Summary
Inspired by a construction by Arnaud Beauville of a surface of general type with K 2 = 8, p g = 0, the second author defined Beauville surfaces as the surfaces which are rigid, i.e., without nontrivial deformations, and which admit an unramified covering which is isomorphic to a product of curves of genus at least 2.
In this case the moduli space of surfaces homeomorphic to the given surface consists either of a unique real point, or of a pair of complex conjugate points corresponding to complex conjugate surfaces. It may also happen that a Beauville surface is biholomorphic to its complex conjugate surface, while failing to admit a real structure.
The first aim of this note is to provide series of concrete examples of the second situation, respectively of the third.
The second aim is to introduce a wider audience, in particular group theorists, to the problem of classification of such surfaces, especially with regard to the problem of existence of real structures on them.
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Bauer, I., Catanese, F., Grunewald, F. (2005). Beauville surfaces without real structures. In: Bogomolov, F., Tschinkel, Y. (eds) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol 235. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4417-2_1
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