论文标题
Igusa四分之一和Prym图，带有一些有理模量
The Igusa quartic and the Prym map, with some rational moduli
论文作者
论文摘要
在本文中，Igusa四分之一$ b \ subset \ mathbb p^4 $再次出现的无处不在，这次与prym Map $ \ Mathfrak P：\ Mathcal R_6 \ to \ Mathcal A_5 $有关。我们介绍了那些四倍的三倍$ x $切割$ b $的$ x $ $ x $的模量$ \ mathcal x $。一般的$ x $是$ 30 $ - 诺达尔，其自然降低的中级雅各布$ j（x）$是$ 5 $维的P.P.阿贝里安品种。令$ \ frak j：\ Mathcal x \ to \ Mathcal A_5 $为期间地图，将$ x $发送到$ j（x）$，在论文中，我们研究$ \ frak j $及其与$ \ frak p $的关系。众所周知，$ \ frak p $的学位为$ 27 $，其单莫罗莫（Monodromy Group）赋予了任何光滑的纤维$ f $ $ \ frak p $，其发病率配置为$ 27 $的立方体表面。然后，相同的单片定义了一张地图$ \ mathfrak j'：\ Mathcal d_6 \ to \ Mathcal a_5 $ 36 $，带有纤维的配置$ 36 $'double-Six'一组立方体表面。我们证明$ \ frak j = \ frak j'\ circ ϕ $，其中$ ϕ：\ mathcal x \ to \ mathcal d_6 $是birational。这提供了$ \ frak j'$的几何描述。最后，我们描述了所考虑的不同模量空间之间的关系，并证明包括$ \ Mathcal X $在内的某些空间是理性的。
In this paper the ubiquity of the Igusa quartic $B \subset \mathbb P^4$ shows up again, this time related to the Prym map $\mathfrak p : \mathcal R_6 \to \mathcal A_5$. We introduce the moduli space $\mathcal X$ of those quartic threefolds $X$ cutting twice a quadratic section of $B$. A general $X$ is $30$-nodal and the intermediate Jacobian $J(X)$ of its natural desingularization is a $5$-dimensional p.p. abelian variety. Let $\frak j: \mathcal X \to \mathcal A_5$ be the period map sending $X$ to $J(X)$, in the paper we study $\frak j$ and its relation to $\frak p$. As is well known the degree of $\frak p$ is $27$ and its monodromy group endows any smooth fibre $F$ of $\frak p$ with the incidence configuration of $27$ lines of a cubic surface. Then the same monodromy defines a map $ \mathfrak j': \mathcal D_6 \to \mathcal A_5$ of degree $36$, with fibre the configuration of $36$ 'double-six' sets of lines of a cubic surface. We prove that $\frak j = \frak j' \circ ϕ$, where $ϕ: \mathcal X \to \mathcal D_6$ is birational. This provides a geometric description of $\frak j'$. Finally we describe the relations between the different moduli spaces considered and prove that some, including $\mathcal X$, are rational.
