Quadric rank loci on moduli of curves and K3 surfaces

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) we find 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) we construct an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) we provide 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) construct myriads of effective divisors of small slope on M_g.

43 pages. Numerous changes and improvements. Final version, to appear in Annales Sci. de L'Ecole Normale Superieure

References in corpus (2)

• Relations in the tautological ring of the moduli space of K3 surfaces
• Limit linear series and ranks of multiplication maps

Cited by in corpus (2)

• Geometric invariant theory of syzygies, with applications to moduli spaces
• Variety of Singular Quadrics Containing a Projective Curve