An intersection number for the punctual Hilbert scheme of a surface

We compute the intersection number between two cycles $A$ and $B$ of complementary 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 to the set of finite closed subschemes the support of which has cardinality 1. The $(n-1)$-cycle $B$ consists of the closed subschemes the support of which is one given point of the surface. Since $B$ is contained in $A$, indirect methods are needed. The intersection number is $A.B=(-1)^{n-1}n$, answering a question by H. Nakajima.