Gavril Farkas : Quadric rank loci on moduli of curves and K3 surfaces

Posted by Kevin Tucker , part of the Algebraic Geometry Seminar.

At

Oct. 14, 2019, 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.