Bases of the Homology Spaces of the Hibert Scheme of Points in an Algebraic Surface

Abstract

For a complex surface S , proper, smooth and connected, the authors find two bases of the spaces of rational homology H n (Hilb d S) Q of the Hilbert scheme of subschemes of S of length d . The idea of the proof of the main theorem is to prove that the elements of the two candidates have as cardinalities the known Betti numbers of Hilb d S and to show that both intersect in a triangular matrix of nonzero diagonal entries. Papers on the subject which have a close connection with the present one are by B. Fantechi [Base of the homology groups of the Hilbert scheme of points on a surface'', Preprint; per bibl.] and L. Gottsche [Math. Ann. 286 (1990), no. 1-3, 193–207 ].