G-invariant Hilbert Schemes on Abelian Surfaces and Enumerative Geometry of the Orbifold Kummer Surface

2020/11/27 by Stephen Pietromonaco, Pietromonaco, Stephen
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2011.14020

openalex publication_date 2020/11/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

For an Abelian surface A with a symplectic action by a finite group G, one can define the partition function for G-invariant Hilbert schemes Z_{A, G}(q) = ∑_d=0^∞ e(Hilb^d(A)^G)q^d. We prove the reciprocal Z_A,G^-1 is a modular form of weight (1)/(2)e(A/G) for the congruence subgroup Γ₀(|G|), and give explicit expressions in terms of eta products. Refined formulas for the χ_y-genera of Hilb(A)^G are also given. For the group generated by the standard involution τ: A → A, our formulas arise from the enumerative geometry of the orbifold Kummer surface [A/τ]. We prove that a virtual count of curves in the stack is governed by χ_y(Hilb(A)^τ). Moreover, the coefficients of Z_{A, τ} are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.

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