Separating Semigroup of Genus 4 Curves

Abstract

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.