On finiteness properties of separating semigroup of real curve

Abstract

A real morphism $f$ from a real algebraic curve $X$ to ${\mathbb{P}}^1$ is called separating if $f^{-1}(\mathbb{R}{\mathbb{P}}^1)=\mathbb{R}X$. A separating morphism defines a covering $\mathbb{R}X\to \mathbb{R}{\mathbb{P}}^1$. Let $X_1,\dots ,X_r$ denote the components of $\mathbb{R}X$. M. Kummer and K. Shaw defined the separating semigroup of a curve $X$ as the set of all vectors $d(f)=(d_1(f),\dots ,d_r(f))\in {\mathbb{N}}^r$ where $f$ is a separating morphism $X\to {\mathbb{P}}^1$ and $d_i(f)$ is the degree of the restriction of $f$ to $X_i$. In the present paper we prove that for a non-negative integer number $g$ the set of all separating semigroups of genus $g$ curves is finite.

Minor updates, some typos fixed