The Chow ring of punctual Hilbert schemes on toric surfaces

Let X be a smooth projective toric surface, and H(X) the Hilbert scheme parametrising the length d zero-dimensional subschemes of X . We compute the rational Chow ring A(H(X))Q. More precisely, if T ⊂ X is the twodimensional torus contained in X , we compute the rational equivariant Chow ring AT (H (X))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. Introduction LetX be a smooth projective surface and H(X) the Hilbert scheme parametrising the zero-dimensional subschemes of length d of X . The problem is to compute the rational cohomology H(H(X)). The additive structure of the cohomology is well understood. First, Ellingsrud and Strømme computed in [6] the Betti numbers bi(H (X)) when X is a plane or a Hirzebruch surface Fn. The Betti numbers bi(H (X)) for a smooth surface X were computed by Göttsche [11] who realised them as coefficients of an explicit power series in two variables depending on the Betti numbers of X . This nice and surprising organisation of the Betti numbers as coefficients of a power series was explained by Nakajima in terms of a Fock space structure on the cohomology of the Hilbert schemes [19]. Grojnowski announced similar results [12]. As to the multiplicative structure of H(H(X)), the picture is not as clear. There are descriptions valid for general surfaces X but quite indirect and unexplicit, and more explicit descriptions for some special surfaces X . The first steps towards the multiplicative structure were performed again by Ellingsrud and Strømme [7] (see also Fantechi-Göttsche [9]) . They gave an indirect description of the ring structure in the case X = P in terms of the action of the Chern classes of the tautological bundles. Explicitly, Ellingsrud and Strømme constructed a variety Y whose cohomology is computable, an embedding i : H(P) → Y , and proved the isomorphism H(H(P)) ≃ H(Y )/Ann(i∗(1)) where Ann is the annihilator.