Unirationality of Hurwitz spaces of coverings of degree <= 5

TL;DR

This paper proves that Hurwitz spaces of degree 3, 4, or 5 coverings of a genus ≥ 1 curve are unirational for large enough branch points, and rational under certain coprimality conditions.

Contribution

It establishes the unirationality and rationality of Hurwitz spaces for degrees up to 5, extending known results to new cases with explicit bounds.

Findings

Hurwitz spaces of degree 3, 4, 5 are unirational for large branch points.

Under certain coprimality conditions, these spaces are rational.

Explicit bounds on the number of branch points are provided.

Abstract

Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ 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 $det(p_{*}O_X/O_Y)$ 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.