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

英文摘要：

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