Quadric rank loci on moduli of curves and K3 surfaces

Gavril Farkas; Richard Rimanyi

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arXiv · 1707.00756

Quadric rank loci on moduli of curves and K3 surfaces

Gavril Farkas·Richard Rimanyi

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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.

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BibTeX RIS

Gavril Farkas, Richard Rimanyi. 2017-07-03. Quadric rank loci on moduli of curves and K3 surfaces. https://arxiv.org/abs/1707.00756

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