In [Dürr, Kabanov, Okonek, Topology 46: 225–294, 2007] we constructed virtual fundamental classes for Hilbert schemes of divisors of topological type m on a surface V, and used these classes to define the Poincaré invariant of V:
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 V induces relations between some of these virtual fundamental classes . The corresponding relations for the Poincaré invariant can be considered as algebraic analoga of the fundamental relations obtained in [Ozsváth, Szabó, Ann. of Math. 151: 93–124, 2000].