spaces.
Being MDS's, we are interested in their effective cones so we then will give the general strategy for computing effective cones. We will apply these methods in this setting. Time permitting, we will outline the exact method for computing the effective cones of these moduli spaces.
Oct. 14, 2015
Daniel McLaury :
3 p.m. in SEO 712
Abstract
Oct. 21, 2015
Seckin Adali :
3 p.m. in SEO 712
Abstract
Research on the Kodaira dimension of the moduli space of curves has motivated an extensive study of the cone of effective divisors. Higher codimension cycles are relatively unexplored. This talk is designed as an introduction to extremal higher codimension cycles on the moduli space of curves. We will review the result of Faber on the generators of A$^2(\overline{\mathcal{M}}_3)$ and study the result of Chen and Coskun on the extremal divisors in Eff$^2(\overline{\mathcal{M}}_3)$.
Oct. 28, 2015
Tabes Bridges :
3 p.m. in SEO 712
Abstract
In my talk a few weeks ago, I sketched the proof that $\mathcal{M}_g$ is of general type (i.e. maximal Kodaira dimension) modulo the major non-formal input, the computation of the class of the Brill-Noether divisor (which, by comparing, allows us to conclude that the canonical class is big). My goal today is to give a reasonably detailed overview of this calculation, and in doing so demonstrate many of the standard techniques for studying $\mathcal{M}_g$.
Nov. 4, 2015
Shuhang Yang :
3 p.m. in SEO 712
Abstract
All the types singular fibres of a minimal elliptic surface are identified.
In Andrea Cattaneo's paper he tries to describe some types of singular fibres of a ellptic threefold.
For an elliptic threefold, it is a resolution of its Weierstrass model. If the resolution is crepant, then
its singular fibre can be identified as contraction of Kodaira fibre.
Nov. 11, 2015
Jay Kopper :
3 p.m. in SEO 712
Nov. 18, 2015
Janet Page :
3 p.m. in SEO 712
Dec. 2, 2015
Tim Ryan :
3 p.m. in SEO 712
Abstract
Assuming a working knowledge of category theory and scheme theory, we will introduce stacks.
First, we will motivate them from the perspective of moduli theory.
We will then see how they are a natural generalization of sheaves using the definition of a stack as a functor.
Following this, we will redefine stacks in the more standard way (as categories) and will proceed to define Deligne-Mumford and Artin stack.
Finally, we will conclude with a couple brief examples.
Jan. 25, 2016
Janet Page :
4 p.m. in SEO 712
Feb. 1, 2016
Janet Page :
4 p.m. in SEO 712
Feb. 8, 2016
Jeremy Kun :
4 p.m. in SEO 712
Abstract
I will discuss the relationship between algebraic geometry and computational complexity, give a short outline of Mulmuley's Geometric Complexity Program, and illustrate the ideas specifically for the determinant versus permanent problem.
Feb. 22, 2016
Sam Shideler :
4 p.m. in SEO 712
Abstract
Seshadri constants are invariants measuring the "local positivity" of a line bundle on a projective variety. I'll start with some motivation, give several equivalent definitions, and discuss results about bounds on Seshadri constants. The main reference is Lazarsfeld's Positivity in Algebraic Geometry I, Chapter 5.
Feb. 29, 2016
Seckin Adali :
4 p.m. in SEO 712
Abstract
Restriction varieties in the orthogonal Grassmannian are subvarieties of $OG(k, n)$ defined by rank conditions given by a flag that is not necessarily isotropic with respect to the relevant symmetric bilinear form. In particular orthogonal Schubert varieties are examples of restriction varieties. In this talk, I will describe a resolution of singularities for restriction varieties and give a smoothness criterion. The image of the exceptional locus of this resolution gives the singular locus of a restriction variety which allows an algorithm for the singular locus. I will illustrate these results with examples.
March 7, 2016
Tim Ryan :
4 p.m. in SEO 712
Abstract
We will discuss the reconstruction theorems of Gabriel, Bondal-Orlov, and Palmer. We also discuss examples due to Bridgeland, Căldăraru, and Lesieutre showing the necessity of the hypotheses of the Bondal-Orlov reconstruction theorem.
March 28, 2016
Joe Berner :
4 p.m. in SEO 712
April 4, 2016
Jay Kopper :
4 p.m. in SEO 712
Abstract
Hurwitz schemes parametrize degree d, genus g covers of the projective line. They are objects of classical interest and have also been a area of modern research. I will describe their basic properties and some recent results.
April 18, 2016
Tabes Bridges :
4 p.m. in SEO 712
Abstract
I will recall the notion of a minimal algebraic surface and how one "runs the MMP" in the surface case. I will then outline the difficulties that arise starting in dimension $3$, state some of the basic structural theorems, and describe some of the ideas due to Mori, Shokurov, etc. which help to circumvent those difficulties.
April 25, 2016
Alexander Stathis :
4 p.m. in SEO 712
Abstract
I will describe Nakajima's basis and relations for the cohomology of the
Hilbert Scheme of points on an algebraic surface with examples from
$\mathbb{P}^{2}$. If there's time, I'll talk a bit about my recent work
in trying to understand the relations for $\mathbb{P}^{2[n]}$.
Aug. 24, 2016
Tabes Bridges :
3 p.m. in SEO 512
Aug. 31, 2016
Jay Kopper :
3 p.m. in SEO 712
Abstract
In this talk I will discuss some recent results on the positivity of cycles on blow-ups of Grassmannians. We will see some interesting pathologies, including an effective cone which is not finitely generated (assuming the SHGH conjecture).
Sept. 7, 2016
Sam Shideler :
3 p.m. in SEO 712
Abstract
I'll discuss the basics of test ideals. This will be followed (next week) by a discussion of connections to algebraic geometry in characteristic 0.
Sept. 14, 2016
Sam Shideler :
3 p.m. in SEO 712
Abstract
I'll continue the discussion from last week. In particular, I'll introduce multiplier ideals and talk about connections to test ideals.
Sept. 21, 2016
Tabes Bridges :
3 p.m. in SEO 712
Abstract
A Fano variety is one with ample anticanonical class. I will prove a theorem of Mori that Fano varieties are covered by rational curves of bounded degree. The proof is a marvelous blend of methods geometric (the famous bend-and-break lemmas) and algebraic (reduction mod p).
Sept. 28, 2016
Dylan Moreland :
3 p.m. in SEO 712
Abstract
Two decades ago Kontsevich introduced the space of a stable maps in order to check enumerative predictions coming from string theory. We will sketch how to do this in the simplest possible example and, if time permits, mention some applications to birational geometry.
No food will be provided, but there will be plenty of waffling.
Oct. 5, 2016
Sayanta Mandal :
3 p.m. in SEO 712
Abstract
I will explain the proof of Grothendieck Riemann Roch as described in the
paper by Borel and Serre titled "Le theoreme de Riemann-Roch".
Oct. 12, 2016
Shuhang Yang :
3 p.m. in SEO 712
Oct. 19, 2016
Janet Page :
3 p.m. in SEO 712
Oct. 26, 2016
Janet Page :
3 p.m. in SEO 712
Abstract
I'll introduce the ring of Frobenius operators and talk about some results on its generation.
Nov. 2, 2016
Wes Townsend :
3 p.m. in SEO 712
Abstract
We examine the quotient varities of the form $\C^2/G$ where G is a finite subgroup of $SL^2$. These varieties have reasonably simple singularities and we examine how they transform under blowup.
Nov. 9, 2016
Joel Stapleton :
3 p.m. in SEO 712
Abstract
We introduce the K-Theory of Schemes. We prove the Resolution, Devissage and Localization theorems for K-Theory. We then utilize them to describe a connection to the Chow Ring of a Smooth Scheme.
Nov. 16, 2016
Fumiaki Suzuki :
3 p.m. in SEO 712
Abstract
I'll discuss the paper by Colliot-thelene and Pirtuka on non-stable rationality of quartic 3-folds. I'll introduce the notion of universal CH_0 triviality and the decomposition of the diagonal, and explain how they play a key role in their proof.
Jan. 18, 2017
Tabes Bridges :
3 p.m. in SEO 712
Abstract
I will briefly review some history and state some relevant definitions, in particular the notion of a stable variety due to Kollár and Shepherd-Barron. I will then illustrate by example the greater subtleties that arise in studying families of surfaces. Finally, I will describe a few special cases in which progress has been made.
Jan. 25, 2017
Jay Kopper :
3 p.m. in SEO 712
Abstract
Hilbert's fourteenth problem asks whether certain subrings of polynomial rings over a field $k$ must be finitely generated over $k$. Nagata first found a counterexample in 1958. Since then, considerable progress has been made in producing many new interesting counterexamples. In this two part seminar, I will discuss another counterexample of Nagata that exploits a surprising relationship between algebra and geometry.
Feb. 1, 2017
Jay Kopper :
3 p.m. in SEO 712
Abstract
Last week we saw that finite generation of a particular ring depended on finite generation of the effective cone of the blow-up of the plane at a set of points. In this (hopefully) self-contained talk, we will prove that the effective cone of the blow-up of the plane at 9 or more general points is not finitely generated.
Feb. 22, 2017
Tabes Bridges :
3 p.m. in SEO 712
Abstract
I will briefly motivate the study of cones of divisors in algebraic geometry, introduce the Kontsevich Moduli Spaces, and describe results of Coskun-Harris-Starr computing effective and ample cones of these spaces.
March 1, 2017
Wes Townsend :
3 p.m. in SEO 712
Abstract
The Hilbert-Kunz Function and F-Signature are two key invariants of rings with positive characteristic. In a 2008 paper, Craig Huneke, Paul Monsky, and Moira A. McDermott established a tighter estimate on the asymptotic behavior of the Hilbert-Kunz Function for suitably geometric rings. There are strong connections between the F-Signature and the Hilbert-Kunz Function, but it remains an open question whether it is possible to obtain the same estimates for F-Signature of excellent local normal rings in positive characteristic. We will discuss Huneke, Monsky, and McDermott's result and attempts to extend it to the F-Signature.
March 15, 2017
Sam Shideler :
3 p.m. in SEO 712
Abstract
I'll start by recalling the definition of regularity, and state the Eisenbud-Goto conjecture. I'll then discuss the proof in some known cases. Finally, I'll talk about recent counterexamples by McCullough and Peeva.
April 5, 2017
Fumiaki Suzuki :
3 p.m. in SEO 712
Abstract
The rationality of smooth Fano complete intersections will be studied in the context of birational geometry. We review Sarkisov program of Mori fiber spaces and discuss birational rigidity. Then we introduce the work by de Fernex on birational rigid hypersurfaces and its extension to complete intersections.
April 19, 2017
Khashayar Sartipi :
3 p.m. in SEO 712
Aug. 30, 2017
Wes Townsend :
1 p.m. in SEO 712
Sept. 6, 2017
Wes Townsend :
1 p.m. in SEO 712
Abstract
We begin by covering some historical motivation for the Hilbert-Kunz function, namely Kunz's 1969 result that showed that flatness of the Frobenius endomorphism characterized positive characteristic regular local rings. We will then introduce the Hilbert-Kunz function, and discuss its properties and significance.
Sept. 13, 2017
Joel Stapleton :
1 p.m. in SEO 712
Abstract
We define the intersection product on the Chow Ring of a scheme X. Along the way, we will prove Chow's Moving Lemma.
Sept. 20, 2017
Fumiaki Suzuki :
1 p.m. in SEO 712
Abstract
Unramified Galois cohomology groups of a field are stably birational invariants of its smooth projective models.
I want to explain how to compute them.
Sept. 27, 2017
Tabes Bridges :
1 p.m. in SEO 712
Abstract
In the space of an hour, I will endeavor to briefly introduce a number of fun moduli problems (curves, abelian varieties, the Hilbert scheme, Kontsevich space), the relations between them, and some of the interesting geometry that arises.
Oct. 4, 2017
See-Hak Seong :
1 p.m. in SEO 712
Oct. 11, 2017
Jay Kopper :
1 p.m. in SEO 712
Abstract
Classical Brill-Noether theory asks how many global sections a line bundle on a (general) smooth curve can have. This was solved (completely, in some sense) by Griffiths-Harris several decades ago. A number of natural generalizations remain open: what if we ask about (semistable) vector bundles instead of line bundles? What if our curves are not Brill-Noether general but nonetheless lie in an interesting family? In this talk I will pose but fail to answer some of these very interesting questions.
Oct. 25, 2017
Janet Page :
1 p.m. in SEO 712
Abstract
In this talk, I will discuss Hilbert's 14th problem, starting with its history and motivation. I will give some related questions and some counterexamples.
Nov. 1, 2017
Jack Hafer :
1 p.m. in SEO 712
Abstract
Intersection theory on varieties is done in the Chow ring, the ring of classes of
rationally equivalent algebraic cycles graded by codimension, where multiplication
is given by intersection of cycles. Following C. Soulé et. al., I will outline the
construction of the Chow ring in a more general setting, on regular separated noetherian
schemes, using the methods of algebraic K-theory.
Nov. 8, 2017
Gregory Taylor :
1 p.m. in SEO 712
Abstract
We discuss the utility of étale maps in algebraic geometry, stressing examples and analogies to topology and differential geometry.
Nov. 15, 2017
Micah Darrell :
1 p.m. in SEO 712
Nov. 29, 2017
Wes Townsend :
1 p.m. in SEO 712
Dec. 6, 2017
Joe Berner :
1 p.m. in SEO 712
Abstract
I will give an introduction to adic spaces. Since the foundations are technical we will try to limit ourselves to their flavor and their applications. Adic spaces are the non-archimedean foundation of perfectoid spaces, which are an exciting and active area of research.
Jan. 31, 2018
Wes Townsend :
3 p.m. in SEO 712
Abstract
We will continue working through the first chapter of 3264 & All That.
Feb. 7, 2018
Joel Stapleton :
3 p.m. in SEO 712
Abstract
We will cover a few of the sections of Chapter 2 including the computation of the Chow Ring of $\mathbb{P}^n$ blown up at a point.
Feb. 21, 2018
Gregory Taylor :
3 p.m. in SEO 712
Feb. 28, 2018
Gregory Taylor :
3 p.m. in SEO 712
April 11, 2018
Sayanta Mandal :
3 p.m. in SEO 712
Abstract
I'll try to describe the nef cone, effective cone etc. Of the Hilbert scheme of points in plane.
April 18, 2018
Jay Kopper :
3 p.m. in SEO 712
Abstract
I introduce various notions of stability for sheaves. I will pay special attention to the case of sheaves on surfaces and discuss some modern results regarding moduli spaces of sheaves and cohomology.
Aug. 29, 2018
Fumiaki Suzuki :
2 p.m. in 712 SEO
Sept. 5, 2018
Fumiaki Suzuki :
2 p.m. in 712 SEO
Abstract
We study algebraic cycles on a smooth cubic 5-fold X.
In particular, we compute its Chow groups and unramified cohomology groups.
It turns out that many interesting stably birational invariants given by algebraic cycles on X are trivial.
Fu-Tian, "2-cycles sur les hypersurfaces cubiques de dimension 5", https://arxiv.org/abs/1801.03995
Sept. 12, 2018
Gregory Taylor :
2 p.m. in 712 SEO
Abstract
We present a survey of recent work of Lehmann-Sengupta-Tanimoto which gives a candidate for the exceptional set Manin's conjecture.
Sept. 19, 2018
Micah Darrell :
2 p.m. in 712 SEO
Abstract
I'll talk about Grothendieck's dream of having a universal Weil cohomology theory, and Scholze's idea for how one might construct such a theory
Sept. 26, 2018
Jay Kopper :
2 p.m. in 712 SEO
Abstract
The Hilbert scheme of points parameterizes length $n$ zero-dimensional subschemes of a variety. A recent paper by Bolognese-Huizenga-Lin-Riedl-Schmidt-Woolf-Zhao computes the nef cone of these spaces. I will give an overview of their work, including necessary background on Hilbert schemes and nefness.
Oct. 3, 2018
See-Hak Seong :
2 p.m. in 712 SEO
Abstract
There is a Voisin's conjecture, which states that there would be a bounds for the degree of rational curves, when those curves lie on a very general hypersurface of general type. Recently, Riedl and Yang showed that all the rational curves in hypersurface will be lines, when we work on specific degree of hypersurfaces. I will talk how they worked on this problem.
Oct. 10, 2018
Joel Stapleton :
2 p.m. in 712 SEO
Abstract
We follow a primer by Antieau-Elmanto in order to give a brief idea of recent work done by Asok-Hoyois-Wendt and Asok-Fasel.
Oct. 31, 2018
Gregory Taylor :
2 p.m. in 712 SEO
Nov. 7, 2018
Luke Jaskowiak :
2 p.m. in 712 SEO
Abstract
For this talk the special linear group will serve as our guide through the book of I. Satake.
Nov. 14, 2018
Ben Gould :
2 p.m. in 712 SEO
Nov. 28, 2018
Shravan Patankar :
2 p.m. in 712 SEO
Abstract
We will study the proof of Kunz's theorem given by Bhatt and Scholze in their paper "Projectivity of the Witt vector affine Grassmannian" and generalize some of their results
Dec. 5, 2018
Matt DeVilbiss :
2 p.m. in 712 SEO
Abstract
Wherein I will attempt to introduce ultraproducts in the context of rings and summarize a recent proof of Stillman's conjecture by Erman, Sam, and Snowden.
Jan. 16, 2019
Greg Taylor and Fumiaki Suzuki :
3 p.m. in 712 SEO
Jan. 23, 2019
Fumiaki Suzuki :
3 p.m. in 712 SEO
Abstract
We review Artin-Mumford’s construction of a 3-fold which is unirational and non-rational. We start from the motivation, give a construction, and explain it in terms of Severi-Brauer schemes
Feb. 6, 2019
Jay Kopper :
3 p.m. in 712 SEO
Abstract
The Brill-Noether theorem describes the number of maps a general smooth curve admits to projective space of given degree. A complete proof was given by Griffiths-Harris in 1980, building on work of Kleiman-Laksov and others. In 1982, Gieseker gave a proof of Petri's conjecture, a strictly stronger result. The proofs of these theorems require very delicate manipulations of transverse intersections and degenerations to special (reducible!) curves in the boundary of the Deligne-Mumford compactification of the moduli space. In 1986, Lazarsfeld gave a comparitively simple proof of the Gieseker-Petri theorem by explicitly constructing curves on K3 surfaces. In this talk I will describe the relevant background and give an overview of Lazarsfeld's proof.
Feb. 13, 2019
See-Hak Seong :
3 p.m. in 712 SEO
Abstract
Hartshorne proves that every Hilbert scheme of projective space is connected, and it is well-known that Hilbert scheme of points on connected set is connected. In this talk I will give details about the connectivity of Hilbert scheme, and also talk about other properties of Hilbert Scheme of points on an affine space.
Feb. 20, 2019
No Seminar :
3 p.m. in 712 SEO
Feb. 27, 2019
No Seminar :
3 p.m. in 712 SEO
March 6, 2019
Gregory Taylor :
3 p.m. in 712 SEO
Abstract
This talk is an introduction to main ideas of Boucksom-Demailly-Păun-Peternell's paper "The Pseudoeffective Cone of a Compact Kähler Manifold and Varieties of Negative Kodaira Dimension". In particular, we will discuss the pseudoeffective and movable cones, volume of big divisors, and Fujita approximations.
March 13, 2019
Ben Gould :
3 p.m. in 712 SEO
Abstract
We will cover Chapter I of Miles Reid's 'A Young Person's Guide to Canonical Singularities.' The paper begins with the motivation for the notion of a (complex quasiprojective) variety to have canonical singularities, and then surveys the general theory, most of which is valid in all dimensions, and some of which is valid only up to dimension 3. We will assume as known much of the material from Hartshorne's II.6 and will overlap somewhat with the ongoing seminar on resolution of singularities.
March 20, 2019
Shravan Patankar :
3 p.m. in 712 SEO
Abstract
We will prove some cohomology vanishing theorems for Schubert Varieties using Frobenius splitting techniques of Mehta and Ramanathan.
April 3, 2019
Adam Pratt :
3 p.m. in 712 SEO
Abstract
I will give an overview of the theory of Grothendieck topologies, sites, and topoi, along with the theory of étale cohomology as laid out in SGA 4.5.
April 10, 2019
Tian Wang :
3 p.m. in 712 SEO
April 17, 2019
Jacob Mayle :
3 p.m. in 712 SEO
April 24, 2019
Matt DeVilbiss :
3 p.m. in 712 SEO
Aug. 28, 2019
Fumiaki Suzuki and Greg Taylor :
2 p.m. in 712 SEO
Sept. 4, 2019
Yaoxiong Wen :
5 p.m. in 712 SEO
Abstract
The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kahler manifold X. Recently a 3d lift (N=2 supersymmetric 3d gauge theory) of this correspondence has been proposed and explored, and Givental's permutation equivariant quantum K-theory gives the IR limit. In this talk, we will explain what K-theoretic Gromov-Witten theory is, and some other relevant topics.
Sept. 11, 2019
Greg Taylor :
5 p.m. in 712 SEO
Abstract
Kodaira vanishing and its generalizations are powerful tools in birational geometry. Unfortunately, they fail in positive characteristic. In this talk, we construct counterexamples to the Kodaira vanishing theorem and, if time permits, discuss strategies for replacing vanishing theorems with Frobenius methods.
Sept. 18, 2019
Fumiaki Suzuki :
5 p.m. in 712 SEO
Abstract
Based on computations over finite fields, Serre asked in a letter to Grothendieck whether a projective variety over the function field of a curve always has a rational point if it is O-acyclic, that is, H^i(X,O)=0 for all i>0. Then Grabber-Harris-Mazur-Starr gave the negative answer to this question by constructing an Enriques surface without rational points. Later, more explicit constructions of such Enriques surfaces were given by Lafon and Starr. We are going to discuss these examples and related problems.
Oct. 2, 2019
Kuang-Yu Wu :
5 p.m. in 712 SEO
Abstract
This talk is a basic introduction of toric varieties. I will first talk about how every toric variety is associated with a combinatorial object called fan. Then, if time permits, I will use divisors as an example to show how algebro-geometric properties of toric varieties can be understood using the combinatorial data.
Oct. 9, 2019
Mark Hertz :
4 p.m. in 712 SEO
Abstract
The first known counterexample to Hilbert's 14th problem was introduced by Nagata in 1958. Since then, much work has been done to simplify and generalize his idea. In this talk, we will present Hilbert's problem and some early results, an elementary counterexample (a simplified version of Nagata's), and connect the construction to the total coordinate rings of certain blow-ups of $\mathbb{P}^2$.
Oct. 16, 2019
Ben Gould :
4 p.m. in 712 SEO
Nov. 6, 2019
Shijie Shang :
5 p.m. in 712 SEO
Abstract
The theory of Seshadri constants originates with Demailly's idea for quantifying how much of the positivity of an ample line bundle can be localized at a given point of a variety. First, I will talk about a theorem of Ein and Lazarsfeld, which gives a universal lower bound of Seshadri constants at a very general point of smooth complex projective surfaces. Then I will introduce Miranda's example, which illustrates that there cannot exist lower bounds on Seshadri constants holding at every point of every surface. At last, I will discuss a generalized version of Miranda's example constructed by Bauer.
Nov. 13, 2019
Joel Stapleton :
5 p.m. in 712 SEO
Abstract
We describe an example from Addington-Antieau-Frei-Honigs showing derived equivalences need not preserve the existence of rational points.
Nov. 20, 2019
See-Hak Seong :
5 p.m. in 712 SEO
Abstract
We will talk about geometry of curves with fixed degree and arithmetic genus, and talk about irreducible components of the Hilbert scheme of curves.
Dec. 4, 2019
Shravan Patankar :
5 p.m. in 712 SEO
Abstract
We will define K0 of categories and try to prove Grothendieck's resolution theorem along with examples from algebraic geometry which justify the existence of definitions and theorems.
Jan. 15, 2020
Fumiaki Suzuki and Greg Taylor :
4 p.m. in 712 SEO
Jan. 22, 2020
Greg Taylor :
4 p.m. in 512 SEO
Abstract
The collection of birational self-maps of an algebraic variety X forms a group Bir(X). When X is a surface, this group acts on an infinite hyperbolic space, constructed via divisors on blowups of the surface. In this talk, we introduce this construction and use it to understand Bir(X). Time permitting, we will discuss very recent work of Lonjou and Urech on Bir(X) for higher dimensional X. The main source for this talk is the paper "Sur les groupes de transformations birationnelles des surfaces" by Serge Cantat.
Jan. 29, 2020
Fumiaki Suzuki :
4 p.m. in 512 SEO
Abstract
In this talk, we explain a recent result of Totaro proving that the integral Hodge conjecture holds on 3-folds of Kodaira dimension zero with non-zero global 3-forms, which strengthens an earlier work of Voisin. This makes a contrast with a result of Benoist-Ottem proving that the conjecture can fail on a 3-fold with 2K=0. We start from a review on the IHC and its relation with birational geometry.
Feb. 5, 2020
Ben Gould :
4 p.m. in 512 SEO
Abstract
Building on foundational work of Brendan Hassett and recent work of Lai on the connections between K3 surfaces and cubic fourfolds, Farkas-Verra have proven the rationality of the universal K3 surface of genus 14. This talk will describe their (surprisingly explicit) work and survey the relevant past results.
Feb. 19, 2020
Jack Hafer :
4 p.m. in 512 SEO
Abstract
In this talk, we will discuss recent work of Xing Gu, building on the work of Vistoli, Vezzosi, and others in computing the structure of the cohomology ring and Chow ring of BPGL_n. In particular, we will use tools from homotopy theory, algebraic geometry, and group cohomology to construct p-torsion classes in the cohomology ring of BPGL_n for p a prime divisor of n, and if p^2 does not divide n, we can further construct corresponding p-torsion in the Chow ring of BPGL_n.
Feb. 26, 2020
Shijie Shang :
4 p.m. in 512 SEO
Abstract
The degree of irrationality is one of the high-dimensional analogues of gonality of curves. In this talk, I will discuss the paper Measures of irrationality for hypersurfaces of large degrees by Bastianelli, De Poi, Ein, Lazarsfeld and Ullery. If time permits, I will talk about some recent progress on computing or bounding degree of irrationality for hypersurfaces under different conditions.
March 4, 2020
Mark Hertz :
4 p.m. in 512 SEO
Abstract
TBD
March 11, 2020
Gregory Taylor :
4 p.m. in 512 SEO
Abstract
The gonality of a curve can be read off from the syzygies of any line bundle of sufficiently large degree. We discuss the proof of this fact following Ein and Lazarsfeld's paper "The gonality conjecture of syzygies of algebraic curves of large degree".
April 1, 2020
Mark Hertz :
4 p.m. in 512 SEO
Abstract
TBA
April 15, 2020
Mengxue Yang :
4 p.m. in 512 SEO
Abstract
TBA
April 22, 2020
Tian Wang :
4 p.m. in 512 SEO
Abstract
TBA
April 29, 2020
Ben Gould :
4 p.m. in 512 SEO
Abstract
TBA
Sept. 2, 2020
Ben Gould :
11 a.m. in Zoom
Abstract
First of a pair of talks with Shijie Shang.
Sept. 9, 2020
Shijie Shang :
11 a.m. in Zoom
Abstract
Second of a pair of talks with Ben Gould.
Sept. 16, 2020
Greg Taylor :
11 a.m. in Zoom
Abstract
We take up the subject of minimal free resolutions of section rings of algebraic curves. After introducing the basic definitions and examples, we prove Green's "2g+p+1" Theorem. This talk serves as a primer for the next three talks in the seminar
1) The gonality conjecture (Jiamin Li 9/23)
2) Green's conjecture for generic curves (Greg Taylor 9/30)
3) Stability of syzygy bundles on surfaces (Shijie Shang 10/7)
Sept. 30, 2020
Gregory Taylor :
11 a.m. in Zoom
Abstract
We give an introduction to Green's conjecture for syzygies of canonical curves. After some motivation for the conjecture, we discuss approaches to Green's conjecture, focusing on the recent proof of the generic case by Michael Kemeny.
Oct. 7, 2020
Shijie Shang :
11 a.m. in Zoom
Dec. 9, 2020
Shravan Patankar :
11 a.m. in Zoom
Abstract
In her thesis, Karen Smith showed that a singularity in characteristic zero is rational provided its `reduction' to a prime which is `large enough' is F-rational. Astonishingly, using recent advances in mixed characteristic commutative algebra, Ma and Schwede show that the former is true for *any* prime p. Apart from being of independent interest, this has substantial algorithmic consequences to determine whether a char 0 singularity is rational.
We will isolate the part of the above proposition which requires `perfectoid theory' and study their general `spreading out' technique which has its roots in a first course in algebraic geometry.
Sept. 8, 2021
Kuang Yu Wu :
4 p.m. in 636 SEO
Abstract
Toric vector bundles are vector bundles on toric varieties equipped with an equivariant toric action. In this talk, I will demonstrate how the data of a toric vector bundle can be encoded in a vector space together with a collection of filtrations (Klyachko's Classification Theorem), and then talk about how this can be used to study various invariants of toric vector bundles, such as positivity and chern classes.
Oct. 10, 2022
Yeqin Liu :
4 p.m. in 427 SEO
Abstract
On K3 surfaces, spherical vector bundles are the ones without deformations. In this talk I introduce a way to compute their exact numbers of global sections by using Bridgeland stability. I will show many examples.
Oct. 31, 2022
Ben Gould :
4 p.m. in 427 SEO
Abstract
We introduce the results of Hartshorne's paper 'Stable bundles of rank 2 on P3' (91 cm)˜, which describes a precise relationship between slope-stable vector bundles of rank 2 on P3 and curves in P3. This is the first of a series of talks on the topic of rank 2 vector bundles on projective spaces.
Jan. 23, 2023
Sixuan Lou :
4 p.m. in 427 SEO
Abstract
I will prove Green's theorem, which states a curve embedded by a line bundle of high degree has more "reasonable" syzygies as we increase the degree. This is the first talk in a series of talks where we explore Larzasfeld's note "A Sampling of Vector Bundle Techniques in the Study of Linear Series".
Jan. 30, 2023
Henry Fontana :
4 p.m. in 427 SEO
Feb. 6, 2023
Henry Fontana :
4 p.m. in 427 SEO
Feb. 13, 2023
Junyan Zhao :
4 p.m. in 427 SEO
Feb. 20, 2023
Junyan Zhao :
4 p.m. in 427 SEO
Feb. 27, 2023
Gregory Taylor :
4 p.m. in 427 SEO
Abstract
We discuss Ein and Lazarsfeld's proof of the Gonality Conjecture. We will present Voisin's interpretation of Koszul cohomology via the Hilbert scheme, the duality theorem, and the cohomological techniques required for the proof.
March 6, 2023
Yeqin Liu :
4 p.m. in 427 SEO
Abstract
Exceptional vector bundles are a special class of rigid vector bundles. In particular, the moduli space of exceptional bundles with a fixed Chern character is discrete. On P^3, it has been conjectured that such moduli space consists of a single point. In this talk, we prove this conjecture.
March 13, 2023
Ben Gould :
4 p.m. in 427 SEO
April 10, 2023
Junyan Zhao :
4 p.m. in 427 SEO
Abstract
A general curve C of genus 6 can be embedded into the unique quintic del Pezzo surface X_5 as a divisor of class -2K_{X_5}. This embedding is unique up to the action of the symmetric group S_5. Taking a double cover of X_5 branched along C yields a K3 surface Y. Thus the K-moduli spaces of the pair (X_5, cC) can be studied via wall-crossing and by relating them to the Hassett-Keel program for C and the HKL program for Y. On the other hand, X_5 can be embedded in P^1 \times P^2 as a divisor of class O(1,2), under which -2K_X is linearly equivalent to O_X(2,2). One can study the VGIT-moduli spaces in this setting. In this talk, I will compare these four types of compactified moduli spaces and their different birational models given by wall-crossing.
April 17, 2023
Sixuan Lou :
4 p.m. in 427 SEO
Abstract
Given a stable vector bundle on a projective variety, it is natural to ask what it looks like when we restrict to a complete intersection subvariety. The theorem of Grauert and Mülich gives us a numerical constraint for the Harder-Narasimhan filtration of the restricted bundle on a general such complete intersection. On the other hand, one can ask what the locus of complete intersections on which our bundle exhibit non-generic filtration looks like. In this talk we will investigate this question in the simplest setting of restriction to lines in P^2 and conic sections on a smooth quadric.