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 M0,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 P1 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.