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Symmetry types of hyperelliptic Riemann surfaces


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


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
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:02 Mar 2016 15:16

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