## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Numerical Godeaux surfaces with an involution in positive characteristic

Soonyoung Kim

#### 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 $G$.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 8 (2014), 113-118.

Dates
First available in Project Euclid: 3 October 2014

https://projecteuclid.org/euclid.pja/1412341994

Digital Object Identifier
doi:10.3792/pjaa.90.113

Mathematical Reviews number (MathSciNet)
MR3266744

Zentralblatt MATH identifier
1338.14042

Subjects
Primary: 14J29: Surfaces of general type

#### Citation

Kim, Soonyoung. Numerical Godeaux surfaces with an involution in positive characteristic. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 8, 113--118. doi:10.3792/pjaa.90.113. https://projecteuclid.org/euclid.pja/1412341994

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