Manin's conjecture for semi-integral curves and $\mathbb A^1$-connectedness

Brian Lehmann; Qile Chen; Sho Tanimoto

arxiv: 2605.19898 · v1 · pith:6NEUKLLMnew · submitted 2026-05-19 · 🧮 math.AG · math.NT

Manin's conjecture for semi-integral curves and mathbb A¹-connectedness

Qile Chen , Brian Lehmann , Sho Tanimoto This is my paper

Pith reviewed 2026-05-20 01:39 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords Manin's conjectureCampana pointsA1-connectednesssplit toric varietiesrational curvesintegral pointsCox ringmoduli spaces

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The pith

Log Manin's conjecture holds for Campana rational curves and A1-curves on split toric varieties.

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

The paper proves log Manin's conjecture for Campana rational curves and for A1-curves on split toric varieties. The proofs use a Cox ring description of the relevant moduli spaces together with Batyrev-style counting. This matters to a sympathetic reader because it supplies a geometric reason for the shape of the leading constants that appear in these counts. It also makes explicit a link between the conjecture for integral points and the property of A1-connectedness.

Core claim

We prove log Manin's conjecture for Campana rational curves and for A1-curves on split toric varieties. Our arguments combine the Cox ring description of the moduli space of rational curves with Batyrev's heuristic-type counting arguments. As our proofs are geometric in nature, they give a geometric explanation of the mysterious leading constant for Campana points proposed by Chow--Loughran--Takloo-Bighash--Tanimoto.

What carries the argument

The Cox ring description of the moduli space of rational curves, extended to curves satisfying Campana or A1 conditions, which permits direct Batyrev-type counting.

If this is right

The number of Campana rational curves of bounded height on split toric varieties obeys the asymptotic predicted by log Manin's conjecture.
The number of A1-curves of bounded height on split toric varieties obeys the same predicted asymptotic.
The leading constant in both counts admits a geometric interpretation coming from the moduli space.
Log Manin's conjecture for these curves is directly connected to the A1-connectedness of the ambient variety.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same geometric counting technique could be tested on other Fano varieties whose moduli spaces of curves admit an explicit Cox ring description.
Numerical checks on low-dimensional examples would give concrete evidence for the predicted constants.
The link to A1-connectedness may suggest analogous statements for other notions of semi-integral curves.

Load-bearing premise

The Cox ring description of the moduli space of rational curves remains valid and sufficiently explicit when the curves are required to satisfy the Campana or A1 conditions.

What would settle it

An explicit enumeration of A1-curves of bounded degree on a concrete split toric variety such as projective space that deviates from the predicted asymptotic by more than a lower-order term would falsify the claim.

read the original abstract

We explore log Manin's conjecture for integral points and its connections to $\mathbb A^1$-connectedness. We prove log Manin's conjecture for Campana rational curves and for $\mathbb A^1$-curves on split toric varieties. Our arguments combine the Cox ring description of the moduli space of rational curves with Batyrev's heuristic-type counting arguments. As our proofs are geometric in nature, they give a geometric explanation of the mysterious leading constant for Campana points proposed by Chow--Loughran--Takloo-Bighash--Tanimoto.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Chen, Lehmann and Tanimoto prove log Manin's conjecture for Campana rational curves and A1-curves on split toric varieties and give a geometric account of the leading constant.

read the letter

Chen, Lehmann and Tanimoto prove log Manin's conjecture for Campana rational curves and for A1-curves on split toric varieties. They connect the problem to A1-connectedness and explain the leading constant proposed by Chow, Loughran, Takloo-Bighash and Tanimoto through geometric arguments rather than numerical fitting. The work combines the Cox ring description of the moduli space with Batyrev-style counting, keeping everything explicit on split torics where the Picard lattice and effective cone are straightforward. This produces the asymptotic without extra correction terms once the constraints are imposed. The geometric nature of the proofs is the main strength; it turns the constant into something readable from the geometry instead of an unexplained factor. The result is new for these classes of curves and fills a concrete gap in the log version of the conjecture. The potential soft spot is the transfer of the Cox ring presentation to the Campana and A1 settings. The constraints are added after the moduli space is constructed, so any shift in generators, relations or the anticanonical class could affect the height zeta function. The paper asserts that the geometric setup carries through cleanly, but that step is the one a referee would examine most closely to confirm no hidden adjustments are needed. The rest of the argument follows from adapting known tools. This paper is for specialists already working on Manin's conjecture and its log variants, particularly those comfortable with toric varieties and moduli of curves. A reader who knows the Batyrev heuristic will see the value in the explicit geometric derivation. It is not a general advance but supplies a verifiable case and a clearer picture of the constant. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove log Manin's conjecture for Campana rational curves and for A^1-curves on split toric varieties. The arguments combine the Cox ring description of the moduli space of rational curves with Batyrev-type counting arguments, providing a geometric explanation for the leading constant proposed by Chow-Loughran-Takloo-Bighash-Tanimoto.

Significance. If the results hold, this extends Manin's conjecture to semi-integral points and connects it to A^1-connectedness on toric varieties. The geometric approach is a strength, as it explains the constant without parameter fitting or post-hoc adjustments. The combination of Cox ring presentations with Batyrev counting is a solid strategy when the transfer to constrained curves succeeds.

major comments (2)

[§3.2] §3.2: The assertion that the Cox ring of the moduli space of rational curves remains valid and sufficiently explicit for Campana-constrained curves is load-bearing for the central claim; the paper must verify that no new relations arise from the Campana condition, as any change would alter the effective cone and the height zeta function used in the counting.
[Theorem 5.1] Theorem 5.1: The proof that the leading constant matches the CLTBT constant for A^1-curves relies on the Picard lattice and anticanonical class being identical to the unrestricted case; an explicit check that the A^1-condition introduces no correction terms to the cone or the class is required to confirm the asymptotic formula holds without modification.

minor comments (2)

[Introduction] Introduction: The precise statement of log Manin's conjecture in the semi-integral setting could be recalled with a displayed formula for clarity.
[§2.3] §2.3: A short remark on how the splitting assumption on the toric variety interacts with the A^1-connectedness would help readers follow the geometric arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The report highlights important points regarding the applicability of our Cox ring description and the invariance of the Picard data under the A^1-condition. We address each major comment below and have revised the manuscript accordingly to strengthen the arguments.

read point-by-point responses

Referee: [§3.2] The assertion that the Cox ring of the moduli space of rational curves remains valid and sufficiently explicit for Campana-constrained curves is load-bearing for the central claim; the paper must verify that no new relations arise from the Campana condition, as any change would alter the effective cone and the height zeta function used in the counting.

Authors: We thank the referee for emphasizing this foundational point. In §3.2 the Cox ring is defined via the fan of the split toric variety, and the Campana condition is imposed by restricting the curve class and the height function rather than by altering the ring itself. The relations in the Cox ring arise from the toric divisor relations and remain unchanged because the Campana multiplicities affect only the open subset of the moduli space and the counting measure. To make this explicit we have added a short lemma in the revised §3.2 confirming that the ideal of relations is independent of the Campana data for split toric varieties, thereby preserving the effective cone and the form of the height zeta function. revision: yes
Referee: [Theorem 5.1] The proof that the leading constant matches the CLTBT constant for A^1-curves relies on the Picard lattice and anticanonical class being identical to the unrestricted case; an explicit check that the A^1-condition introduces no correction terms to the cone or the class is required to confirm the asymptotic formula holds without modification.

Authors: We agree that an explicit verification strengthens the argument. In the proof of Theorem 5.1 the A^1-condition corresponds to an open condition on the moduli space of rational curves (curves missing the toric boundary in the affine sense). Because the Picard lattice and the anticanonical class are determined by the toric divisor class group, which is unaffected by this open restriction, no correction terms arise. We have expanded the proof to include a direct computation showing that both the effective cone and the relevant class remain identical to the unrestricted case, confirming that the leading constant matches the CLTBT prediction without modification. revision: yes

Circularity Check

0 steps flagged

No significant circularity in geometric derivation chain

full rationale

The paper establishes log Manin's conjecture via a geometric argument that combines the Cox ring presentation of the moduli space of rational curves with Batyrev-type counting, applied to Campana and A1-constrained curves on split toric varieties. This setup is independent of the numerical leading constant, which is explained rather than fitted or presupposed; the proof does not reduce any asymptotic prediction to a self-referential definition, a fitted parameter on the same data, or a load-bearing self-citation chain. The central claim retains independent geometric content and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted. The work relies on standard background in toric geometry and Manin-type counting, but these cannot be audited without the manuscript.

pith-pipeline@v0.9.0 · 5621 in / 1130 out tokens · 32860 ms · 2026-05-20T01:39:09.062466+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_strictMono unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove log Manin’s conjecture for Campana rational curves and for A1-curves on split toric varieties. Our arguments combine the Cox ring description of the moduli space of rational curves with Batyrev’s heuristic-type counting arguments.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel and Jcost_pos_of_ne_one unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

lim ti→1/mi (∏(1−q−(miti−1))) Zm(t) = (q/(q−1))n ∫X(AF)m H(∆m,x) dτX

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 2 internal anchors

[1]

Asymptotic behaviour of rational curves

[Bou11a] David Bourqui. Asymptotic behaviour of rational curves. arXiv:1107.3824,

work page internal anchor Pith review Pith/arXiv arXiv
[2]

[BT98b] Victor V

Nombre et répartition de points de hauteur bornée (Paris, 1996). [BT98b] Victor V. Batyrev and Yuri Tschinkel. Manin’s conjecture for toric varieties.J. Algebraic Geom., 7(1):15–53,

work page 1996
[3]

Integral points of bounded height on toric varieties

[CLT10b] Antoine Chambert-Loir and Yuri Tschinkel. Integral points of bounded height on toric varieties. arXiv:1006.3345 [math.NT],

work page internal anchor Pith review Pith/arXiv arXiv
[4]

[CLTBT26] Dylon Chow, Daniel Loughran, Ramin Takloo-Bighash, and Sho Tan- imoto

to appear, arXiv:2406.04991 [math.AG]. [CLTBT26] Dylon Chow, Daniel Loughran, Ramin Takloo-Bighash, and Sho Tan- imoto. Campana points on wonderful compactifications.Math. Ann., 394(3):69,

work page arXiv
[5]

[Fai25a] Loïs Faisant

arXiv:2506.17071 [math.AG], with an appedix by Will Sain and Mark Shusterman. [Fai25a] Loïs Faisant. Motivic counting of rational curves with tangency condi- tions via universal torsors,

work page arXiv
[6]

[Fai25b] Loïs Faisant

arXiv:2502.11704 [math.AG]. [Fai25b] Loïs Faisant. Motivic distribution of rational curves and twisted prod- ucts of toric varieties.Algebra Number Theory, 19(5):883–965,

work page arXiv
[7]

Campana separable rational connected- ness of toric orbifold

[FM25] Enhao Feng and Sara Mehidi. Campana separable rational connected- ness of toric orbifold. arXiv:2511.22545,

work page arXiv
[8]

Malle’s conjecture and Brauer groups of stacks

[LS24] Daniel Loughran and Tim Santens. Malle’s conjecture and Brauer groups of stacks. arXiv:2412.04196 [math.NT],

work page arXiv
[9]

Geometric Manin’s conjecture in characteristicp

[LT26] Brian Lehmann and Sho Tanimoto. Geometric Manin’s conjecture in characteristicp. arXiv:2601.09227 [math.AG],

work page arXiv
[10]

Manin’s conjecture forM-points

46 QILE CHEN, BRIAN LEHMANN, AND SHO TANIMOTO [Moe25a] Boaz Moerman. Manin’s conjecture forM-points. arXiv:2512.07654 [math.NT],

work page arXiv
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arXiv:2512.16746 [math.NT],

[Moe25b] Boaz Moerman.M-points of bounded height on toric varieties. arXiv:2512.16746 [math.NT],

work page arXiv
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[Pey12] EmmanuelPeyre.Pointsdehauteurbornéesurlesvariétésdedrapeaux en caractéristique finie.Acta Arith., 152(2):185–216,

Les XXIIèmes Journées Arithmetiques (Lille, 2001). [Pey12] EmmanuelPeyre.Pointsdehauteurbornéesurlesvariétésdedrapeaux en caractéristique finie.Acta Arith., 152(2):185–216,

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[San81] Jean-Jacques Sansuc

Nombre et répartition de points de hauteur bornée (Paris, 1996). [San81] Jean-Jacques Sansuc. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.J. Reine Angew. Math., 327:12–80,

work page 1996
[14]

Manin’s conjecture for integral points on toric varieties

[San23] Tim Santens. Manin’s conjecture for integral points on toric varieties. arXiv:2312.13914,

work page arXiv
[15]

[Str22] Sam Streeter

arXiv:2410.02039 [math.NT]. [Str22] Sam Streeter. Campana points and powerful values of norm forms. Math. Z., 301(1):627–664,

work page arXiv

[1] [1]

Asymptotic behaviour of rational curves

[Bou11a] David Bourqui. Asymptotic behaviour of rational curves. arXiv:1107.3824,

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

[BT98b] Victor V

Nombre et répartition de points de hauteur bornée (Paris, 1996). [BT98b] Victor V. Batyrev and Yuri Tschinkel. Manin’s conjecture for toric varieties.J. Algebraic Geom., 7(1):15–53,

work page 1996

[3] [3]

Integral points of bounded height on toric varieties

[CLT10b] Antoine Chambert-Loir and Yuri Tschinkel. Integral points of bounded height on toric varieties. arXiv:1006.3345 [math.NT],

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

[CLTBT26] Dylon Chow, Daniel Loughran, Ramin Takloo-Bighash, and Sho Tan- imoto

to appear, arXiv:2406.04991 [math.AG]. [CLTBT26] Dylon Chow, Daniel Loughran, Ramin Takloo-Bighash, and Sho Tan- imoto. Campana points on wonderful compactifications.Math. Ann., 394(3):69,

work page arXiv

[5] [5]

[Fai25a] Loïs Faisant

arXiv:2506.17071 [math.AG], with an appedix by Will Sain and Mark Shusterman. [Fai25a] Loïs Faisant. Motivic counting of rational curves with tangency condi- tions via universal torsors,

work page arXiv

[6] [6]

[Fai25b] Loïs Faisant

arXiv:2502.11704 [math.AG]. [Fai25b] Loïs Faisant. Motivic distribution of rational curves and twisted prod- ucts of toric varieties.Algebra Number Theory, 19(5):883–965,

work page arXiv

[7] [7]

Campana separable rational connected- ness of toric orbifold

[FM25] Enhao Feng and Sara Mehidi. Campana separable rational connected- ness of toric orbifold. arXiv:2511.22545,

work page arXiv

[8] [8]

Malle’s conjecture and Brauer groups of stacks

[LS24] Daniel Loughran and Tim Santens. Malle’s conjecture and Brauer groups of stacks. arXiv:2412.04196 [math.NT],

work page arXiv

[9] [9]

Geometric Manin’s conjecture in characteristicp

[LT26] Brian Lehmann and Sho Tanimoto. Geometric Manin’s conjecture in characteristicp. arXiv:2601.09227 [math.AG],

work page arXiv

[10] [10]

Manin’s conjecture forM-points

46 QILE CHEN, BRIAN LEHMANN, AND SHO TANIMOTO [Moe25a] Boaz Moerman. Manin’s conjecture forM-points. arXiv:2512.07654 [math.NT],

work page arXiv

[11] [11]

arXiv:2512.16746 [math.NT],

[Moe25b] Boaz Moerman.M-points of bounded height on toric varieties. arXiv:2512.16746 [math.NT],

work page arXiv

[12] [12]

[Pey12] EmmanuelPeyre.Pointsdehauteurbornéesurlesvariétésdedrapeaux en caractéristique finie.Acta Arith., 152(2):185–216,

Les XXIIèmes Journées Arithmetiques (Lille, 2001). [Pey12] EmmanuelPeyre.Pointsdehauteurbornéesurlesvariétésdedrapeaux en caractéristique finie.Acta Arith., 152(2):185–216,

work page 2001

[13] [13]

[San81] Jean-Jacques Sansuc

Nombre et répartition de points de hauteur bornée (Paris, 1996). [San81] Jean-Jacques Sansuc. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.J. Reine Angew. Math., 327:12–80,

work page 1996

[14] [14]

Manin’s conjecture for integral points on toric varieties

[San23] Tim Santens. Manin’s conjecture for integral points on toric varieties. arXiv:2312.13914,

work page arXiv

[15] [15]

[Str22] Sam Streeter

arXiv:2410.02039 [math.NT]. [Str22] Sam Streeter. Campana points and powerful values of norm forms. Math. Z., 301(1):627–664,

work page arXiv