Unirationality of Hurwitz spaces of coverings of degree <= 5

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Abstract

Let Y be a smooth, projective curve of genus g>=1. 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, 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.
Lingua originaleEnglish
pagine (da-a)3006-3052
Numero di pagine47
RivistaInternational Mathematics Research Notices
Volume2013
Stato di pubblicazionePublished - 2013

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Hurwitz Space
Covering
Monodromy Group
Line Bundle
Genus
Isomorphic
Curve

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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title = "Unirationality of Hurwitz spaces of coverings of degree <= 5",
abstract = "Let Y be a smooth, projective curve of genus g>=1. 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, 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.",
keywords = "Hurwitz spaces; unirationality; coverings; vector bundles.",
author = "Vassil Kanev",
year = "2013",
language = "English",
volume = "2013",
pages = "3006--3052",
journal = "International Mathematics Research Notices",
issn = "1073-7928",
publisher = "Oxford University Press",

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T1 - Unirationality of Hurwitz spaces of coverings of degree <= 5

AU - Kanev, Vassil

PY - 2013

Y1 - 2013

N2 - Let Y be a smooth, projective curve of genus g>=1. 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, 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.

AB - Let Y be a smooth, projective curve of genus g>=1. 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, 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.

KW - Hurwitz spaces; unirationality; coverings; vector bundles.

UR - http://hdl.handle.net/10447/64970

UR - http://arxiv.org/abs/1106.1006

M3 - Article

VL - 2013

SP - 3006

EP - 3052

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

ER -