Gap sequences on Klein surfaces

Abstract

In this work we provide a possible definition for the gap sequence at a point of a compact Klein surface in an attempt to generalize the notion of Weierstrass gap sequence at a point of a compact Riemann surface. We obtain some results about the properties of these gap sequences and use them to study the G n sets consisting of the points which have n as its first non-gap. We prove that these sets are invariant under the action of the automorphisms of the surface. We show that there are Klein surfaces of arbitrary genus such that the set G 1 is non-empty (if this is the case, it is a semialgebraic subset of real dimension one). If a surface has this property, then it must be hyperelliptic. In this case, we find that the topology of the sets G n determine the topological type of the surface.