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Real forms of a Riemann surface of even genus. (English) Zbl 0913.20029
Let \(X\) be a Riemann surface of genus \(g\geq 2\). By a symmetry of \(X\) is meant an anticonformal involution \(\sigma\) of \(X\) with fixed points and a surface admitting a symmetry is said to be symmetric. In the group \(\operatorname{Aut}^\pm(X)\), of all conformal and anticonformal automorphisms of \(X\), non-conjugate symmetries correspond to different real models of the curve. Natanzon proved that a Riemann surface \(X\) of genus \(g\geq 2\) has at most \(2(\sqrt g+1)\) conjugacy classes of symmetries, and this bound is attained for infinitely many odd genera \(g\).
The aim of this note is to prove that a Riemann surface of even genus \(g\) has at most \(4\) conjugacy classes of symmetries and this bound is attained for an arbitrary even \(g\) as well. An equivalent formulation is that a Riemann surface of even genus \(g\) is the complex double of at most \(4\) bordered Klein surfaces, or in terms of algebraic curves, that a complex curve of an even genus \(g\) has at most four real forms which are not birationally equivalent. Corollary 3.4. Let \(X\) be a Riemann surface of even genus and let \(G\) be a subgroup of \(\operatorname{Aut}^\pm(X)\) generated by non-conjugate symmetries \(\sigma_1,\sigma_2,\sigma_3\) and \(\sigma_4\). Then \(G=D_n\times Z_2\).

MSC:
20F36 Braid groups; Artin groups
30F10 Compact Riemann surfaces and uniformization
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
14H55 Riemann surfaces; Weierstrass points; gap sequences
30F50 Klein surfaces
14P25 Topology of real algebraic varieties
20F05 Generators, relations, and presentations of groups
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