Intersection Theory on Quotients of Algebraic Varieties
R. E. Briney · American Journal of Mathematics · 1962
Introduction. Given two cycles Z, Z' carried by an ambient algebraic variety X, we can assign to each component C of their intersection which is simple on X a well-defined multiplicity (C;Z Z')x. However, no general definition has yet been given for multiplicities of intersection at components which are singular on the ambient variety. Certain special cases have been studied: for example, if X is projectively embedded, Samuel ([11], ch. II, 6, no. 5) has defined intersection multiplicities for those cycles which locally are complete intersections of X with cycles in the projective space. This applies in particular to the intersection of two generators of a quadratic cone in 3-space. In the present paper, we study another particular case of this problem. Let a finite group g act as a group of biregular transformations on a nonsingular variety X, and assume that the quotient space Y =X/ is a variety. If elements of g other than the identity have fixed points on X, the variety Y will in general have singularities. We shall define an intersection multiplicity (C; Z Z') y for any component C of the proper dimension, dim Z + dim Z' -dimY; when C is simple on Y, our definition will coincide with the usual one. We shall in fact establish a global intersection theory, not only on quotients of nonsingular varieties but on varieties of a possibly more general type-those which result from nonsingular varieties after a finite number of repetitions of the process of forming quotients. The precise result is stated as Theorem 3. T. In defining intersection multiplicities on quotient varieties, we shall be forced to use cycles with rational coefficients and to allow rational multiplicities. The example of the two generators on a ouadratic cone ([11. loc. cit.)