Rational maps between moduli spaces of curves and Gieseker-Petri divisors

TL;DR

This paper investigates the intersection theory of rational maps between moduli spaces of stable curves, focusing on Brill-Noether loci and Gieseker-Petri divisors, with applications to the cone of moving divisors and Prym varieties.

Contribution

It introduces new intersection theoretic methods to analyze rational maps between moduli spaces and describes the cone of moving divisors on M_g, also studying Gieseker-Petri loci.

Findings

Describes the cone of moving divisors on M_g.

Shows the Gieseker-Petri locus has codimension 1 in M_g.

Provides applications to Prym varieties.

Abstract

We perform an intersection theoretic study of the rational map between two different moduli spaces of stable curves which associates to a curve its corresponding Brill-Noether locus (in the case this locus has virtual dimension 1). We then use these results to describe the cone of moving divisors on M_g. Several other applications to moduli spaces of Prym varieties are presented. In a different direction, we prove that the locus in M_g of curves failing to satisfy the Gieseker-Petri theorem is supported in codimension 1 for every possible type of linear series.