The separating gonality of a separating real curve

TL;DR

This paper investigates the separating gonality of smooth real curves, establishing that its possible values are only constrained by known bounds, with no additional restrictions.

Contribution

The paper proves that the separating gonality of a separating real curve can attain any value within the established bounds, showing no further restrictions.

Findings

Separating gonality is bounded below by the number of real components.

Ahlfors' upper bound of g+1 is improved by Gabard.

No additional restrictions on separating gonality beyond known bounds.

Abstract

A smooth real curve is called separating in case the complement of the real locus inside the complex locus is disconnected. This is the case if there exists a morphism to the projective line whose inverse image of the real locus of the projective line is the real locus of the curve. Such morphism is called a separating morphism. The minimal degree of a separating morphism is called the separating gonality. The separating gonality cannot be less than the number s of the connected components of the real locus of the curve. A theorem of Ahlfors implies this separating gonality is at most the g+1 with g the genus of the curve. A better upper bound depending on s is proved by Gabard. In this paper we prove that there are no more restrictions on the values of the separating gonality.