Compositio Mathematica

Table of Contents - March 2012 - Volume 148, Issue 02  

Research Article

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

Vivek Shendea1

a1 Department of Mathematics, Princeton University, Princeton NJ, 08540, USA (email: vivek.vijay.shende@gmail.com)

Abstract

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.

(Received October 07 2010)

(Accepted July 29 2011)

(Online publication January 26 2012)

2010 Mathematics Subject Classification

  • 14H20