Unirationality of Hurwitz spaces of coverings of degree <= 5

Abstract

Let $Y$ be a smooth, projective curve of genus $g\ge 1$ over the complex numbers. Let $H_{d,A}^0(Y)$ be the Hurwitz space which parametrizes coverings $p:X\to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{\ast }{O}_X/O_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3,4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.

arXiv commentProposition 2.11 and Lemma 2.13 are corrected. The corrections do not affect the other statements of the paper. Corrigendum submitted to IMRN

References in corpus (3)

• A note on Hurwitz schemes of covers of a positive genus curve
• Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds
• Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of abelian threefolds A_3(1,1,4)