Homology of Hilbert schemes of points on a locally planar curve

2013/08/19 by Jørgen Vold Rennemo, Rennemo, Jørgen Vold · 3 citations
Mathematics · #14C05 #14H20 #14H40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1308.4104

openalex publication_date 2013/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let C be a proper, integral, locally planar curve, and consider its Hilbert schemes of points C^[n]. We define 4 creation/annihilation operators acting on the rational homology groups of these Hilbert schemes and show that the operators satisfy the relations of a Weyl algebra. The action of this algebra is similar to that defined by Grojnowski and Nakajima for a smooth surface. As a corollary, we compute the cohomology of C^[n] in terms of the cohomology of the compactified Jacobian of C together with an auxiliary grading on the latter. This recovers and slightly strenghtens a formula recently obtained in a different way by Maulik and Yun and independently Migliorini and Shende.

Cited by

• Cohomology of compactified Jacobians for locally planar integral curves
• Birational Chow-Künneth decompositions
• Abel maps for integral curves via a derived perspective

Related

• Homology of Hilbert schemes of reducible locally planar curves
• Macdonald formula for curves with planar singularities
• Generating series in the cohomology of Hilbert schemes of points on surfaces
• Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces
• Cohomology of compactified Jacobians for locally planar integral curves