Unirationality of Hurwitz spaces of coverings of degree <= 5
Unirationality of Hurwitz spaces of coverings of degree <= 5
复制标题
次数 <= 5 的覆盖 Hurwitz 空间的非理性
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
V. Kanev
中科院分区：
文献类型：
--
作者：
V. Kanev
设Y是复数域上亏格为ggeq 1的光滑射影曲线.设$H^0_{d，A}（Y）$是参数化覆盖$p：X的Hurwitz空间 o d次Y$，在n= 2 e $点上单分支，单值群等于S_d$，且det（p_{*}O_X/O_Y）$同构于一个-e$的固定线丛A^{-1}$。证明了当d=3，4或5且n足够大（给出了精确的界）时，这些Hurwitz空间是单有理的.如果另外$（e，2）=1$（当$d=3$），$（e，6）=1$（当$d=4$）和$（e，10）=1$（当$d=5$），则这些Hurwitz空间是有理的。
Let $Y$ be a smooth, projective curve of genus $ggeq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X o Y$ 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.