Relations for virtual fundamental classes of Hilbert schemes of curves on surfaces

Abstract

In [DKO] we constructed virtual fundamental classes $[[Hil{b}_V^m]]$ for Hilbert schemes of divisors of topological type m on a surface V, and used these classes to define the Poincare invariant of V: (P^+_V,P^-_V): H^2(V,Z) --> Λ^* H^1(V,Z) x Λ^* H^1(V,Z) We conjecture that this invariant coincides with the full Seiberg-Witten invariant computed with respect to the canonical orientation data. In this note we prove that the existence of an integral curve $C\subset V$ induces relations between some of these virtual fundamental classes $[[Hil{b}_V^m]]$. The corresponding relations for the Poincare invariant can be considered as algebraic analoga of the fundamental relations obtained in [OS].

13 pages