On finiteness properties of separating semigroup of real curve

TL;DR

This paper investigates the separating semigroup of real algebraic curves, proving that for any fixed genus, the set of all such semigroups is finite, thus revealing a key finiteness property in real algebraic geometry.

Contribution

It establishes the finiteness of the set of separating semigroups for real algebraic curves of a fixed genus, a new result in the study of real morphisms.

Findings

Finiteness of separating semigroups for fixed genus curves

Characterization of separating morphisms and their degrees

Implications for real algebraic curve classification

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 $X_1, \ldots, X_r$ 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 $d(f) = (d_1(f), \ldots, d_r(f)) \in \mathbb{N}^{r}$ 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.