Proceedings of the Japan Academy, Series A, Mathematical Sciences

Numerical Godeaux surfaces with an involution in positive characteristic

Soonyoung Kim

Full-text: Open access


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

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

First available in Project Euclid: 3 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J29: Surfaces of general type

Godeaux surface involution positive characteristic action of group scheme


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.

Export citation


  • E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219.
  • E. Bombieri and D. Mumford, Enriques' classification of surfaces in char. $p$. II, in Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1974, pp. 23–42.
  • D. M. Burns, Jr. and J. M. Wahl, Local contributions to global deformations of surfaces, Invent. Math. 26 (1974), 67–88.
  • A. Calabri, C. Ciliberto and M. Mendes Lopes, Numerical Godeaux surfaces with an involution, Trans. Amer. Math. Soc. 359 (2007), no. 4, 1605–1632 (electronic).
  • F. Catanese, Pluricanonical-Gorenstein-curves, in Enumerative geometry and classical algebraic geometry (Nice, 1981), 51–95, Progr. Math., 24, Birkhäuser Boston, Boston, MA, 1982.
  • T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 97–144.
  • H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Seminar, 20, Birkhäuser, Basel, 1992.
  • N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), no. 5, 981–996.
  • J. Keum and Y. Lee, Fixed locus of an involution acting on a Godeaux surface, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 2, 205–216.
  • J. Kollár and S. Mori, Birational geometry of algebraic varieties, translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, 134, Cambridge Univ. Press, Cambridge, 1998.
  • W. E. Lang, Classical Godeaux surface in characteristic $P$, Math. Ann. 256 (1981), no. 4, 419–427.
  • Y. Lee and N. Nakayama, Simply connected surfaces of general type in positive characteristic via deformation theory, Proc. Lond. Math. Soc. (3) 106 (2013), no. 2, 225–286.
  • Y. Lee and J. Park, A simply connected surface of general type with $p_{g}=0$ and $K^{2}=2$, Invent. Math. 170 (2007), no. 3, 483–505.
  • C. Liedtke, Algebraic surfaces of general type with small $c_{1}^{2}$ in positive characteristic, Nagoya Math. J. 191 (2008), 111–134.
  • C. Liedtke, Non-classical Godeaux surfaces, Math. Ann. 343 (2009), no. 3, 623–637.
  • R. Miranda, Nonclassical Godeaux surfaces in characteristic five, Proc. Amer. Math. Soc. 91 (1984), no. 1, 9–11.
  • Y. Miyaoka, Tricanonical maps of numerical Godeaux surfaces, Invent. Math. 34 (1976), no. 2, 99–111.
  • M. Reid, Godeaux and Campedelli surfaces. ( masda/surf/Godeaux.pdf).
  • M. Reid, Surfaces with $p_{g}=0$, $K^{2}=1$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978), no. 1, 75–92.
  • Q. Xie, Kawamata-Viehweg vanishing on rational surfaces in positive characteristic, Math. Z. 266 (2010), no. 3, 561–570.