Relations for virtual fundamental classes of Hilbert schemes of curves   on surfaces
  In [DKO] we constructed virtual fundamental classes $[[ Hilb^m_V ]]$ 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) --> \Lambda^* H^1(V,Z) x \Lambda^* 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 $[[Hilb^m_V ]]$. The corresponding relations for the Poincare invariant can be considered as algebraic analoga of the fundamental relations obtained in [OS]. 
M. Duerr and Ch. Okonek