Purity of generalized affine Springer fibers from generic planar curve singularities

Taiwang Deng; Tao Su

paper

Purity of generalized affine Springer fibers from generic planar curve singularities

arXiv:2509.20800

Abstract

We prove the cohomological purity of punctual Hilbert schemes of points on generic irreducible planar curve singularities, by constructing an explicit affine paving. Via their identification with generalized $G L_{N}$ -affine Springer fibers attached to the direct sum of the adjoint and standard representations, this establishes a new case of the purity conjecture for generalized affine Springer fibers. The combinatorics of the paving - cell indices and dimensions - are controlled by $(d n, d m)$ -Dyck paths extending results of Gorsky-Mazin-Oblomkov on compactified Jacobians. As a byproduct, we also give a simpler proof of their bijection between admissible $(d n, d m)$ -invariant subsets and $(d n, d m)$ -Dyck paths.

v2: 48 pages, 4 figures