Maximal gonality on strata of differentials and uniruledness of strata in low genus

TL;DR

This paper proves that generic curves in certain strata of differentials have maximal gonality and uses this to show that specific low-genus strata are uniruled, revealing geometric properties of these moduli spaces.

Contribution

It establishes maximal gonality for generic elements in nonhyperelliptic strata and demonstrates uniruledness of low-genus strata components, extending known geometric results.

Findings

Generic elements in nonhyperelliptic strata have maximal gonality.

Certain low-genus strata components are uniruled.

Results apply to both abelian and quadratic strata.

Abstract

We prove that for a generic element in a nonhyperelliptic component of an abelian stratum $\mathcal{H}_g(\mu)$ in genus $g$, the underlying curve has maximal gonality. We extend this result to the case of quadratic strata when the partition $\mu$ has positive entries. As a consequence we deduce that all nonhyperelliptic components of $\mathcal{H}_9(\mu)$ are uniruled when $\mu$ is a positive partition of 16 and all nonhyperelliptic components of $\mathcal{H}^2_g(\mu)$ are uniruled when $\mu$ is a positive partition of $4g-4$ and either $3\leq g\leq5$ or $g=6$ and $l(\mu)\geq 4$.