FA-modules of holomorphic forms on M_g,n

2025/09/10 by Samir Canning, Canning, Samir, Hannah Larson +5
Mathematics · #14H10 #14H15 #14J15 #18G85 #32G15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2509.08774

openalex publication_date 2025/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For fixed genus g and varying finite marking set A, the gluing and forgetful maps give the spaces of holomorphic forms on the moduli space of stable A-marked curves of genus g has the structure of an FA-module, i.e., a functor from the category of finite sets to vector spaces. We prove that the resulting FA-modules of holomorphic k-forms are simple, for k less than or equal to 18, whenever they are nonzero. Conditional upon the conjectured vanishing of holomorphic 19-forms and 20-forms in genus 3, for 15 and 16 marked points, respectively, this extends to k less than or equal to 20.

Citations

• The Kodaira classification of the moduli space of pointed curves in genus 3
• A Serre spectral sequence for the moduli space of tropical curves
• Holomorphic differential forms on moduli spaces of stable curves
• Siegel modular forms of degree two and three and invariant theory
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Related

• Holomorphic differential forms on moduli spaces of stable curves
• Holomorphic forms and non-tautological cycles on moduli spaces of curves
• Holomorphic 1-forms on some coverings of the moduli space of curves
• Stability in the homology of Deligne-Mumford compactifications
• Cycles of finite type