Hilbert schemes of elliptic surfaces: group actions and derived categories

2025/08/12 by David Zhiyuan Bai, Bai, David Zhiyuan
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2508.09065

openalex publication_date 2025/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X→ C be an elliptic surface with integral fibers and a section. The Hilbert scheme X^[n] fibers over C^[n]. We construct a commutative group scheme over the entire base C^[n] that embeds as an open subscheme of the Hilbert scheme, such that its action on itself extends to the entirety of X^[n]. We show that the action is δ-regular in the sense of Ngô. Using the derived McKay correspondence, we construct an exact autoequivalence of D^bCoh(X^[n]) whose kernel is a maximal Cohen-Macaulay sheaf on the fiber product. We show that this Fourier-Mukai transform intertwines with our group action, i.e. theorem of the square holds. We also discuss the case without a section using the theory of Tate-Shafarevich twists.

Citations

• Hitchin fibrations are Ngô fibrations
• The Néron model of a higher-dimensional Lagrangian fibration
• Ngô support theorem and polarizability of quasi-projective commutative group schemes
• Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles
• Hilbert schemes as moduli of Higgs bundles and local systems
• Autoduality of compactified Jacobians for curves with plane singularities
• Good moduli spaces for Artin stacks
• Sur le Théorème Principal de Zariski en Géométrie Algébrique et Géométrie Analytique
• Equivariant autoequivalences for finite group actions
• The compactified Picard scheme of the compactified Jacobian
• Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
• Moduli spaces of stable sheaves on abelian surfaces
• Fourier-Mukai transforms for elliptic surfaces

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• Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme
• Toroidal analogues of the Grothendieck-Springer map
• The Euclid-Fourier-Mukai algorithm for elliptic surfaces
• Fourier-Mukai transforms for K3 and elliptic fibrations