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Purity of generalized affine Springer fibers from generic planar curve singularities

Taiwang Deng, Tao Su

TL;DR

The paper proves cohomological purity for punctual Hilbert schemes of generic irreducible planar curve singularities by constructing an explicit affine paving, and identifies these schemes with generalized GL_{dn}-affine Springer fibers attached to the adjoint plus standard representations. The paving is controlled by $(dn,dm)$-Dyck paths, extending GMO’s results for compactified Jacobians, and two complementary bijections between admissible $(dn,dm)$-invariant subsets and $(dn,dm)$-Dyck paths are established via a streamlined, sweep-map-compatible approach. The authors connect these geometric results to the Oblomkov–Rasmussen–Shende and Cherednik–Danilenko–Philipp conjectures through the Hilbert $L$-function and perverse filtration, and provide a clear combinatorial realization of cell indices and dimensions for the paving. A new combinatorial take yields simpler constructions of the key bijections and clarifies the relationship between Dyck-path data and invariant subsets, with potential independent interest beyond this specific geometric context.

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 $GL_N$-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 $(dn,dm)$-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 $(dn,dm)$-invariant subsets and $(dn,dm)$-Dyck paths.

Purity of generalized affine Springer fibers from generic planar curve singularities

TL;DR

The paper proves cohomological purity for punctual Hilbert schemes of generic irreducible planar curve singularities by constructing an explicit affine paving, and identifies these schemes with generalized GL_{dn}-affine Springer fibers attached to the adjoint plus standard representations. The paving is controlled by -Dyck paths, extending GMO’s results for compactified Jacobians, and two complementary bijections between admissible -invariant subsets and -Dyck paths are established via a streamlined, sweep-map-compatible approach. The authors connect these geometric results to the Oblomkov–Rasmussen–Shende and Cherednik–Danilenko–Philipp conjectures through the Hilbert -function and perverse filtration, and provide a clear combinatorial realization of cell indices and dimensions for the paving. A new combinatorial take yields simpler constructions of the key bijections and clarifies the relationship between Dyck-path data and invariant subsets, with potential independent interest beyond this specific geometric context.

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 -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 -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 -invariant subsets and -Dyck paths.

Paper Structure

This paper contains 28 sections, 37 theorems, 274 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Planar curve singularities
  4. Hilbert schemes of points
  5. Semigroup and semimodules of valuations
  6. (Local) compactified Jacobians
  7. The (local) Abel--Jacobi map
  8. Affine paving and purity
  9. The affine paving for the local compactified Jacobian
  10. Cell decomposition via semimodules of valuations
  11. Alternative description via Schubert cells
  12. Admissibility and affine paving
  13. The affine paving for the Hilbert scheme of points
  14. Cell indices and dimensions via $(dn,dm)$-Dyck paths
  15. Admissible $(dn,dm)$-invariant subsets revisited
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $(C,0)$ be a generic irreducible planar curve singularity: Then $\mathrm{Hilb}^{[\bullet]}(C,0) \cong \mathrm{Spr}_{\mathfrak{v}}$ admits an explicit affine paving: In particular, its (co)homology is pure.

Figures (3)

  • Figure 1: The correspondence between $(y_{i,j})\in \mathcal{Y}(dn,dm)$ and a $(dn,dm)$-Dyck path. Here $(n,m,d)=(2,3,2)$. The box $z$ has top-left vertex $(3,2)$, with $(\mathfrak{a},\mathfrak{l})(z)=(1,2)$ and $\mathrm{rk}(z)=5$.
  • Figure 3: The enhanced rank for $D_k$, $1\leq k\leq 23$, with $(n,m,d) = (2,3,2)$. Left figure: $k\neq 5$; Right figure: $k=5$.
  • Figure 4: The box $z = (p,y) \in \mathcal{R}_{dn,dm}$ and the pair $(p,y_p)$, $(x,y)$ with $y_x < y \leq y_{x+1}$ determine each other. The red paths correspond to part of the Dyck path. Left: $x<p$, $\Leftrightarrow$$z\in D$; Right: $x>p$, $\Leftrightarrow$$z\in \mathcal{R}_{dn,dm}\setminus D$.

Theorems & Definitions (113)

  • Conjecture 1.1
  • Theorem 1.2: Cor. \ref{['cor:affine_paving_for_Hilbert_scheme']}, Thm. \ref{['thm:affine_paving_for_Hilbert_scheme_over_generic_curve_singularity']}
  • Conjecture 1.3: see ORS18 or Conjecture \ref{['conj:ORS_conjecture']}
  • Conjecture 1.4: CD16CP18
  • Conjecture 1.5: Conj. \ref{['conj:combinatorial_two-variable_CDP_conjecture_for_generic_curve_singularity']}
  • Definition 2.1: GMO25
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: GK23
  • Definition 2.5
  • ...and 103 more