Bases for the homology groups of the Hilbert scheme of points in the plane

Abstract

Let Hilb d P 2 =Hilb d P 2 C denote the Hilbert scheme parametrizing 0-dimensional, length-d subschemes of the plane. The authors determine an additive basis for the Chow group A.(Hilb d P 2 ) consisting of classes of closures of locally closed subschemes of P 2 . A key feature of this basis is that it consists of cycle classes whose generic points correspond to reduced length-d subschemes, making it convenient for use in enumerative problems. This is in contrast to the basis given by G. Ellingsrud and S. A. Strømme that arises from a cell decomposition [Invent Math. 91 (1988), no. 2, 365–370]. The authors' method of proof involves showing that their basis specializes (via an intermediate basis) to that of Ellingsrud and Strømme. As applications of their main results and methods, the authors employ a G m -action on P 2 and the Ellingsrud-Strømme basis for A.(Hilb d P 2 ) to give a basis for the Chow group of the variety W ∗ of Schubert triangles. Their results agree with work of J. Roberts and R. Speiser [Comm. Algebra 15 (1987), no. 9, 1929–1966] and A. Collino and W. Fulton [Mém. Soc. Math. France (N.S.) No. 38 (1989), 75–117]. The paper concludes with enumerative applications using the authors' basis for A.(Hilb d P 2 ) to verify formulas conjectured by Schubert for the number of bitangent contacts between members of two families of moving plane curves.