Resolving the Singularities of Splitting Loci

TL;DR

This paper constructs modular resolutions for splitting loci singularities, proving rational singularities for tame cases and general $k$-gonal curves, and recovers classical theorems with new cohomology vanishing results.

Contribution

It introduces modular resolutions for splitting loci singularities and establishes rational singularities for tame cases and general $k$-gonal curves, also recovering classical theorems.

Findings

Tame splitting loci have rational singularities.

Components of $W^r_d(C)$ have rational singularities for general $k$-gonal curves.

Proves a new cohomology vanishing statement for tautological vector bundles.

Abstract

We construct modular resolutions of singularities for splitting loci, and use them to show that tame splitting loci have rational singularities. As a corollary of our results and Hurwitz-Brill-Noether theory, we prove that if $C$ is a general $k$-gonal curve, the components of $W^r_d(C)$ have rational singularities. We also recover the classical Gieseker-Petri theorem. Along the way, we prove a cohomology vanishing statement for certain tautological vector bundles on $\operatorname{Quot}^{r,d}_{\mathbb{P}^1}(\mathcal{O}^{\oplus N})$, which may be of independent interest.