Complete Symmetric Varieties II Intersection theory
Complete Symmetric Varieties II Intersection theory
复制标题
完全对称簇 II 交集理论
DOI：
10.2969/aspm/00610481
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发表时间：
1985
期刊：
影响因子：
--
通讯作者：
C. Procesi
中科院分区：
文献类型：
--
作者：
C. Concini;C. Procesi
本文是我们“完全对称变”[5]的延续。本文给出了一种适用于求解对称齐次空间上一般枚举问题的方法。在经典的圆锥数列理论中，研究了以下一类问题。有人给出了圆锥曲线的一个“条件”，即经过一点的条件，与一条直线相切的条件，与给定族中的曲线相切的条件等。满足这类条件的二次曲线的集合称为代数子变量，它的余维称为条件的维数。如果给定若干独立条件，使其维数之和等于5（二次曲线空间的维数），则满足给定条件的二次曲线集合是有限的，计算其基数是二次曲线列举几何的主要问题。当然，这里我们讨论的是二次曲线的例子，但至少对于一般的方法，一般的齐次变量适用于讨论。Chasles， Schubert经典使用的方法是构造一个带有一组条件的代数；将任何条件计算为基本条件的线性组合，并将任何枚举问题简化为涉及基本条件的问题。例如，Chasles计算与5个普通圆锥相切的圆锥数为。遵循。他证明了与圆锥曲线相切的条件是。2(a+p)，其中a是“经过一个点”，p是“与一条直线相切”，那么请求的数字是25(a+p)5，每个单项aip5-i可以通过直接的几何参数计算。条件等价的思想是由数守恒原理证明的，我们通过这个原理把一个枚举问题转化为具有相同解数的另一个问题。关于这一理论的进一步评论和信息，我们参考克莱曼的论述。要将这一套思想形式化并不难，尽管我们将会看到，
This paper is a continuation of our "Complete symmetric varieties" [5]. We explain here a method suitable to solve general enumerative problems on a symmetric homogeneous space. In the classical enumerative theory of conics the following class of problems was studied. One gives a "condition" on a conic, i.e. the condition of passing through a point, being tangent to a line, being osculating to a curve in a given family etc. The set of conics satisfying this type of conditions will be an algebraic subvariety, its codimension is called the dimension of the condition. If one imposes a number of independent conditions, such that the sum of their dimensions is equal to 5 (the dimension of the space of conics), then the set of conics satisfying the given conditions is finite and the main question of enumerative geometry of conics to compute its cardinality. Of course here we are talking about conics as a way of example but at least for this general approach, a general homogeneous variety fits in the discussion. The method of Chasles, Schubert used classically is to construct an algebra with the set of conditions; compute any condition as the linear combination of basic ones and reduce any enumerative problem to the ones involving the basic conditions. For instance Chasles computes the number of conics tangent to 5 general conics as. follows. He proves that the condition of being tangent to a conic is. 2( a + p)where a is "passing through a point", p is "being tangent to a line" the number requested is then 25(a+p)5 and each monomial aip5-i can be computed by direct geometric arguments. The idea of equivalence of conditions is justified by the principle of conservation of number, we transform an enumerative problem into another one which has the same number of solutions through this principle. We refer to Kleiman's treatment for further comments and informations on this theory [12]. It is not hard to formalize this set of ideas although, as we shall see,