Moduli spaces of curves with linear series and the slope conjecture

TL;DR

This paper studies the moduli space of curves with linear series, extends it over a partial compactification, and provides new counterexamples to the Harris-Morrison slope conjecture through Chow class computations.

Contribution

It introduces a compactified moduli space of curves with linear series and computes Chow class push-forwards to produce counterexamples to the slope conjecture.

Findings

Counterexamples to the Harris-Morrison slope conjecture

Infinite sequence of potential counterexamples

Chow class computations for the moduli space

Abstract

We describe the moduli space G^r_d of triples consisting of a curve C, a line bundle L on C of degree d, and a linear system V on L of dimension r. This moduli space extends over a partial compactification {\tilde M_g} of M_g inside {\bar M_g}. For the proper map h : G^r_d --> \tilde M_g, we compute the push-forward on Chow 1-cocyles in the case where h has relative dimension zero. As a consequence we obtain another counterexample to the Harris-Morrison slope conjecture as well as an infinite sequence of potential counterexamples.