Modifications of Hodge bundles and enumerative geometry : the stable hyperelliptic locus

TL;DR

This paper investigates the stable hyperelliptic locus within the moduli space of stable curves, constructing a bundle map on a blowup of the Hilbert scheme to compute its class using intersection theory and Porteous' formula.

Contribution

It introduces a novel bundle map on a blowup of the Hilbert scheme to analyze the hyperelliptic locus and computes its class via advanced intersection theory techniques.

Findings

Constructed a degree-2 Brill-Noether bundle map.

Computed the class of the hyperelliptic locus using Porteous' formula.

Provided new intersection-theoretic tools for moduli space analysis.

Abstract

We study the stable hyperelliptic locus, i.e. the closure, in the Deligne- Mumford moduli space of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated to a family of stable curves, we construct a bundle map ('degree-2 Brill-Noether') from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton-Macpherson residual intersection theory, the class of the structure sheaf and various other sheaves supported on the stable hyperelliptic locus can be computed by Porteous' formula.