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

Abstract

We prove that for a generic element in a nonhyperelliptic component of an abelian stratum ℋ𝑔(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 ℋ9(𝜇) are uniruled when 𝜇 is a positive partition of 16 and all nonhyperelliptic components of ℋ2𝑔(𝜇) are uniruled when 𝜇 is a positive partition of 4𝑔−4 and either 3⩽𝑔⩽5 or 𝑔=6 and 𝑙(𝜇)⩾4.