On q-n-gonal Klein Surfaces

We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism ϕ of prime order n with quotient space X/〈ϕ〉 of algebraic genus q. These Klein surfaces are called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the (admissible) values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given: q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.