Cones and Segre Classes

If X is a proper subvariety of a variety Y, the Segre class (s X, Y) of X in Y is the class in A * X defined as follows: let C = C x Y be the normal cone to X in Y, P (C) the projectivized normal cone, p the projection from P (C) to X. Then $$ s\left( {X,Y} \right) = {\sum\limits_{i \geqslant 0} {{p_*}\left({{c_1}\left( \theta \right)\left( 1 \right)} \right)} ^i} \cap \left[ {p\left( C \right)} \right] $$ When Xis regularly imbedded in Y, C = N is a vector bundle, and $$ s\left( {X,Y} \right) = c{\left( N \right)^{ - 1}} \cap \left[ X \right] $$ These Segre classes have a fundamental birational invariance: if f: Y’ → Y is a birational proper morphism, and X’= f -1 (X), then s (X’, Y’) pushes forward to s (X, Y). The coefficient of [X] in s (X, Y) is the multiplicity of Y along X. Segre classes will be used in one of our later constructions of intersection products, and in several intersection formulas. This chapter contains the construction of Segre classes for general cones, and for general closed subschemes of a scheme. The birational invariance is a special case of a general proposition describing the behavior of Segre classes under proper push-forward and flat pull-back. Segre classes arise naturally in many areas of algebraic geometry. Some of these occurrences are discussed in the examples and in the last two sections, which are not required for later chapters.