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October 2014 Numerical Godeaux surfaces with an involution in positive characteristic
Soonyoung Kim
Proc. Japan Acad. Ser. A Math. Sci. 90(8): 113-118 (October 2014). DOI: 10.3792/pjaa.90.113

Abstract

A numerical Godeaux surface $X$ is a minimal surface of general type with $\chi(\mathcal{O}_{X})=K_{X}^{2}=1$. Over $\mathbf{C}$ such surfaces have $p_{g}(X)=h^{1}(\mathcal{O}_{X})=0$, but $p_{g}=h^{1}(\mathcal{O}_{X})=1$ also occurs in characteristic $p>0$. Keum and Lee~[9] studied Godeaux surfaces over $\mathbf{C}$ with an involution, and these were classified by Calabri, Ciliberto, and Mendes Lopes~[4]. In characteristic $p\ge 5$, we obtain the same bound $|\mathrm{Tors}\,X|\le 5$ as in characteristic 0, and we show that the quotient $X/\sigma$ of $X$ by its involution is rational, or is birational to an Enriques surface. Moreover, we give explicit examples in characteristic 5 of quintic hypersurfaces $Y$ with an action of each of the group schemes $G$ of order 5, and having extra symmetry by $\mathrm{Aut}\,G\cong\mathbf{Z}/4\mathbf{Z}$, hence by the \textit{holomorph} $H_{20}=\mathrm{Hol}\,G=G\rtimes\mathbf{Z}/4\mathbf{Z}$ of $G$.

Citation

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Soonyoung Kim. "Numerical Godeaux surfaces with an involution in positive characteristic." Proc. Japan Acad. Ser. A Math. Sci. 90 (8) 113 - 118, October 2014. https://doi.org/10.3792/pjaa.90.113

Information

Published: October 2014
First available in Project Euclid: 3 October 2014

zbMATH: 1338.14042
MathSciNet: MR3266744
Digital Object Identifier: 10.3792/pjaa.90.113

Subjects:
Primary: 14J29

Keywords: action of group scheme , Godeaux surface , involution , positive characteristic

Rights: Copyright © 2014 The Japan Academy

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Vol.90 • No. 8 • October 2014
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