Intersections of projective varieties and generic projections
Intersections of projective varieties and generic projections
复制标题
投影簇和通用投影的交集
DOI：
10.1007/bf02678194
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发表时间：
1997
影响因子：
0.6
通讯作者：
M. Manaresi
中科院分区：
文献类型：
--
作者：
H. Flenner;M. Manaresi
设X，YC P~分别是n维和m维的闭子簇. Stiickrad和Vogel在[SVo]中证明了一个关于不适当交的Bezout定理，并在XNY上引入了维数为k的圈vk”= vk（X，Y），在X和Y的规则并簇J：= J（X，Y）上引入了维数为k的圈vk“= vk（X，Y），它们是通过一个简单的算法得到的.本文用一般投影Pk：pN~ pn+ mkl解释了这些循环。为此，我们引入了一个相对分歧轨迹R（Pk，X，Y）的Pk，这是最多的维数为k和一般的分歧循环的情况下，X= Y。我们证明了这个循环就是Vk，0< k< dimXCIY-1。此外，圈flk+1（对于-1 < k< dimXCtY-1）可以几何地解释为与规则连接J中的所有（x：y）的集合的闭包相关联的Pk的双点的圈，使得（pk（x）：Pk（Y））在j（pn+ n-1，p 'n + n-1）的对角线A~，+~ k_：中。
Let X, YC P~ be closed subvarieties of dimensions n and m respectively. Proving a Bezout theorem for improper intersections Stiickrad and Vogel [SVo] introduced cycles vk"= vk (X, Y) of dimension k on XNY and/~ k on the ruled join variety J:= J (X, Y) of X and Y which are obtained by a simple algorithm.. In this paper we give an interpretation of these cycles in terms of generic projections Pk: pN~ pn+ mkl. For this we introduce a relative ramification locus R (Pk, X, Y) of Pk which is of dimension at most k and generalizes the usual ramification cycle in the case X= Y. We prove that this cycle is just Vk for 0< k< dimXCIY-1. Moreover, the cycles flk+ l (for-1< k< dimXCtY-1) may be interpreted geometrically as the cycle of double points of Pk associated to the closure of the set of all (x: y) in the ruled join J such that (pk (x): Pk (Y)) is in the diagonal A~,+~ _k_: of j (pn+ mk-1, p'n+ mk-1).