2010/09/30 by Vivek Shende
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Computer science #Conjecture #Context (archaeology) #Euler's formula #Geometry #Hilbert scheme #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Planar #Point (geometry) #Pure mathematics #Simple (philosophy) #Singular point of a curve #Singularity #math.AG
paper · pdf · doi:10.1112/s0010437x11007378
published as Compositio Math. 148 (2012) 531-547 · 16 pages
arxiv created 2011/09/01 · openalex publication_date 2012/01/26 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.