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Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

Part of: Curves

Published online by Cambridge University Press:  26 January 2012

Vivek Shende
Vivek Shende*
Affiliation:
Department of Mathematics, Princeton University, Princeton NJ, 08540, USA (email: vivek.vijay.shende@gmail.com)
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Abstract

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Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande–Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar–Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a conjecture of Oblomkov and the present author identifies the Euler numbers of the Hilbert schemes with the ‘U(∞)’ invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.

MSC classification

Secondary: 14H20: Singularities, local rings
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 531 - 547
Copyright
Copyright © Foundation Compositio Mathematica 2012

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