Macdonald formula for curves with planar singularities

2011/07/12 by Davesh Maulik, Maulik, Davesh, Zhiwei Yun +1 · 6 citations
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #14F43 #14H20 #14H40 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1107.2175

openalex publication_date 2011/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes an essential use of Ngo's support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we generalize part of the Weil conjectures to the Hilbert-zeta function of curves with planar singularities.

Citations

• The deformation theory of local complete intersections

Cited by

• Cohomology of compactified Jacobians for locally planar integral curves
• Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces
• HOMFLY polynomials, stable pairs and motivic Donaldson-Thomas invariants
• Purity of generalized affine Springer fibers from generic planar curve singularities
• Abel maps for integral curves via a derived perspective
• On the Perversity of Chern Classes for Compactified Jacobians

Related

• Homology of Hilbert schemes of points on a locally planar curve
• On the geometry of punctual Hilbert schemes on singular curves and their motivic zeta functions
• Cohomology of compactified Jacobians for locally planar integral curves
• The Hilbert schemes of locally Cohen-Macaulay curves in P³ may after all be connected
• Motivic Hilbert zeta functions for curve singularities and related invariants