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Bujalance, E. and Cirre, F.J. and Gamboa, J. M. and Gromadzki, G.
(2001)
*Symmetry types of hyperelliptic Riemann surfaces.*
Mémoires de la Société Mathématique de France. Nouvelle Série., 86
.
pp. 1-122.
ISSN 0249-633X

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Official URL: http://smf4.emath.fr/Publications//Memoires/2001/86/pdf/smf_mem-ns_86.pdf

## Abstract

Let $X$ be a compact hyperelliptic Riemann surface which admits anti-analytic involutions (also called symmetries or real structures). For instance, a complex projective plane curve of genus two, defined by an equation with real coefficients, gives rise to such a surface, and complex conjugation is such a symmetry. In this memoir, the real structures $\tau$ of $X$ are classified up to isomorphism (i.e., up to conjugation). This is done as follows: the number of connected components of the set of fixed points of $\tau$ together with the connectedness or disconnectedness of the complementary set in $X$ classifies $\tau$ topologically; they determine the species of $\tau$, which only depends on the conjugacy class of $\tau$ (however, different conjugacy classes may have identical species). On these grounds, for a given genus $g\ge2$, the authors first give a list of all full groups of analytic and anti-analytic automorphisms of genus $g$ compact hyperelliptic Riemann surfaces. For every such group $G$, the authors compute polynomial equations for a surface $X$ having $G$ as full group and then find the number of conjugacy classes containing symmetries; they also compute a representative $\tau$ in every such class. Finally, they compute the species corresponding to such classes. This memoir is an exhaustive piece of work, going through a case-by-case analysis. The problem for general compact Riemann surfaces dates back to 1893, when {\it F. Klein} [Math. Ann. 42, 1--29 (1893)] first studied it. For zero genus, it is easy. For genus one, that is, for elliptic surfaces, it was solved by {\it N. Alling} ["Real elliptic curves" (1981)]. Partial results for hyperelliptic surfaces of genus two were obtained by {\it E. Bujalance} and {\it D. Singerman} [Proc. Lond. Math. Soc. 51, 501--519 (1985)].

Item Type: | Article |
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Uncontrolled Keywords: | Riemann surface, symmetry, automorphism group, real form, real algebraic curve |

Subjects: | Sciences > Mathematics > Algebra |

ID Code: | 17252 |

Deposited On: | 28 Nov 2012 11:55 |

Last Modified: | 20 Feb 2017 11:54 |

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