On finiteness properties of separating semigroup of real curve

TL;DR

The paper studies the separating semigroup Sep$(X)$ of a real curve $X$, defined by the degree vectors on the real components of $\mathbb{R}X$ associated to separating morphisms $f:X\to P^1$. It introduces the notion of a finitely covered additive subsemigroup of $\mathbb{N}^r$ and proves that Sep$(X)$ is finitely covered, with an explicit bound on the finite part given by $\binom{2g-3}{r}+\binom{g-1}{r-1}$. The work further shows that when the real locus has at least two connected components ($r\ge 2$), Sep$(X)$ is not finitely generated, contrasting with the $r=1$ case. The proofs combine divisor-theoretic arguments—splitting separating divisors into special and non-special parts, properties of canonical divisors with even degrees on $\mathbb{R}X$—with Dickson's lemma to obtain a finite generating framework and a construction that demonstrates non-finite generation.

Abstract

A real morphism $f$ from a real algebraic curve $X$ to $\mathbb{P}^1$ is called \emph{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~\cite{kummer_separating_2020} 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 $X_i$. Let us call an additive subsemigroup of $\mathbb{N}^{r}$ \emph{finitely covered} if it can be written as $S = S_0 \cup \bigcup_{i=1}^{m} (s_i + \mathbb{N}_{0}^{r})$, where $S_0$ is a finite set. In the present paper, we prove that the separating semigroup of a real curve $X$ is finitely covered, but not finitely generated when $\mathbb{R} X$ has at least two connected components.

Theorems & Definitions (11)

• Definition 1
• Lemma 1: Prop. 3.2. in kummer_separating_2020
• Lemma 2
• proof
• Theorem 1
• proof
• Remark 1
• Theorem 2
• Remark 2
• proof : Proof of Theorem \ref{['infin_gen']}