A classification of toric varieties with few generators
A classification of toric varieties with few generators
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具有少量生成器的复曲面品种的分类
DOI：
10.1007/bf01830946
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发表时间：
1988
影响因子：
0.8
通讯作者：
P. Kleinschmidt
中科院分区：
文献类型：
--
作者：
P. Kleinschmidt
d维环变是环面T=(k*) A （k可以是C或任何代数闭域）的T不变补全的T不变子变。这种变化可以用扇形来描述，扇形是R d中由整数格点张成的锥的有限系统。近年来，环面变分理论已被证明对解决由组合学和凸性理论引起的问题是有用的。另一方面，凸多面体理论的方法已被应用于特殊环形品种[4]、[5]、[10]、[14]的分类。在b[10]中，所有具有多达8个“发生器”的三维光滑环形品种都被分类。本文给出了具有任意维数但产生子相对较少的完全光滑环变的分类。我们认为，我们的线性变换方法，结合（定向）拟阵理论，将产生更复杂的分类品种。通过我们的分类得到的品种可以被认为是Hirzebruch-varieties[8]的自然推广。使用[4]中的判据，我们将证明我们的变种都是投影的。利用风图理论[7]对我们的品种进行了分类，这些品种是Fano品种。
A d-dimensional toric variety is a T-invariant subvariety of a T-invariant completion of the torus T=(k*) a (k can be C or any algebraically closed field). Such a variety can be described by a fan, a finite system of cones in R d spanned by integer lattice points. Recently, the theory of toric varieties has proved useful for the solution of problems originating from combinatorics and the theory of convexity [12],[13]. On the other hand, methods from the theory of convex polytopes have been applied to the classification of special toric varieties [4],[5],[10],[14]. In [10] all 3-dimensional smooth toric varieties with up to 8" generators" have been classified. In this paper a classification of complete smooth toric varieties with arbitrary dimension but relatively few generators is obtained. We think that our approach of linear transforms, combined with the theory of (oriented) matroids, will yield a classification of more complicated varieties. The varieties obtained by our classification can be thought of as natural generalizations of Hirzebruch-varieties [8]. Using a criterion from [4] we will show that our varieties are all projective. Using the theory of Gale-diagrams [7] those of our varieties which are Fano varieties will be classified.