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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.18545 |
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Table of Contents:
- 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.