Fabrizio Catanese, Paola Frediani
Real hyperelliptic surfaces and the orbifold fundamental group
Preprint series: Mathematica Gottingensis
MSC:
14P25 Topology of real algebraic varieties
Abstract: In this paper we finish the topological classification of
real algebraic surfaces of Kodaira dimension zero and we make a step
towards the Enriques classification of real algebraic surfaces, by
describing in detail the structure of the moduli space of real
hyperelliptic surfaces.

Moreover, we point out the relevance in real geometry of the notion
of the orbifold fundamental group of a real variety, and we discuss
related questions on real varieties $(X, \sigma)$ whose underlying
complex manifold $X$ is a $K ( \pi, 1)$.

Our first result is that if $(S, \sigma)$ is a real hyperelliptic
surface, then the differentiable type of the pair $(S, \sigma)$ is completely determined by the orbifold fundamental group
exact sequence. \\ This result allows us to determine all the
possible topological types of
$(S, \sigma)$, and to prove that they are exactly 78.

It follows also as a corollary that there are exactly eleven cases
for the topological type of the real part of S.

Finally, we show that once we fix the topological type of $(S, \sigma)$ corresponding to a real hyperelliptic surface, the
corresponding moduli space is irreducible (and connected).

We also give, through a series of tables, explicit analytic
representations of the 78 components of the moduli space.
Keywords: real varieties, hyperelliptic surfaces, orbifold fundamental group