A note on the unirationality of a moduli space of double covers
A note on the unirationality of a moduli space of double covers
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DOI：
10.1002/mana.201010060
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发表时间：
2011-12
影响因子：
1
通讯作者：
NN Iyer Jaya;S. Müller–Stach
中科院分区：
文献类型：
--
作者：
NN Iyer Jaya;S. Müller–Stach
在这篇注记中，我们研究了亏格为3条曲线的双覆盖的模空间\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3，2}$End{Document}，它们沿4个不同的点分支。Bardelli，Ciliberto和Verra在1中研究了这个空间。它允许一个控制态射\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3，2}\RIGROW{\数学A}_4$\END{Document}到Siegel空间。证明了作为两个\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal簇的乘积的群商，存在一个Grassmannian群的二元模型。这给出了\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3，2}$\end{Document}的单调性的一个证明，从而为\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal A}_4$\end{Document}的单调性提供了一个新的证明。
In this note we look at the moduli space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3,2}$\end{document} of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra in 1 . It admits a dominating morphism \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3,2} \rightarrow {\mathcal A}_4$\end{document} to Siegel space. We show that there is a birational model of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3,2}$\end{document} as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3,2}$\end{document} and hence a new proof for the unirationality of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal A}_4$\end{document}.