Compactifications of the period space of Enriques surfaces Part I
Compactifications of the period space of Enriques surfaces Part I
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Enriques 曲面周期空间的紧化第一部分
DOI：
10.1007/bf02571372
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发表时间：
1991
影响因子：
0.8
通讯作者：
H. Sterk
中科院分区：
文献类型：
--
作者：
H. Sterk
本文是关于几何积分Enrique曲面周期空间紧化的两篇论文中的第一篇。这项工作的动机来自两个来源。一个是Shah[24]关于Enrique曲面的射影退化的工作。另一个是Looijenga[14，15]关于局部对称簇的新紧化技术的工作。Shah的论文可以被解释为处理模方面(它是根据Mumford的几何不变量理论[16]来表述的)，而Looijenga的工作涉及周期空间方面。因此，我们的最终目的是描述Enrique曲面的周期空间的紧致，这解释了Shah的结果。事实证明，这是Satake-Baly-Borel紧致化沿着描述‘特殊’Enrique曲面的周期的除数的闭包的归一化爆破(关于‘特殊’Enrique曲面的概念，见w)。我们使用的周期空间是D/F的形式，其中D是第IV型有界对称域，维度为10，F是某个算术群(见w，在Horikawa[11，12]和纳米比亚kawa[17]关于Enrique曲面的工作中也出现了这种情况。然而，应该指出的是，我们的小组比他们的小组小。这是因为我们希望考虑到Shah方法的特定几何背景，即，我们认为我们的Enrique曲面被赋予(几乎)极化(次数为2)，它将K3覆盖实现为Y‘，0=p1xP~或二次锥~cp3的分支双覆盖的最小分辨率。得到的两个空间之间映射的次数，27 9 17-31(w表示这样一个事实，即一个“一般的”Enrique曲面(在[2]的意义上)可以以27 9 17-31不同的方式给出(几乎)2次的极化。然后，我们期望这种几何背景至少部分地反映在周期空间([21，22]和[1])的Satake-Baly-Borel紧化中。我们
This paper is the first of two papers dealing with compactifications of the period space of Enriques surfaces which are of geometric intcrcst. The motivation for this work comes from two origins. One is the work of Shah [24] on projective degenerations of Enriques surfaces. The other is the work of Looijenga [14, 15] on new compactification techniques for locally symmetric varieties. Shah's paper can be interpreted as dealing with the moduli aspect (it is phrased in terms of Mumford's geometric invariant theory [16]), whereas Looijenga's work refers to the period space aspect. Our ultimate aim then is to describe a compactification of the period space of Enriques surfaces which accounts for Shah's results. This turns out to be a normalized blow-up of the Satake-Baily-Borel compactification along the closure of the divisor describing periods of'special'Enriques surfaces (see w for the notion of a'special'Enriques surface). The period space which we use is of the form D/F where D is a bounded symmetric domain of type IV and dimension 10 and F is some arithmetic group (see w Such a situation also occurred in the work of Horikawa [11, 12] and Namikawa [17] on Enriques surfaces. It should be pointed out, however, that our group is smaller than theirs. The reason is that we wish to take into account the specific geometric setting of Shah's approach ie, we think of our Enriques surfaces as being endowed with an (almost) polarization (of degree 2), which realizes the K3 cover as the minimal resolution of a branched double cover of Y', 0= p1 x P~ or a quadratic cone~ c p3. The degree of the resulting map between the two spaces, 27 9 17-31 (w expresses the fact that a'general'Enriques surface (in the sense of [2]) can be given, up to automorphisms, an (almost) polarization of degree 2 in 27 9 17-31 different ways. We then expect this geometric background to be at least partially reflected in, for instance, the Satake-Baily-Borel compactification of the period space ([21, 22] and [1]). We
影响因子：
3.1
作者：
I. Dolgachev;I. Dolgachev
通讯作者：
I. Dolgachev;I. Dolgachev