Wonderful Compactification of an Arrangement of Subvarieties
Wonderful Compactification of an Arrangement of Subvarieties
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亚品种排列的精彩紧凑化
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Li Li
中科院分区：
文献类型：
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作者：
Li Li
本文的目的是定义所谓的亚簇排列的奇紧化，证明它的期望性质，给出一系列爆破的构造，并讨论爆破的执行顺序。在(具有任意特征的)代数闭域上固定一个非奇异代数簇Y。S亚种的排列是非奇异亚种的有限集合，使得S中亚种的所有非空图式理论交又在S中，或者等价地，使得任意两个亚种干净地相交，并且交集要么是空的，要么是这个集合中的一个亚种(见定义2.1)。设S是Y的子种的排列，子集G⊆S称为S的构建集，如果对所有S∈S G，{G∈G：G⊇S}中的极小元横截相交且交为S，则称G为构建集，如果G中所有可能的子种的交形成一个排列S(称为G的诱导排列)，G是S的构建集(见定义2.2)。对于任一建筑集G，G的奇紧化定义如下。定义1.1。设G是非空建筑集，Y◦=Y∖⋃G∈G。自然局部闭嵌入Y◦↪→∏的像的闭包。
The purpose of this paper is to define the so-called wonderful compactification of an arrangement of subvarieties, to prove its expected properties, to give a construction by a sequence of blow-ups, and to discuss the order in which the blow-ups can be carried out. Fix a nonsingular algebraic variety Y over an algebraically closed field (of arbitrary characteristic). An arrangement of subvarieties S is a finite collection of nonsingular subvarieties such that all nonempty scheme-theoretic intersections of subvarieties in S are again in S or, equivalently, such that any two subvarieties intersect cleanly and the intersection is either empty or a subvariety in this collection (see Definition 2.1). Let S be an arrangement of subvarieties of Y. A subset G ⊆ S is called a building set of S if, for all S ∈ S G, the minimal elements in {G∈ G : G ⊇ S} intersect transversally and the intersection is S. A set of subvarieties G is called a building set if all the possible intersections of subvarieties in G form an arrangement S (called the induced arrangement of G) and G is a building set of S (see Definition 2.2). For any building set G, the wonderful compactification of G is defined as follows. Definition 1.1. Let G be a nonempty building set and Y ◦ = Y ∖⋃G∈G G. The closure of the image of the natural locally closed embedding Y ◦ ↪→ ∏