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

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 Added by Markus D\\\"urr
 Publication date 2004
  fields
and research's language is English




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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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