On the homology of the Hilbert scheme of points in the plane

Abstract

Geir Ellingsrud 1 and Stein Arild Stromme 2 i Matematisk institutt, Universitetet i Oslo, Blindern, N-Oslo 3, Norway 2 Matematisk institutt, Universitetet i Bergen, N-5014 Bergen, Norway Although several authors have been interested in the Hilbert scheme Hilba(IP z) parametrizing finite subschemes of length d in the projective plane ([I1, I2, F 1, F2, Br] among others) not much is known about the topological properties of this space. The Picard group has been calculated I-F2], and the groups of Hilb3(Ip 2) have been computed [HI. In this paper we give a precise description of the additive structure of the of Hilba(Ip2), applying the results of Birula-Bialynicki [B1, B2] on the cellular decompositions de- fined by a torus action to the natural action of a maximal torus of SL(3) on Hilbe(IPZ). A rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the Chow groups and the groups are isomorphisms. In particular there is no odd homology, and the groups are all free. The main objective of this work is to compute their ranks: the Betti numbers of Hilba(Ipz). As a byproduct of our method we get similar results on the of the punctual Hilbert scheme and of the Hilbert scheme of points in the affine plane. It seems natural to generalize our results to any toric smooth surface. However, we give the results only for the rational ruled surfaces IF. with an indication of the necessary changes in the proofs. For simplicity we work over the field of complex numbers, but with an appropriate interpretation of the word homology our results remain valid over any base field.