Revista Matemática Iberoamericana

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Volume 33, Issue 1, 2017, pp. 183–194
DOI: 10.4171/RMI/932

Published online: 2017-02-22

The double of the doubles of Klein surfaces

Antonio F. Costa[1], Paola Cristofori[2] and Ana M. Porto[3]

(1) UNED, Madrid, Spain
(2) Università di Modena e Reggio Emilia, Modena, Italy
(3) UNED, Madrid, Spain

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface $X$ is a Klein surface $Y$ such that there is a degree two morphism (of Klein surfaces) $Y \to X$. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if $X$ is a non-orientable Klein surface with non-empty boundary, the three natural doubles, although distinct Klein surfaces, share a common double: “the double of doubles” denoted by $DX$. We describe how to use the double of doubles in the study of both moduli spaces and automorphisms of Klein surfaces. Furthermore, we show that the morphism from $DX$ to $X$ is not given by the action of an isometry group on classical surfaces.

Keywords: Klein surface, Riemann surface, automorphism, real algebraic curve, moduli space

Costa Antonio, Cristofori Paola, Porto Ana: The double of the doubles of Klein surfaces. Rev. Mat. Iberoam. 33 (2017), 183-194. doi: 10.4171/RMI/932