Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Abstract

Let Y be a smooth, projective curve of genus g≥1 over the complex numbers. Let be the Hurwitz space that parameterizes equivalence classes of coverings π:X→Y of degree d simply branched in n=2e points, such that the monodromy group is Sd and is isomorphic to a fixed line bundle A of degree e. We prove that, when d=3,4, or 5 and n is sufficiently large (precise bounds are given), the Hurwitz space is unirational. If, in addition, (e,2)=1 (when d=3), (e,6)=1 (when d=4), and (e,10)=1 (when d=5), then is rational.