The separating gonality of a separating real curve

Abstract

Let X be a smooth real curve of genus g such that the real locus has s connected components. We say X is separating if the complement of the real locus is disconnected. In case there exists a morphism f from X to \({\mathbb{P}^1}\) such that the inverse image of the real locus of \({\mathbb{P}^1}\) is equal to the real locus of X then X is separating and such morphism is called separating. The separating gonality of a separating real curve X is the minimal degree of a separating morphism from X to \({\mathbb{P}^1}\). It is proved by Gabard that this separating gonality is between s and (g + s + 1)/2. In this paper we prove that all values between s and (g + s + 1)/2 do occur.