On finiteness properties of separating semigroup of real curve

Matthew Magin

REVIEW 1 major objections 5 minor 9 references

On finiteness properties of separating semigroup of real curve

T0 review · 1 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For each genus, only finitely many separating semigroups exist

desk verdict First finiteness theorem for separating semigroups of real curves, with a genuinely nice deletion lemma — but it leans on an unproved strengthening of Orevkov's lemma that the author flags only in a footnote. read the letter →

arxiv 2511.18545 v4 pith:OK6IGPM6 submitted 2025-11-23 math.AG

Matthew Magin This is my paper · ORCID

classification math.AG MSC 14H5514P2514H05

keywords separatingsemigrouprealalgebraiccurvemorphismdivisorsAbel-Jacobimapspecialdivisorfinitenesstheoremgonality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a finiteness theorem for separating semigroups of real algebraic curves: for a fixed genus g, there are only finitely many distinct separating semigroups among all real curves of that genus. This settles a structural question about how the degrees of separating morphisms to the projective line can vary. The proof shows that every separating semigroup is determined by two finite pieces of data—a finite set of special divisors and a finite set of minimal non-special separating divisors—and bounds the degrees of the latter by 4g−3. The key deletion theorem says that from any sufficiently large separating divisor one can remove at least half its points while preserving the separating property.

What carries the argument

The load-bearing tool is the infinitesimal Abel-Jacobi criterion for separating divisors: a totally real divisor P = p_1 + ... + p_n is separating if and only if there are positive tangent vectors v_i at the points p_i (positive with respect to a complex orientation) such that ∑_i ω_k(v_i)=0 for every holomorphic one-form ω_k. This converts the separability condition into linear algebra over the reals. The other central mechanism is the deletion theorem: for n ≥ g+2, linear dependence among the vectors u_i=(ω_1(v_i),...,ω_g(v_i)) forces a proper non-empty sub-sum with positive coefficients involving at least half the points, so the corresponding subdivisor is again separating. This yields th

What would settle it

Take a special separating divisor P of degree at least g+2 on a real curve of genus g and check whether P admits positive tangent vectors v_i at its points with ∑ω_k(v_i)=0 for all holomorphic one-forms ω_k. If some such divisor fails this condition, the paper's strengthened tangent-vector criterion is false, and the deletion theorem and the bound deg P_i ≤ 4g−3 no longer follow. Alternatively, exhibit a separating semigroup whose minimal non-special separating divisor has degree exceeding 4g−3.

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Extended reading notes

Core claim

The paper's main theorem, Theorem 2, states that for every non-negative integer g, the set of all separating semigroups of real curves of genus g is finite. To reach this, the paper proves a structural decomposition: the separating semigroup of any real curve is the union of a finite set of degree partitions coming from special separating divisors together with finitely many cofinal tails d(P_i) + N_0^r attached to minimal non-special separating divisors. The degrees of these minimal divisors are uniformly bounded by 4g−3. The bound follows from a deletion theorem: if a separating divisor has at least g+2 points, then some proper subdivisor containing at least half the points is still separa

Load-bearing premise

The argument rests on the assertion that the tangent-vector criterion is an if-and-only-if even for special divisors—the paper cites this strengthened converse as removable but gives no proof; if it is false, the deletion theorem's output subdivisor may not be separating and the degree bound 4g−3 collapses.

Editorial extensions

If this is right

Every separating curve of genus g admits a separating morphism of degree at most g+1, recovering the classical sharp bound on separating gonality.
For real curves with at least two real components, the separating semigroup is not finitely generated; only the finite special-degree part and finitely many additive tails exist.
The uniform degree bound 4g−3 places all minimal non-special separating divisors inside a finite box, so the full separating semigroup can in principle be found by a finite search.
For fixed genus g and number of real components r, the number of possible separating semigroups is bounded by an explicit expression in g and r, namely (4g−3)^r times a combinatorial factor for special partitions.
Known classifications of separating semigroups for small genera (up to 4) are consistent with the new uniform finiteness statement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the strengthened tangent-vector criterion holds in full generality, the bound 4g−3 is probably far from optimal—the author notes known examples have much smaller degrees, so a sharper bound near 2g or even g+1 may hold.
The finiteness theorem suggests that separating semigroups are stable invariants that could, in principle, be enumerated by scanning finitely many divisor classes; a computer search for small g would test how close the 4g−3 bound is to the true maximum.
The dependence on the strengthened converse of the tangent-vector criterion is the fragile point: if that converse fails for special divisors, the deletion theorem may fail exactly at the degrees needed and the main theorem would not follow from the given arguments.
A similar finiteness statement may hold for separating semigroups of real curves with marked points or for separating maps to higher-genus targets, since the underlying tangent-vector linear algebra is the same.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is the first finiteness theorem for separating semigroups of real curves of fixed genus, and the proof strategy — deletion lemma plus decomposition into a finite special part and finitely many cones — is fresh and mostly clean. I would send it to a referee, but I would not certify the main theorem as it stands, because the one genuinely load-bearing step is an unproved footnote assertion.

What's new: Theorem 2 is not in the existing literature. Kummer–Shaw and Orevkov give explicit descriptions for M-curves, low genus, hyperelliptic curves, and plane quintics, but no finiteness for all genus g. Theorem 1 (the deletion lemma) is a nice linear-algebra argument using Abel–Jacobi: from a separating divisor of degree n, you get a proper separating subdivisor of degree at least n/2. Lemma 4's decomposition of Sep(X) into a finite set plus finitely many translated orthants is a good structural way to think about the semigroup, and Corollary 2's infinite-generation argument is elegant. The degree bound deg P_i ≤ 4g−3 is then straightforward.

The soft spot is exactly where the stress-test note points: Lemma 1 is stated as an iff, with the (⇐) direction attributed to Orevkov's Lemma 3.2. The footnote admits Orevkov's version assumed non-specialness and says the assumption 'can be removed,' but gives no proof. Theorem 1 applies that converse to the subdivisor Q, which need not be non-special. If the strengthening is false, Theorem 1, Corollary 1, and Lemma 6 all collapse. This is not a cosmetic gap; it is the load-bearing point. It may well be true — the Abel–Jacobi picture suggests a perturbative proof — but the paper does not supply one. A referee should ask for it before the main theorem is accepted.

Minor things: the count for canonical degree partitions in Lemma 4 looks off by one — positive even r-tuples summing to 2g−2 should be binomial(g−2, r−1), not binomial(g−1, r−1) — but it does not affect finiteness. Also, the ⊔ notation in Lemma 4's proof is too strong if a single degree partition can be realized by both special and non-special divisors, which the paper itself raises in Question 1.

For whom: anyone working on real algebraic curves, separating morphisms, or the Kummer–Shaw invariant. The paper is short, clearly written, and the main idea is worth knowing. I'd send it to a serious referee with the explicit request to prove or source the strengthened converse of Orevkov's lemma. If the author fills that gap, the paper is solid.

Referee Report

1 major / 5 minor

Summary. This paper studies the separating semigroup Sep(X) of a real algebraic curve X, consisting of the degree vectors of separating morphisms to P^1. The main result (Theorem 2) is that for every genus g there are only finitely many possible separating semigroups of genus-g curves. The proof scheme is as follows: Lemma 4 expresses Sep(X) as the union of a finite set Sep_s(X) of degree partitions of special separating divisors and finitely many cones d(P_i)+N_0^r attached to minimal non-special separating divisors; Theorem 1 is a deletion result showing that every separating divisor of degree n≥g+2 contains a proper separating subdivisor of degree at least ⌈n/2⌉; and Lemma 6 uses Theorem 1 to prove the uniform bound deg P_i≤4g−3 for the minimal non-special divisors, which yields finiteness. The paper also proves that Sep(X) is never finitely generated when b0(RX)≥2 and illustrates the decomposition on examples.

Significance. If the proof is completed, the main result is a valuable finiteness theorem: it reduces the classification of separating semigroups to finite data and gives a uniform explanation for the known case-by-case results. The linear-algebra deletion argument is elegant and potentially applicable to other questions about real divisors. The paper is clearly written and the structural decomposition is a useful conceptual contribution. However, the central argument currently rests on an unproved strengthening of Orevkov's Lemma 3.2 (the converse for special divisors), so the claims are not fully established in the present form.

major comments (1)

[§2, Lemma 1 (footnote 1) and Theorem 1, Eq. (4)] The equivalence in Lemma 1 is stated without qualification, but footnote 1 concedes that the (⇐) direction of Orevkov's Lemma 3.2 was proved only for non-special divisors, and asserts the assumption 'can be removed' without proof or reference. This is load-bearing: in the proof of Theorem 1, the divisor Q obtained from Eq. (3)-(4) has degree only ≥⌈n/2⌉ and can be special; Lemma 1(⇐) is then applied to Q. Since Lemma 6 and hence Theorem 2 invoke Theorem 1, the current proof depends on an unverified strengthening. I note that in Lemma 6 the subdivisor Q_i has degree ≥2g−1 and is therefore non-special, so that application could be repaired by using Orevkov's original non-special converse; but Theorem 1 as stated and Corollary 1 still require the stronger form. Please supply a proof of the strengthened converse, or modify the statement/proofs to avoid it.

minor comments (5)

[§3.1, Lemma 4, inequality (⋄)] The second binomial term counts tuples with sum 2g−2 and all entries even. Writing d_i=2e_i with e_i≥1 and ∑e_i=g−1 gives C(g−2,r−1), not C(g−1,r−1). For example, g=3,r=2 gives only (2,2), yet the displayed binomial equals 2. The bound is still finite, so this does not affect the main theorem, but the formula is incorrect.
[§3.2, Corollary 2 proof] The assertion that d=d(1)+(0,k) cannot be written as a sum of elements of Sep(X) involving Sep_n(X) is not evident and as written appears false, since d∈Sep_n(X) by Lemma 2 (d(1) is the degree partition of a non-special separating divisor). The ratio argument that follows only excludes representations using elements of Sep_s(X). Please repair the argument or clarify the intended notion of generation.
[§2, Theorem 1 proof, Eqs. (1)-(3)] The phrase 'not all of the same sign' is ambiguous when some coefficients are zero. Please state explicitly that one chooses a relation with at least one positive and at least one negative coefficient (this can be achieved by subtracting a suitable multiple of the all-ones relation), so that β_s>0 and the set Γ in (3) is proper.
[§4, Theorem 2 proof] The sentence 'it suffices to obtain a uniform bound on the degrees of the minimal non-special divisors' should also recall that all elements of Sep_s(X) have |d|≤2g−2 by the proof of Lemma 4, so that for fixed g,r the set Sep_s(X) ranges over only finitely many subsets of N^r. This is implicit but should be stated.
[§1, Notation] In 'We always equip the sets N r a N r 0' the symbol 'a' appears to be a typo for 'and'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the finiteness proof is derived from external infinitesimal criteria and counting; the unproved strengthening of Orevkov's Lemma 3.2 is a soundness gap, not a circular step.

full rationale

I find no significant circularity. The derivation chain is: Theorem 2 is reduced via Lemma 4 to bounding the degrees of minimal non-special separating divisors; Lemma 6 bounds those degrees using the deletion Theorem 1; Theorem 1 is proved from Lemma 1 (Orevkov's infinitesimal criterion) and linear algebra among the vectors u_i = (omega_1(v_i), ..., omega_g(v_i)). The decomposition in Lemma 4 is derived from the definition of special/non-special divisors, Serre duality, Dickson's lemma, and Lemma 2 from [KS20]; it does not assume the finiteness it later proves. The bound on |Sep_s(X)| is a combinatorial count of integer tuples, not an input. The only self-citation, [MO25], appears in the introductory survey list ('plane quintics (see [MO25])') and is not used in any proof, so it is not load-bearing. The footnote to Lemma 1 ('In [Ore19] the (⇐) part of Lemma 3.2. assumes the divisor is non-special. In fact, this assumption can be removed.') is flagged here: it is a genuine unproved strengthening and it is load-bearing, since Theorem 1 applies the (⇐) direction to subdivisors Q that may be special. However, this is a correctness/soundness gap, not circularity: the conclusion is not assumed as an input, and the strengthening is attributed to an external lemma rather than derived from the target result. If the strengthening is false, the proof collapses, but no step reduces to its own conclusion by construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof introduces no fitted numerical parameters and no new entities. Its assumptions are classical theorems plus one unproved strengthening of a cited lemma; that strengthening is the main burden.

assumptions (7)

ad hoc to paper Strengthened Orevkov Lemma 3.2: for arbitrary divisors with distinct real points, separating ⇔ existence of positive tangent vectors with ∑ω_k(v_i)=0, even for special divisors.
The footnote removes the non-special assumption from the (⇐) direction without proof. Theorem 1 uses this for subdivisors Q that may be special.
standard math Ahlfors: X is separating iff it admits a separating morphism X→P^1 with f^{-1}(RP^1)=RX.
Background definition of separating morphisms and separating semigroup, cited [Ahl50].
standard math Abel–Jacobi infinitesimal criterion: condition (†) is equivalent to the tangent tuple lying in ker d_P φ.
Used in Lemma 1 and Theorem 1; standard Abel–Jacobi theory.
standard math Serre duality and the fact that effective divisors of degree > 2g−2 are non-special.
Used in Lemma 4 and Lemma 6 to classify special/non-special separating divisors.
standard math Dickson's lemma: N^r has no infinite antichains under pointwise order.
Used in Lemma 4 to conclude there are finitely many minimal non-special degree partitions.
standard math Harnack's inequality: b_0(RX) ≤ g+1 for real curves of genus g.
Used in Theorem 2 to restrict to finitely many possible values of r.
domain assumption Kummer–Shaw results: Sep(X) is an additive semigroup and Lemma 2 about adding points to non-special separating divisors.
Background results from [KS20] used to decompose Sep(X) into special and non-special parts.

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Pith. "Pith review of On finiteness properties of separating semigroup of real curve." pith.science (2026). https://pith.science/paper/OK6IGPM6

@misc{pith2026251118545,
  author       = {Pith},
  title        = {Pith review of: On finiteness properties of separating semigroup of real curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OK6IGPM6}},
  note         = {Machine review of arXiv:2511.18545}
}

abstract

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.

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Reference graph

Works this paper leans on

9 extracted references · 4 canonical work pages

[1]

Lars L. Ahlfors. Open Riemann surfaces and extremal problems on compact subregions. Commentarii Mathematici Helvetici , 24(1):100--134, December 1950. https://doi.org/10.1007/BF02567028 doi:10.1007/BF02567028

work page doi:10.1007/bf02567028 1950
[2]

Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors

Leonard Eugene Dickson. Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. American Journal of Mathematics , 35(4):413--422, 1913. URL: http://www.jstor.org/stable/2370405

arXiv 1913
[3]

J. Huisman. Non-special divisors on real algebraic curves and embeddings into real projective spaces. Annali di Matematica Pura ed Applicata , 182(1):21--35, April 2003. https://doi.org/10.1007/s10231-002-0053-1 doi:10.1007/s10231-002-0053-1

work page doi:10.1007/s10231-002-0053-1 2003
[4]

The separating semigroup of a real curve

Mario Kummer and Kristin Shaw. The separating semigroup of a real curve. Annales de la Faculté des sciences de Toulouse : Mathématiques , 29(1):79--96, July 2020. arXiv:1707.08227 [math]. URL: http://arxiv.org/abs/1707.08227, https://doi.org/10.5802/afst.1624 doi:10.5802/afst.1624

arXiv 2020
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Separating semigroup of plane quintics

Matthew Magin and Stepan Orevkov. Separating semigroup of plane quintics. 2025. URL: https://www.math.univ-toulouse.fr/ orevkov/sepd5-e.pdf

2025
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In preparation., 2025

Grigory Mikhalkin and Stepan Orevkov. In preparation., 2025

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S. Yu. Orevkov. Separating semigroup of hyperelliptic curves and of genus 3 curves. St. Petersburg Mathematical Journal , 31(1):81--84, December 2019. URL: http://www.ams.org/spmj/2020-31-01/S1061-0022-2019-01586-2/, https://doi.org/10.1090/spmj/1586 doi:10.1090/spmj/1586

work page doi:10.1090/spmj/1586 2019
[8]

Yu Orevkov

S. Yu Orevkov. Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves. Geometric and Functional Analysis , 31(4):930--947, August 2021. arXiv:2010.09130 [math]. URL: http://arxiv.org/abs/2010.09130, https://doi.org/10.1007/s00039-021-00569-1 doi:10.1007/s00039-021-00569-1

arXiv 2021

Show all 9 references

[9]

Separating semigroup of genus 4 curves

Stepan Yur'evich Orevkov. Separating semigroup of genus 4 curves. Funktsional. Anal. i Prilozhen , 59(4):40--51, 2025. URL: http://arxiv.org/abs/2412.02460, https://doi.org/10.48550/arXiv.2412.02460 doi:10.48550/arXiv.2412.02460

2025 doi arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Lars L. Ahlfors. Open Riemann surfaces and extremal problems on compact subregions. Commentarii Mathematici Helvetici , 24(1):100--134, December 1950. https://doi.org/10.1007/BF02567028 doi:10.1007/BF02567028

work page doi:10.1007/bf02567028 1950

[2] [2]

Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors

Leonard Eugene Dickson. Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. American Journal of Mathematics , 35(4):413--422, 1913. URL: http://www.jstor.org/stable/2370405

arXiv 1913

[3] [3]

J. Huisman. Non-special divisors on real algebraic curves and embeddings into real projective spaces. Annali di Matematica Pura ed Applicata , 182(1):21--35, April 2003. https://doi.org/10.1007/s10231-002-0053-1 doi:10.1007/s10231-002-0053-1

work page doi:10.1007/s10231-002-0053-1 2003

[4] [4]

The separating semigroup of a real curve

Mario Kummer and Kristin Shaw. The separating semigroup of a real curve. Annales de la Faculté des sciences de Toulouse : Mathématiques , 29(1):79--96, July 2020. arXiv:1707.08227 [math]. URL: http://arxiv.org/abs/1707.08227, https://doi.org/10.5802/afst.1624 doi:10.5802/afst.1624

arXiv 2020

[5] [5]

Separating semigroup of plane quintics

Matthew Magin and Stepan Orevkov. Separating semigroup of plane quintics. 2025. URL: https://www.math.univ-toulouse.fr/ orevkov/sepd5-e.pdf

2025

[6] [6]

In preparation., 2025

Grigory Mikhalkin and Stepan Orevkov. In preparation., 2025

2025

[7] [7]

S. Yu. Orevkov. Separating semigroup of hyperelliptic curves and of genus 3 curves. St. Petersburg Mathematical Journal , 31(1):81--84, December 2019. URL: http://www.ams.org/spmj/2020-31-01/S1061-0022-2019-01586-2/, https://doi.org/10.1090/spmj/1586 doi:10.1090/spmj/1586

work page doi:10.1090/spmj/1586 2019

[8] [8]

Yu Orevkov

S. Yu Orevkov. Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves. Geometric and Functional Analysis , 31(4):930--947, August 2021. arXiv:2010.09130 [math]. URL: http://arxiv.org/abs/2010.09130, https://doi.org/10.1007/s00039-021-00569-1 doi:10.1007/s00039-021-00569-1

arXiv 2021