Unirationality of Hurwitz spaces of coverings of degree <= 5

Details Unirationality of Hurwitz spaces of coverings of degree <= 5 Unirationality of Hurwitz spaces of coverings of degree <= 5

2011-06-06

arxiv.org/abs/1106.1006v3

Mathematics

Algebraic Geometry

v1: 32 pages, Latex, v2: Typos corrected; minor changes; references added; proof of Lemma 1.2 skipped; added Lemma 2.4 and Pro

Scientific paper

Let Y be a smooth, projective curve of genus g>=1 over the complex numbers. Let H^0_{d,A}(Y) be the Hurwitz space which parametrizes coverings p:X --> 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.

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