Real Algebraic and Analytic Geometry |

Gap sequences on Klein surfaces.

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Submission: 2004, April 30.

*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 sets Gn 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 G1 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 Gn determine the topological type of the surface.*

Mathematics Subject Classification (2000): 14H55, 30F50.

Keywords and Phrases: Gap sequence, Weierstrass point, Klein surface, hyperelliptic Klein surface.

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