On finiteness properties of separating semigroup of real curve

2025/11/23 by Magin, Matthew
#14P99 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.18545

Abstract

A real morphism f from a real algebraic curve X to ℙ¹ is called separating if f^-1(ℝ ℙ¹) = ℝ X. A separating morphism defines a covering ℝ X → ℝ ℙ¹. Let X₁, …, X_r denote the components of ℝ X. M. Kummer and K. Shaw~\citekummer_separating₂020 defined the separating semigroup of a curve X as the set of all vectors d(f) = (d₁(f), …, d_r(f)) ∈ ℕ^r where f is a separating morphism X → ℙ¹ and d_i(f) is the degree of the restriction of f to X_i. Let us call an additive subsemigroup of ℕ^r finitely covered if it can be written as S = S₀ ∪ \bigcup_i=1^m (s_i + ℕ₀^r), where S₀ 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 ℝ X has at least two connected components.

Citations

• Separating semigroup of genus 4 curves
• Separating semigroup of hyperelliptic curves and of genus 3 curves
• The separating gonality of a separating real curve
• Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors

Related

• Separating semigroup of genus 4 curves
• Separating semigroup of hyperelliptic curves and of genus 3 curves
• Obstructions for the existence of separating morphisms and totally real pencils
• The separating gonality of a separating real curve
• Covering semigroups