Intersection Multiplicities

William Fulton ()
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William Fulton: University of Michigan, Department of Mathematics

Chapter Chapter 7 in Intersection Theory, 1998, pp 119-129 from Springer

Abstract: Abstract As in Chap. 6, consider a fibre square $$ \begin{array}{*{20}{c}} W \to V \\ \downarrow {}{ \downarrow {}^f} \\ X{\mathop \to \limits_i }Y \end{array} $$ with i a regular imbedding of codimension d, V a k-dimensional variety. If Z is an irreducible component of W of dimension k — d, the intersection multiplicity i (Z, X·V; Y) is defined to be the coefficient of Z in the intersection class X·V ∈ A k-d (W). The intersection multiplicity is a positive integer, satisfying $$ i\left({Z,X \cdot V;Y} \right) \leqq length\left({{\upsilon _{z,w}}} \right) $$ Examples show that this inequality may be strict; equality holds, however, if $$ {\mathcal{O}_{z,v}} $$ is a Cohen-Macaulay ring. On the other hand, the criterion of multiplicity one asserts that i (Z, X·V;Y) is one precisely when $$ {\mathcal{O}_{z,v}} $$ is a regular local ring with maximal ideal generated by the ideal of X in Y. The standard properties of intersection multiplicities, worked out in the examples, follow from the basic properties of the general intersection product which were proved in Chapter 6.

Date: 1998
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DOI: 10.1007/978-1-4612-1700-8_8

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