Separating Semigroup of a Real Curve

• The paper introduces the concept of the separating semigroup as a combinatorial invariant, linking real morphisms and degree vectors of a separating real algebraic curve.
• It classifies the semigroup for various cases—such as M-curves, hyperelliptic curves, and plane quartics—using canonical embeddings, meromorphic forms, and residue theory.
• The work highlights obstructions and open problems, including the non-finite generation of the semigroup and connections to Ahlfors’ theorem and divisor theory.

A separating semigroup is a fundamental combinatorial invariant attached to a real algebraic curve of dividing type: for such a curve $C$, equipped with an antiholomorphic involution and real locus $\mathbb{R}C$, the separating semigroup $\operatorname{Sep}(C)$ enumerates all possible degree vectors arising from separating morphisms—real maps $f:C\to\mathbb{P}^1$ satisfying $f^{-1}(\mathbb{R}\mathbb{P}^1)=\mathbb{R}C$. The structure of $\operatorname{Sep}(C)$ depends on the curve's real topological type and its embedding, with deep connections to classical results (Ahlfors' theorem), obstruction theory, and modern techniques involving meromorphic forms and residue theory. Separation properties reflect the arithmetic of unramified coverings of each real component $c_i$ to the real projective line, with rich geometric and arithmetic implications.

1. Separating Morphisms and Semigroup Definition

A real algebraic curve $C$ is a smooth projective complex curve with an antiholomorphic involution. The real locus $\mathbb{R}C$ is a union of $r$ disjoint circles, each called an oval or pseudoline: $\mathbb{R}C = c_1\cup\cdots\cup c_r$. $C$ is termed separating (or of type I) if $C\setminus \mathbb{R}C$ has two connected components. A real morphism $f:C\to\mathbb{P}^1$ is separating if $f^{-1}(\mathbb{R}\mathbb{P}^1)=\mathbb{R}C$, making the restriction $f|_{c_i}$ an unramified covering of degree $d_i\geq 1$ onto $\mathbb{R}\mathbb{P}^1$. The separating semigroup is defined as the set of all degree vectors of separating morphisms:

$$\operatorname{Sep}(C) = \{ (d_1,\ldots,d_r)\in\mathbb{N}^r \mid \exists\ f:C\to\mathbb{P}^1\ \text{separating},\ d_i=\deg(f|_{c_i}) \}.$$

This semigroup is closed under coordinate-wise addition (Kummer et al., 2017, Orevkov, 2017, Magin, 23 Nov 2025).

2. Structural Classification in Low Genus and Canonical Cases

The description of $\operatorname{Sep}(C)$ is highly sensitive to both genus and real topology.

• M-curves (maximal, $r=g+1$): $\operatorname{Sep}(C)=\mathbb{N}^{g+1}$—all degree vectors occur (Kummer et al., 2017, Orevkov, 3 Dec 2024).
• Hyperelliptic, non-M: For $g$ even ($r=2$), $$\operatorname{Sep}(C)=\{(k,k):k\in\mathbb{N}\} \cup \{ (d_1,d_2): d_1\ge (g+1)/2, d_2\ge (g+1)/2\}$$; for $g$ odd ($r=1$), two types—either $d$ is even, or $d\geq g+1$ (Orevkov, 2017, Orevkov, 3 Dec 2024).
• Plane quartics/ genus 3, non-hyperelliptic: If $\mathbb{R}C$ consists of two nested ovals, $$\operatorname{Sep}(C)=\mathbb{N}\times\mathbb{N}_{\geq 2}$$ (Orevkov, 2017).
• Genus 4, non-hyperelliptic: Curves embedded on quadric surfaces (ellipsoid, cone, hyperboloid) have classification driven by $(r,\ell)$—number of real components and count of ovals (bounding disks). For example, on the cone with three non-oval loops, $$\operatorname{Sep}(C)=\{(1,1,1)\} \cup\{(1,2+n,1): n\geq 0\}$$; if two ovals, $(1,2,1)+\mathbb{N}_0^3$ (Orevkov, 3 Dec 2024).

Case	$(r,\ell)$	$\operatorname{Sep}(C)$
M-curve ($g=4$)	(5,5)	$\mathbb{N}^{5}$
Quadratic cone (non-M)	(3,0)	$\{(1,1,1)\}\cup\{(1,2+n,1):n\geq 0\}$
Quadratic cone (non-M)	(3,2)	$(1,2,1)+\mathbb{N}_0^3$
Hyperboloid (1 real comp.)	(1,0)	$3+\mathbb{N}_0$
Hyperboloid (3 real comps.)	(3,0)	$\mathbb{N}^3$
Hyperboloid (3, 2 ovals)	(3,2)	$(1,2,1)+\mathbb{N}_0^3$

3. Finiteness and Generators: Finitely Covered but Not Finitely Generated

In dimension one, any additive subsemigroup of $\mathbb{N}$ is finitely generated; for higher rank $\mathbb{N}^r$ with $r\ge 2$, $\operatorname{Sep}(C)$ is not finitely generated (Magin, 23 Nov 2025). Instead, it is finitely covered: after excluding finitely many "small exceptional degrees," the remaining part is a finite union of orthants shifted by finitely many degree vectors. Precisely,

$$\operatorname{Sep}(C) = S_0 \cup \bigcup_{i=1}^m \left( s_i + \mathbb{N}_0^r \right),$$

with $S_0$ finite and $s_i$ minimal non-special separating divisors. This means eventual behavior is unbounded along infinitely many independent "extremal rays," so standard finite generation fails as soon as $r\ge 2$. For instance, for an M-curve,

$$\operatorname{Sep}(C) = \{ (d_1,\ldots,d_{g+1}) \in \mathbb{N}^{g+1} : d_i\geq 1\ \forall i,\ |d| -(g+1) \text{ even} \}$$

is finitely covered but not finitely generated (Magin, 23 Nov 2025).

4. Obstructions, Minimal Degrees, and Orientation Comparisons

Ahlfors' theorem proves existence of separating morphisms for all separating curves, and establishes the lower bound $\deg f\geq l$, where $l$ is the number of real components (Manzaroli, 2022). For M-curves and specific hyperelliptic cases, the bound is sharp, but for many separating curves—particularly for those embedded in surfaces with specific numerical invariants (e.g., surfaces with $$(\chi(\mathcal O_X)\geq 1, (-K_X)^2\geq 0, -K_X\cdot D\geq 0)$$)—orientation-comparison obstructions (Orevkov's Theorem 3.2) and Riemann–Roch produce gaps in the semigroup, excluding the smallest candidate degree. For example, on certain quintics with four ovals, degree $4$ is minimal and $5$ excluded by orientation constraints; in plane curves, precise numerical inequalities in terms of $(d,l)$, parity, and intersection numbers determine further gaps (Manzaroli, 2022). Thus, the minimal degree $m(C)=\min \operatorname{Sep}(C)$ can strictly exceed $l$.

5. Canonical Embedding, Residue Theory, and Abel's Theorem

Classification of $\operatorname{Sep}(C)$ for non-hyperelliptic genus $g=4$ curves uses the canonical embedding into a quadric $X\subset\mathbb{P}^3$ and analytic techniques (Orevkov, 3 Dec 2024). For a divisor $D$ on $X$, global 2-forms with $\operatorname{div}(\Omega)=C-D$ induce chessboard boundary orientations. The Poincaré residue construction yields meromorphic 1-forms $\eta=\mathrm{Res}_C \Omega$, which assign local orientations to each component. Abel's theorem applied to $\eta$ enforces universal linear sign constraints on degree vectors via sums $\sum_{p\in P} \eta(v_p)=0$, restricting feasible separating morphisms. In the cone case $(r,\ell)=(3,0)$, only degree vectors $(1,1,1)$ and vectors of the form $(1,2+n,1)$ occur, proven via orientation and residue arguments (Orevkov, 3 Dec 2024).

6. Applications, Examples, and Extensions

• Totally real pencils induce separating morphisms of degree $kd$, but can be obstructed by the arithmetic conditions of Theorem A and B, eliminating $kd$ from $\operatorname{Sep}(C)$ for many plane curves when $kd\leq l$ (Manzaroli, 2022).
• High genus: For $g\ge 5$, canonical embeddings lie in $\mathbb{P}^{g-1}$ as complete intersections of quadrics or in scrolls, with separating semigroups conjecturally determined by higher-dimensional residue and Abel theory (Orevkov, 3 Dec 2024).
• Hyperbolic semigroup $\operatorname{Hyp}(X)$ includes degree vectors arising from linear projections and is strictly smaller than $\operatorname{Sep}(X)$ except in low genera or maximal cases. Its topological characterization involves linking numbers of real components with real planes or lines (Kummer et al., 2017).
• Examples: For planar quartic, $(1,1)\notin\operatorname{Sep}(C)$ but $(1,2)$, $(2,2)$, and all $(d_1,d_2)$ with $d_2\geq 2$ are present (Orevkov, 2017). For a genus $3$ hyperelliptic, $2\mathbb{N}\cup\{d\geq 4\}$ arises.

7. Research Directions and Open Problems

Current open questions include classifying all $(C,l)$ for which $l\notin \operatorname{Sep}(C)$, refining lower bounds $m(C)$ in terms of $(g,l)$ for arbitrary surfaces, and understanding the finer semigroup structure in the vector-valued case—tracking degree distributions across real components (Manzaroli, 2022). Extending Poincaré residue and Abel-theoretic arguments to higher genus and codimension may enable uniform approaches. The interplay of real topology, divisor theory, and analytic obstructions is expected to dominate future advances, while the non-finite-generation property shapes the underlying combinatorics.