The Chow ring of punctual Hilbert schemes on toric surfaces

Abstract

Let X be a smooth projective toric surface, and ${\mathbb H}^d(X)$ the Hilbert scheme parametrizing the length d zero-dimensional subschemes of X. We compute the rational Chow ring $A^*({\mathbb H}^d(X))_{\mathbb Q}$. More precisely, if $T\subset X$ is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring $A_T^*({\mathbb H}^d(X))_{\mathbb Q}$ and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.