A Stratification of a Moduli Space of Abelian Varieties
A Stratification of a Moduli Space of Abelian Varieties
复制标题
阿贝尔簇模空间的分层
DOI：
10.1007/978-3-0348-8303-0_13
复制
发表时间：
2001
影响因子：
1.7
通讯作者：
F. Oort
中科院分区：
文献类型：
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作者：
F. Oort
本文研究了具有正特征维数的极化阿贝尔簇的模空间A0 Fp。我们构建了这个空间的分层。层是由p所杀的群概型的同构类来索引的;如果X [p]属于由某个离散不变量给出的同构类，则极化阿贝尔簇（X，A）在某个层中有其模点。我们通过N = X [p]的一个滤子的数值性质定义了这些不变量，从一个层到另一个层的边界感觉就像是“退化p-结构”。这些层都是准仿射的，这一事实允许我们继续这个过程，直到我们到达唯一的零维层，即超特殊轨迹。我们可以这样表述这个想法：普通轨迹有几个“边界”，一个是阿贝尔簇退化的地方，一个是p结构“变得更特殊”的地方（对于所有非零维层来说，这是一个类似的想法）。这种现象虽然在零特性中不存在，但在正特性中却很有效，并有很强的应用前景。
In this paper we study the moduli space A0 Fp of polarized abelian varieties of dimensiongin positive characteristic. We construct a stratification of this space. The strata are indexed by isomorphism classes of group schemes killed byp;a polarized abelian variety (X, A) has its moduli point in a certain stratum if X [p] belongs to the isomorphism class given by a certain discrete invariant. We define these invariants by a numerical property of a filtration ofN= X [p].Passing from one stratum to a stratum in its boundary feels like “degenerating the p-structure”. The fact that these strata are all quasi-affine allows us to keep going in this process until we arrive at the unique zero-dimensional stratum, the superspecial locus. One can formulate this idea by saying that the ordinary locus has several “boundaries”, one where the abelian variety degenerates, one where the p-structure “becomes more special” (and an analogous idea for all non-zero-dimensional strata). This phenomenon, non-present in this form in characteristic zero, but available and powerful in positive characteristic, is ex-pected to have many applications.