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

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 special case of a conjecture of Oblomkov and Shende identifies the Euler numbers of the Hilbert schemes with the "U(infinity)" invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.

16 pages

References in corpus (1)

• Donaldson-Thomas invariants via microlocal geometry

Cited by in corpus (20)

• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link
• 13/2 ways of counting curves
• DAHA and iterated torus knots
• Refined curve counting on complex surfaces
• A support theorem for the Hitchin fibration: the case of $S{L}_n$
• Compactified Jacobians and q,t-Catalan Numbers, I
• Macdonald formula for curves with planar singularities
• A support theorem for Hilbert schemes of planar curves, II
• Sheaves of maximal intersection and multiplicities of stable log maps
• On the Göttsche Threshold
• HOMFLY polynomials, stable pairs and motivic Donaldson-Thomas invariants
• Log BPS numbers of log Calabi-Yau surfaces
• Homology of Hilbert schemes of points on a locally planar curve
• Invariants for non-reductive group actions
• The multiplicity of generic normal surface singularities
• Counting rational curves on K3 surfaces with finite group actions
• The Kleiman-Piene Conjecture and node polynomials for plane curves in ${\mathbb{P}}^3$
• A support theorem for nested Hilbert schemes of planar curves
• Superpolynomials of algebraic links
• Homology of Hilbert schemes of reducible locally planar curves