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

Mullane, Scott

Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1379-1416.

Nous considérons deux applications des strates de différentielles de second type (tous les résidus sont égaux à zéro) avec des multiplicités fixes de zéros et de pôles :

Positivité : En genre $g=0$, nous montrons que tous les diviseurs de ${\overline{ℳ}}_{0,n}$ obtenus par projection de ces strates sont $F$-nef et donc conjecturalement nef. Nous calculons sa classe pour tous les genres lorsque la projection est divisorielle et n’oublie que des zéros simples. Dans ce cas, nous montrons que les projections en genre $g=0$ sont effectivement nef.

Espaces d’Hurwitz : Nous montrons que les espaces d’Hurwitz des revêtements de degré $d$ de genre $g$ au-dessus de ${\mathbb{P}}^1$ dont les ramifications sont pures (un unique point de ramification au dessus d’un point de branchement) à l’exception d’au plus un point de ramification sont irréductibles s’il existe au moins $3g+d-1$ points de ramification simples ou $d-3$ de points de ramification simples lorsque $g=0$.

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 ${\overline{ℳ}}_{0,n}$ is $F$-nef and hence conjectured to be nef. We compute the class for all genus when the divisorial projection only forgets simple zeroes 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 $g=0$.

Classification : 14C20, 14C25, 14E99, 30F30
Keywords: Riemann surfaces, Moduli spaces, Hurwitz spaces, Algebraic cycles, Birational geometry, Strata of differentials
Mot clés : surfaces de Riemann, espaces des modules, espaces d’Hurwitz, cycles algébriques, géométrie birationnelle, strates de différentielles

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