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 H 2 g (μ) are uniruled when μ is a positive partition of 4g − 4 and either 3 g 5 or g = 6 and l(μ) 4.

On the Kodaira dimension of moduli spaces of Abelian differentials

“…The (projectivized) strata of holomorphic Abelian differentials are uniruled for low genus g ≤ 9 and all zero types µ as well as for g ≤ 11 if moreover the number of zeros n is large ( [Bar18], [Bud21]). On the other hand, when n ≥ g − 1 these strata can be viewed as generically finite covers of the moduli space of pointed curves ( [Gen18]) and thus of general type for large genus.…”