Strata of differentials of the second kind, positivity and irreducibility of certain Hurwitz spaces

TL;DR

This paper explores the positivity properties of divisors related to differentials of the second kind and establishes irreducibility criteria for certain Hurwitz spaces with specified branching conditions.

Contribution

It demonstrates that divisorial projections in genus zero are $F$-nef and computes their classes, and it proves irreducibility of Hurwitz spaces under specific simple branch point conditions.

Findings

Divisorial projections in genus 0 are $F$-nef.

Computed classes for divisorial projections with simple zeros.

Proved irreducibility of Hurwitz spaces with many simple branch points.

Abstract

We consider two applications of the strata of differentials of the second kind (all residues equal to zero) with fixed multiplicities of zeros and poles: Positivity: In genus $g=0$ we show any associated divisorial projection to $\overline{\mathcal{M}}_{0,n}$ is $F$-nef and hence conjectured to be nef. We compute the class for all genus when the divisorial projection forgets only simple zeros and show in these cases the genus $g=0$ projections are indeed nef. Hurwitz spaces: We show the Hurwitz spaces of degree $d$, genus $g$ covers of $\mathbb{P}^1$ with pure branching (one ramified point over the branch point) at all but possibly one branch point are irreducible if there are at least $3g+d-1$ simple branch points or $d-3$ simple branch points when $g=0$.