Minimal Siegel modular threefolds
Minimal Siegel modular threefolds
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最小西格尔模块三重
DOI：
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发表时间：
1995
影响因子：
0.8
通讯作者：
K. Hulek
中科院分区：
文献类型：
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作者：
V. Gritsenko;K. Hulek
本文的出发点是Sp 4（n）的子群Γt的极大扩张Γ*t，它与仿模群共轭.相应地，我们称商[Ascr ]*t=Γ*t <$2为最小三重西格尔模。空间[Ascr ]*t以及[Ascr ]t=Γt <$2（即（1，t）-极化交换曲面空间）与[Ascr ]*t之间的中间空间尚未得到详细的研究。利用Torelli定理证明了[Ascr ]*t可表示为（1，t）-极化交换曲面的库默曲面空间，且[Ascr ]*t上的某个2次商是格极化K3曲面的模空间.利用Γ*t在Jacobi型空间上的作用，我们证明了[Ascr ]t和[Ascr ]*t之间的许多空间具有非平凡的3-形式，即这些空间的科代拉维数是非负的.计算空间[Ascr ]*t本身的科代拉维数似乎是一个困难的问题。作为这个方向上的第一个必要步骤，我们确定有限映射[Ascr ]t→[Ascr ]*t的分歧轨迹的除数部分。这是Humbert曲面的并集，可以解释为Hilbert模曲面。
The starting point of this paper is the maximal extension Γ*t of Γt, the subgroup of Sp4(ℚ) which is conjugate to the paramodular group. Correspondingly we call the quotient [Ascr ]*t=Γ*tℍ2 the minimal Siegel modular threefold. The space [Ascr ]*t and the intermediate spaces between [Ascr ]t=Γtℍ2 which is the space of (1, t)-polarized abelian surfaces and [Ascr ]*t have not yet been studied in any detail. Using the Torelli theorem we first prove that [Ascr ]*t can be interpreted as the space of Kummer surfaces of (1, t)-polarized abelian surfaces and that a certain degree 2 quotient of [Ascr ]t which lies over [Ascr ]*t is a moduli space of lattice polarized K3 surfaces. Using the action of Γ*t on the space of Jacobi forms we show that many spaces between [Ascr ]t and [Ascr ]*t possess a non-trivial 3-form, i.e. the Kodaira dimension of these spaces is non-negative. It seems a difficult problem to compute the Kodaira dimension of the spaces [Ascr ]*t themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map [Ascr ]t→[Ascr ]*t. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces.