The separating semigroup of a real curve

TL;DR

This paper introduces the separating and hyperbolic semigroups of real algebraic curves, characterizes them for maximal curves, and explores their implications for the hyperbolicity locus and embeddings.

Contribution

It defines new semigroups associated with real curves, completely determines them for maximal curves, and analyzes the hyperbolicity locus in relation to embeddings.

Findings

Complete determination of semigroups for maximal curves.

Any high-degree embedding is hyperbolic.

Hyperbolicity locus can be disconnected.

Abstract

We introduce the separating semigroup of a real algebraic curve of dividing type. The elements of this semigroup record the possible degrees of the covering maps obtained by restricting separating morphisms to the real part of the curve. We also introduce the hyperbolic semigroup which consists of elements of the separating semigroup arising from morphisms which are compositions of a linear projection with an embedding of the curve to some projective space. We completely determine both semigroups in the case of maximal curves. We also prove that any embedding of a real curve to projective space of sufficiently high degree is hyperbolic. Using these semigroups we show that the hyperbolicity locus of an embedded curve is in general not connected.