Morphisms of Klein surface

Sin resume

Groups of automorphisms of cyclic trigonal Riemann surfaces

AbstractWe give the list of all groups G acting as a group of automorphisms of some cyclic trigonal compact Riemann surface X of genus g⩾5 and containing the trigonality automorphism group. An abstract group G may act in different ways producing coverings X→X/G with different ramification type; the list of all such different ramification types is also given

Symmetry types of hyperelliptic Riemann surfaces

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)]

On the problem of finding the full automorphism group of a compact Klein surface

The paper under review surveys most known results about the following problem: let $X$ be a compact topological surface of algebraic genus $p>1$, with or without boundary, orientable or not. How to calculate all groups acting as the full automorphism group of some structure of Klein surface having $X$ as underlying topological surface? It must be remarked that from Riemann's uniformization theorem, and since \Aut(X) has no more than 168 $(p-1)$ automorphisms (including the orientation-reversing ones), this problem is of a finite nature. In practice this is an unaccessible task except for low values of $p$ or some extra conditions on the surfaces one is dealing with.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Normal coverings of hyperelliptic real algebraic curves

We consider normal (possibly) branched, finite-sheeted coverings $\pi:X\rightarrow X'$ between hyperelliptic real algebraic curves. We are interested in the topology of such coverings and also in describing them in terms of algebraic equations. In this article we completely solve these two problems in case $X$ has the maximum number of ovals within its genus. We first analyze the topological features and ramification data of such coverings. For each isomorphism class we then describe a representative, with defining polynomial equations for $X$ and for $X'$, formulae for generators of the covering transformation group, and a rational formula for the covering $\pi:X\rightarrow X'$.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Services, citizenship and the country of origin principle

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On Compact Riemann Surfaces With Dihedral Groups Of Automorphisms

We study compact Riemann surfaces of genus g 2 having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group DN. The question of extendability of the action of DN is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined

Symmetry types of hyperelliptic Riemann surfaces

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)].Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu