Separating Semigroup of Genus 4 Curves

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A rational function on a real algebraic curve $$C$$ is called separating if it takes real values only at real points. Such a function defines a covering $$\mathbb R C\to\mathbb{RP}^1$$ . Let $$c_1,\dots,c_r$$ be the connected components of $$\mathbb R C$$ . M. Kummer and K. Shaw defined the separating semigroup of $$C$$ as the set of all sequences $$(d_1(f),\dots,d_r(f))$$ where $$f$$ is a separating function, and $$d_i(f)$$ is the degree of the restriction of $$f$$ to $$c_i$$ . In the present paper, we describe the separating semigroups of all genus 4 curves. For the proofs, we consider the canonical embedding of $$C$$ into a quadric $$X$$ in $$\mathbb P^3$$ , and apply Abel’s theorem to 1-forms on $$C$$ obtained as Poincaré residues of certain meromorphic 2-forms.