Quotients of toric varieties
Quotients of toric varieties
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DOI：
10.1007/bf01459264
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发表时间：
1991-03
影响因子：
1.4
通讯作者：
M. Kapranov;B. Sturmfels;A. Zelevinsky
中科院分区：
文献类型：
--
作者：
M. Kapranov;B. Sturmfels;A. Zelevinsky
代数几何中的一个重要问题是通过作用在X上的代数群H来构造复射影簇X的商，Mumford[13]发展的几何不变量理论(GIT)给出了这个问题的一种解决方案。Git-商定义为分次坐标环C[X]中H-不变量子环的投射谱。集理论上，这个商由X上半稳定H-轨道的等价类组成。GIT的一个缺点是，半稳定的概念和由此产生的商取决于X上H-作用的线性化[13，1.3节]的选择：一般来说，C[X]上有不止一个H-作用扩展了X上的给定作用。因此，我们得到了许多不同的Git-商，它们都不是先验的优先于其他的。在这篇文章中，当X是环面簇，H是其定义环面的子环面时，我们给出了这个问题的一个解决方案。为了解释我们的构造，让我们假设X是射影的。X中每个轨道的闭包Hx是投影子簇，当x在通用点的开H-不变子集UCX内时，这些簇将具有相同的维度和次数。设~g是X中具有这些参数的所有代数圈的Chow簇([10-12，15])。赋值x~H-~定义了拓扑商U/H到射影簇Rg中的嵌入。我们定义周商X//H为集合的闭包
An important problem in algebraic geometry is to construct the quotient of a complex projective variety X by an algebraic group H acting on X. One solution to this problem is provided by geometric invariant theory (GIT) as developed by Mumford [13]. The GIT-quotient is defined as the projective spectrum of the subring of H-invariants in the graded coordinate ring C [X]. Set-theoretically, this quotient consists of the equivalence classes of semistable H-orbits on X. It is a certain drawback of GIT that the concept of semistability and the resulting quotient depend upon the choice of a linearization [13, Sect. 1.3] of the H-action on X: in general, there is more than one H-action on C [X] which extends the given action on X. So, we get a number of different GIT-quotients neither of which is a priori preferable to the others. In this article we suggest a solution to this problem for the case when X is a toric variety and H is a subtorus of its defining torus. To explain our construction, let us assume that X is projective. The closure Hx of each orbit in X is a projective subvariety, and, as x ranges within an open H-invariant subset UCX of generic points, these varieties will have the same dimension and degree. Let~ g be the Chow variety of all algebraic cycles in X having these parameters (of.[10-12, 15]). The assignment x~ H-~ defines an embedding of the topological quotient U/H into the projective variety rg. We define the Chow quotient X//H to be the closure of the set