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

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

2004-08-10

arxiv.org/abs/math/0408130v1

Mathematics

Algebraic Geometry

13 pages

Scientific paper

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].

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