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

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