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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].

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].

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:May 2009
Deposited On:05 Nov 2009 13:36
Last Modified:06 Dec 2017 21:30
Publisher:De Gruyter
ISSN:1615-715X
Publisher DOI:https://doi.org/10.1515/ADVGEOM.2009.014
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2523841

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