Nonclassical Godeaux surfaces in characteristic five
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- by Rick Miranda PDF
- Proc. Amer. Math. Soc. 91 (1984), 9-11 Request permission
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.References
- Sven Toft Jensen, Picard schemes of quotients by finite commutative group schemes, Math. Scand. 42 (1978), no. 2, 197–210. MR 512270, DOI 10.7146/math.scand.a-11748
- William E. Lang, Classical Godeaux surface in characteristic $P$, Math. Ann. 256 (1981), no. 4, 419–427. MR 628223, DOI 10.1007/BF01450537
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 9-11
- MSC: Primary 14J50; Secondary 14J05
- DOI: https://doi.org/10.1090/S0002-9939-1984-0735553-8
- MathSciNet review: 735553