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

Dürr, M; Okonek, C (2009). Relations for virtual fundamental classes of Hilbert schemes of curves on surfaces. Advances in Geometry, 9(2):219-231.

## Abstract

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].

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].

## Altmetrics

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English May 2009 05 Nov 2009 13:36 05 Apr 2016 13:31 De Gruyter 1615-715X 10.1515/ADVGEOM.2009.014 http://www.ams.org/mathscinet-getitem?mr=2523841
Permanent URL: http://doi.org/10.5167/uzh-23485