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On the algebraicity about the Hodge numbers of the Hilbert schemes of algebraic surfaces

Part of: Arithmetic algebraic geometry Algebraic number theory: global fields Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  19 April 2022

Seokho Jin  and
Sihun Jo Open the ORCID record for Sihun Jo [Opens in a new window]
Seokho Jin
Affiliation:
Department of Mathematics, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul 06974, Republic of Korea (archimed@cau.ac.kr)
Sihun Jo
Affiliation:
Department of Mathematics Education, Woosuk University, 443 Samnye-ro, Samnye-eup, Wanju-Gun, Jeollabuk-do 55338, Republic of Korea (sihunjo@woosuk.ac.kr)
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Abstract

Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$. Our result gives a refinement of the algebraicity on Betti numbers.

Keywords

Hodge numberalgebraicityHilbert scheme

MSC classification

Primary: 11R04: Algebraic numbers; rings of algebraic integers
Secondary: 11G16: Elliptic and modular units 11F50: Jacobi forms
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 2 , May 2022 , pp. 392 - 403
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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