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Symmetry types of hyperelliptic Riemann surfaces. (English) Zbl 1078.14044
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 \(g\geq2\), 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 F. Klein [Math. Ann. 42, 1–29 (1893; JFM 25.0689.03)] first studied it. For zero genus, it is easy. For genus one, that is, for elliptic surfaces, it was solved by N. Alling [“Real elliptic curves” (1981; Zbl 0478.14022)]. Partial results for hyperelliptic surfaces of genus two were obtained by E. Bujalance and D. Singerman [Proc. Lond. Math. Soc. 51, 501–519 (1985; Zbl 0545.30032)].

14H55 Riemann surfaces; Weierstrass points; gap sequences
30F10 Compact Riemann surfaces and uniformization
20F05 Generators, relations, and presentations of groups
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
14P25 Topology of real algebraic varieties
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