AN INTERSECTION NUMBER FOR THE PUNCTUAL HILBERT SCHEME OF A SURFACE

We compute the intersection number between two cycles A and B of compl ementary dimensions in the Hilbert scheme H parameterizing subschemes of given finite length n of a smooth projective surface S. The (n + 1) -cycle A corresponds 40 the set of finite closed subschemes the suppor t of which has cardinality 1. The (n - 1)-cycle B consists of the clos ed subschemes the support of which is one given point of the surface. Since B is contained in A, indirect methods are needed. The intersecti on number is A.B = (-1)(n-1)n, answering a question by H. Nakajima.