Purity of generalized affine Springer fibers from generic planar curve singularities Article Swipe

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Taiwang Deng , Tao Su ·

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2509.20800

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.

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2509.20800
PDF: https://arxiv.org/pdf/2509.20800
OA Status: green
OpenAlex ID: https://openalex.org/W4414789364

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DOI: https://doi.org/10.48550/arxiv.2509.20800

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Title: Purity of generalized affine Springer fibers from generic planar curve singularities

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-09-25

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Authors: Taiwang Deng, Tao Su

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Landing page: https://arxiv.org/abs/2509.20800

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