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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Nonclassical Godeaux surfaces in characteristic five


Author: Rick Miranda
Journal: Proc. Amer. Math. Soc. 91 (1984), 9-11
MSC: Primary 14J50; Secondary 14J05
MathSciNet review: 735553
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Abstract: A classical Godeaux surface is a smooth minimal projective surface $ X$, with $ K_X^2 = 1$, $ {p_a} = {p_g} = 0$ and $ {\text{Pi}}{{\text{c}}^\tau }(X) = {\mathbf{Z}}/5{\mathbf{Z}}$. A nonclassical Godeaux surface is a smooth minimal projective surface $ X$ with $ K_X^2 = 1$, $ {p_a} = 0$, $ {p_g} = 1$ and $ {\text{Pi}}{{\text{c}}^\tau }(X) = {\mu _5}$ or $ {\alpha _5}$; such surfaces should exist in characteristic 5. It is the purpose of this note to construct nonclassical Godeaux surfaces in characteristic 5, with $ {\text{Pi}}{{\text{c}}^\tau }(X) = {\mu _5}$. The method is to exhibit a smooth quintic surface on which $ {\mathbf{Z}}/5{\mathbf{Z}}$ acts, so that the quotient is smooth; this quotient is the desired surface.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0735553-8
PII: S 0002-9939(1984)0735553-8
Article copyright: © Copyright 1984 American Mathematical Society