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G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface

G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer... 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 ZA,G(q)=∑d=0∞e(Hilbd(A)G)qd.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} Z_{A, G}(q) = \sum _{d=0}^{\infty } e(\text {Hilb}^{d}(A)^{G})q^{d}. \end{aligned}$$\end{document}We prove the reciprocal ZA,G-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Z_{A,G}^{-1}$$\end{document} is a modular form of weight 12e(A/G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{1}{2}e(A/G)$$\end{document} for the congruence subgroup Γ0(|G|)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _{0}(|G|)$$\end{document} and give explicit expressions in terms of eta products. Refined formulas for the χy\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi _{y}$$\end{document}-genera of Hilb(A)G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text {Hilb}(A)^{G}$$\end{document} are also given. For the group generated by the standard involution τ:A→A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau : A \rightarrow A$$\end{document}, our formulas arise from the enumerative geometry of the orbifold Kummer surface [A/τ]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[A/\tau ]$$\end{document}. We prove that a virtual count of curves in the stack is governed by χy(Hilb(A)τ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi _{y}(\text {Hilb}(A)^{\tau })$$\end{document}. Moreover, the coefficients of ZA,τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Z_{A, \tau }$$\end{document} are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
eISSN
2197-9847
DOI
10.1007/s40687-021-00298-9
Publisher site
See Article on Publisher Site

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 ZA,G(q)=∑d=0∞e(Hilbd(A)G)qd.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} Z_{A, G}(q) = \sum _{d=0}^{\infty } e(\text {Hilb}^{d}(A)^{G})q^{d}. \end{aligned}$$\end{document}We prove the reciprocal ZA,G-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Z_{A,G}^{-1}$$\end{document} is a modular form of weight 12e(A/G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{1}{2}e(A/G)$$\end{document} for the congruence subgroup Γ0(|G|)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _{0}(|G|)$$\end{document} and give explicit expressions in terms of eta products. Refined formulas for the χy\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi _{y}$$\end{document}-genera of Hilb(A)G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text {Hilb}(A)^{G}$$\end{document} are also given. For the group generated by the standard involution τ:A→A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau : A \rightarrow A$$\end{document}, our formulas arise from the enumerative geometry of the orbifold Kummer surface [A/τ]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[A/\tau ]$$\end{document}. We prove that a virtual count of curves in the stack is governed by χy(Hilb(A)τ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi _{y}(\text {Hilb}(A)^{\tau })$$\end{document}. Moreover, the coefficients of ZA,τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Z_{A, \tau }$$\end{document} are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Mar 1, 2022

References