Unirationality of Hurwitz spaces of coverings of degree <= 5

Let Y be a smooth, projective curve of genus g&gt;=1. Let H^0_{d,A}(Y)be the Hurwitz space which parametrizes coverings p:X --&gt; Y of degree d simply branched in n=2e points, such that the monodromy group is S_d and det(P_*O_X/O_Y) is 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.

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. © 2012 The Author(s) 2012. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup. com.