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Homological stability and weak approximation

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Pith's one-line read Homological stability holds for jet-constrained sections of key Fano fibrations

desk verdict New topological stability results, but the bridge theorem's central inequality (5.2) is false as stated; needs revision before the main theorems stand. read the letter →

arxiv 2509.11021 v2 pith:EIMTWMT2 submitted 2025-09-14 math.AG math.ATmath.NT

Sho Tanimoto , Yuri Tschinkel This is my paper · ORCID
classification math.AGmath.ATmath.NT MSC 14D2014H6014F4514J60
keywords homologicalstabilitysectionspacesFanofibrationsweakapproximationjetconstraintssemi-topologicalmodelsprojectivebundlesquadricsurface
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

This paper tries to prove that the low-degree homology of spaces of sections of certain Fano fibrations over a curve is eventually independent of the degree of the section, even when the sections are required to match a fixed finite set of jet conditions. It establishes this homological stability for three classes: projectivizations of vector bundles, smooth conic bundles, and smooth nonsplit quadric surface bundles with at most A1-singular fibers. The setting is the geometric side of weak approximation: jet conditions are local Taylor constraints, and stability means that the homology of the solution space in a given range is unchanged when the degree is raised. If true, the stabilization gives a topological backbone for counting sections of bounded height with prescribed jets, linking to Manin-type asymptotics and equidistribution over function fields.

What carries the argument

The load-bearing object is the semi-topological model Sect^stop(X/B, alpha, Sigma), a compactly generated space of continuous sections matching jets, together with the Abel-Jacobi realization of the algebraic section space as an open subset of a projective bundle over the Picard variety. The comparison Theorem 15 passes through the bar complex B(P, Z_{k,U}) of vanishing-section strata; its homology controls the Gysin map between algebraic and semi-topological section spaces. Three homotopy/isotopy lemmas (Propositions 6, 7, 8) show that, for topological sections, moving jet supports, gluing a vertical rational curve, and changing jet data all give homotopy equivalences, yielding Corollary 10

What would settle it

Evaluate inequality (5.2) for the vanishing locus of a section of a P^1-bundle with a single 0-jet at b_1 and reduced support y={b_1}: gamma=1, deg=1, |Supp|=1, |Sigma|=1 gives kappa=-1, so deg-|Sigma|=0 is not <= -1. If such a combinatorial type occurs in the bar complex for d in the stabilization range, Proposition 14's conclusion fails, and the isomorphisms of Theorems 16-19 would need a different proof or a modified inequality.

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Extended reading notes

Core claim

The central claim is Condition (HS): for a section class alpha, a vertical rational curve class beta, and a nonempty jet datum Sigma, there is a linear function ell(m) such that H_i(Sect(X/B, alpha+m beta, Sigma), Z) is isomorphic to H_i(Sect(X/B, alpha+(m+1)beta, Sigma), Z) for all i <= ell(m). The paper proves this for projective bundles (Theorem 16), conic bundles (Theorem 17), and quadric surface bundles (Theorem 19), with explicit ell, e.g., ell(d)=d-|Sigma|-A(E,deg Sigma)-2 for a projectivized vector bundle. The proof reduces jets to 0-jets by blowups, replaces the algebraic section space by a semi-topological model identified with an open subset of a projective bundle over Pic^d(B) vi

Load-bearing premise

The proof of the bridge theorem (Theorem 15) depends on inequality (5.2), deg(T)-|Sigma| <= 2 gamma(T)-deg(T)-2|Supp(T)|, stated in Section 5, Proposition 14, without proof; it is false for a P^1-bundle with one 0-jet (deg=1, |Sigma|=1 gives 0 <= -1), and the paper does not state the restriction to the vanishing types that may actually satisfy it.

Editorial extensions

If this is right

  • For any fixed jet datum Sigma, the homology H_i of the section space in degrees i up to a linear function of the section degree is the same at degree d and d+1 (or d+2 for conic/quadric bundles).
  • The inclusion of algebraic sections into the semi-topological model induces isomorphisms in a homology range that grows linearly with the degree, so the topology of the algebraic section space can be read from a topological model.
  • The explicit radius ell(d) = d - |Sigma| - A(E, deg Sigma) - 2 for projective bundles makes the stability range effective once the constant A is determined.
  • Since jets encode weak approximation data, the result shows the topology of the space of sections matching a local constraint stabilizes in the height, the regime relevant to counting points over function fields.
  • The open questions about del Pezzo surfaces or toric varieties are left unresolved; the paper's method suggests they would require a similar parametrization by an Abel-Jacobi-type map.

Reading between the lines

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

  • If the hidden inequality (5.2) is restricted to combinatorial types that actually arise from jet-satisfying sections (where jet points cannot lie in the vanishing locus), the bridge theorem may still hold; checking this restricted inequality for all types in the stabilization range would settle the gap.
  • The same semi-topological method could in principle be applied to other Fano fibrations admitting a moduli parametrization, e.g., del Pezzo surfaces, yielding a stability theorem there.
  • Over a finite field, the stable homology of these section spaces would, via the Grothendieck-Lefschetz trace formula, control the number of sections of bounded height with prescribed jets, making the stability results directly arithmetic.
  • The constant A(E, deg Sigma) is not explicit; if it could be bounded uniformly in the fibration, the stabilization radii would become fully effective for applications.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper introduces Condition (HS), a homological-stability conjecture for spaces of sections of Fano fibrations over a curve with prescribed jet data, and proves it for projective bundles, smooth conic bundles, and smooth nonsplit quadric surface bundles with at most A1-singular fibers. The proof proceeds by constructing a semi-topological model of section spaces, proving homotopy equivalences for changing the support or order of jets (Propositions 6–8, Corollary 10), and then establishing a comparison theorem (Theorem 15) between algebraic and topological section spaces via a stratification and bar-complex argument imported from Das–Tosteson [DT24]. The main theorems 16, 17, and 19 assert explicit stabilization radii, e.g. ℓ(d)=d−|Σ|−A(E,degΣ)−2 in the projective-bundle case. The central technical hinge is the inequality (5.2) in Section 5, which is used in Proposition 14 to verify the hypotheses of the bar-complex approximation theorem.

Significance. If correct, the paper would be a significant advance: it would establish homological stability in the presence of jet constraints, a setting not previously covered, and it would connect arithmetic weak approximation to topological stability. The paper contains several genuinely new and appealing ingredients: the semi-topological section spaces with jets, the jet-twisting construction of the Banach bundle V_stop_{d,Σ}, and explicit homotopies for moving jet supports and gluing rational curves. However, the comparison theorem that carries the entire proof rests on an inequality that is asserted but not proved and is in fact false as stated. Because the main theorems inherit this comparison, the central claim is not yet established by the manuscript as written. The structure is plausible and the defect is localized, so the paper is worth revising rather than rejecting outright.

major comments (3)
  1. [Section 5, Eq. (5.2), Proposition 14] The inequality (5.2) is false as stated. For n=1, take one 0-jet at b_1, so |Σ|=deg(Σ)=1, and take y={b_1}, reduced. Then γ(y)=1, deg(y)=1, |Supp(y)|=1, hence κ(T)=2−1−2=−1 while deg(T)−|Σ|=0. More generally, for each R≥0 the reduced type y={b_1}∪{c_1,…,c_{R+1}} satisfies γ=2R+3, deg=R+2, |Supp|=R+2, so κ=R, but deg=R+2>R+|Σ|; thus κ(T)≤R does not imply that T is one of the types in P. The proof of Proposition 14 explicitly uses this implication, and Proposition 14 is the bar-complex input to Theorem 15. The failing types are not excluded by the geometry: the jet condition is ˇs(b_j)∈ε_{b_j}, and 0 belongs to ε_{b_j}, so sections vanishing at b_j do occur in the strata Z_{k,(L,y)}. A corrected inequality—with a proof, or with the necessary restriction on the ℓ_j—is required before the bridge theorem and the radii in Theorems 16–19 can be accepted.
  2. [Section 5, Proposition 13 vs. Proposition 14] Proposition 13 is stated only for reduced y∈Hilb(B), and it gives the expected codimension 2γ(y) only for reduced y and for |I|≤d−A(E,degΣ). Proposition 14, however, applies the bar-complex theorem to the whole downward closed set P, whose elements y_i can be non-reduced. The paper says that all combinatorial types are saturated but does not prove the codimension statement for non-reduced y. The text therefore contains a gap between the statement of Proposition 13 and its use in Proposition 14. The authors should either prove the non-reduced case or modify the bar-complex setup so that only reduced supports are used and show this modification is harmless.
  3. [Section 5, proof of Theorem 15] The proof of Theorem 15 is a six-step sketch that delegates the decisive comparison to external results: Step 3 uses [DT24, Proposition 6.16], Step 6 invokes 'a version of [DT24, Theorem 5.6]', and Proposition 14 imports [DT24, Theorem 5.9]. The hypotheses of these cited results in the present jet-constrained setting are not stated precisely. In particular, the density of ∪_k W_k and the exact bar-complex approximation statement should be spelled out so that the reader can verify that the existing theorems apply. This is not a mere presentation issue: the proof of the central bridge theorem rests on these imported statements, and a false or inapplicable import would void the main results.
minor comments (3)
  1. [Section 6, Theorem 17] The notation A(E_{r(α)}, deg(Σ)+d) should be clarified: the constant A is introduced in Proposition 13 for a projective bundle E and a jet datum of degree deg(Σ), but in Theorem 17 the reader must identify the relevant vector bundle and jet datum after the strict transform. A sentence making this explicit would improve readability.
  2. [Section 6, Lemma 18] The formula h=2d+deg(E)−|d|/2 uses |d| for what appears to be the degree of the discriminant divisor, while d is also used for the degree of a section. This double use of d is confusing; please disambiguate, e.g. by writing deg(D) for the discriminant degree.
  3. [Section 5, equation (5.2)] The phrase 'One can show that (5.2)' is not a proof. Even if a corrected inequality is found, it should be proved or accompanied by a precise reference. As written, the assertion is unverifiable.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the stability theorems rest on new gluing constructions and externally imported bar complexes; self-citations are contextual or published facts. The unproved inequality (5.2) is a correctness gap, not a circularity.

full rationale

The central derivation is self-contained and non-circular. Condition (HS) for projective bundles (Thm 16) is obtained by composing the algebraic-to-topological comparison (Thm 15) with the semi-topological stability (Cor 10). Cor 10 is proved directly via new explicit homotopies: moving jet supports (Prop 6), gluing rational curves (Prop 7), and independence of jets (Prop 8). Thm 15's comparison is not defined into existence: it uses the Abel-Jacobi description (Prop 11, Prop 12) and a bar-complex comparison imported from DT24 (Das-Tosteson), which has no author overlap with this paper. The constant A(E,deg Sigma) in Prop 13 is an unobstructedness constant, not a parameter fitted to the homology groups being predicted, so no fitted input is renamed as prediction. Conic and quadric bundle cases (Thms 17, 19) reduce via Prop 5 and Lemma 18 to the projective-bundle case; these use published geometric facts (HT06, HT12) rather than an unverified self-citation chain. Self-citations to DLTT25 appear in the introduction and in a passing 'as in [DT24, DLTT25]' methods sentence, but the actual proofs cite DT24 for the bar-complex theorems; DLTT25 is not load-bearing. The quadric normalization from HT12 is a prior published result, not a result whose conclusion is equivalent to the present theorem. We therefore find no circular step. We do flag, per the reviewing rule, a serious omitted proof: Section 5, eq. (5.2) states 'One can show that deg(T)−|Σ| ≤ κ(T) := 2γ(T)−deg(T)−2|Supp(T)|'. This inequality is asserted without proof and is false as stated (e.g., n=1, one 0-jet at b_1, reduced y={b_1}: κ=−1 while deg−|Σ|=0). Prop 14's inference 'by (5.2), κ(T)≤R implies that T is the type of P' depends on it, so Thm 15 and the radii in Thms 16-19 inherit this risk. This is a correctness/rigor gap, not a circular reduction of the claimed stability to its own inputs, and it does not raise the circularity score beyond the minor self-citation level.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No 'graviton-type' entities are introduced: the semi-topological models, the twisted bundle Ẽ^∨(-Sigma), and the iterated blowups X̃_Sigma are explicitly constructed and, in the projective bundle case, proven to match the algebraic spaces (Prop 12) and the original jet spaces (Prop 5). The main un-evidenced inputs are imported theorems (DT24, Aum25), a stated but unproved inequality (5.2), and the deformation claim in Lemma 9; these are captured in the axioms and red flags.

free parameters (1)
  • A(E, deg Sigma) family of constants = unspecified (existence asserted only)
    Introduced in Prop 13 to bound the range |I| <= d - A(E,deg Sigma) where vanishing strata have expected codimension; it appears in all three stability radii (Thm 16: d - |Sigma| - A - 2; Thm 17: h/2 - 1/2 - |Sigma| - d/2 - A - 2; Thm 19: h/2 - deg(E)/2 + |disc|/4 - 2|Sigma| - A - 2). Its value is never computed. Condition (HS) only needs existence of some linear ell, so the qualitative claim does
assumptions (5)
  • standard math The bar-complex comparison theorems [DT24, Thm 5.6 and Thm 5.9] and approximation results [DT24, Prop 6.16] and [Aum25, Lemma 7.2] transfer to the jet-twisted setting (twisted bundle Ẽ^∨(-Sigma), jet-constrained sections, base Pic^d(B)); the hypotheses are listed but not fully checked.
    Proposition 14 and Theorem 15 Steps 5-6 reduce the central comparison to 'a version of [DT24, Theorem 5.9] / Theorem 5.6'. The transfer to the new setting is asserted rather than carried out in the text.
  • standard math Local triviality of smooth morphisms in the Euclidean topology ([Voi02, Prop 9.5]) and the existence of a proper smooth deformation over B from X̃_Σ to X̃_Σ' giving a B-homeomorphism (Lemma 9, Prop 8).
    Proposition 8 and Lemma 9 need that iterated blowups at different jets over the same point are B-homeomorphic; the paper asserts the deformation exists without constructing it, and different blowups are in general non-isomorphic as varieties.
  • domain assumption Fano fibration setup: pi: X -> B is a smooth integral model with relative anticanonical class ample on the generic fiber, and sections are measured by the anticanonical degree (height) and by classes alpha in H^2(X(C),Z) with alpha·X_b = 1.
    Section 1 setup; this is what makes sections, heights, and the jet spaces Sect(X/B, alpha, Sigma) well-defined.
  • domain assumption The structural hypotheses of the three fibrations: vector bundles of rank at least 2; smooth conic bundles whose singular fibers are unions of two lines; smooth nonsplit quadric surface bundles with relative Picard rank one and at most A1-singular fibers, with F_1(X) -> D a smooth P^1-bundle in the
    Section 6; these hypotheses are used in the reductions (6.2), Lemma 18, and the jet-transform degree formulas deg(Sigma_{r(alpha)}) = deg(Sigma) + d and deg(Sigma_D) = 2 deg(Sigma_B).
  • ad hoc to paper Inequality (5.2): deg(T) - |Sigma| <= kappa(T) := 2*gamma(T) - deg(T) - 2*|Supp(T)|, asserted with 'One can show' and used in the proof of Proposition 14.
    As stated it is false (counterexample: n=1, one 0-jet, reduced y = {b_1} gives kappa = -1 < 0 = deg - |Sigma|). It likely holds for the types that actually occur (vanishing loci with ell_j = 0), but the paper does not state the restriction, so the claim as written is an unproved, false-in-general assertion.

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Pith. "Pith review of Homological stability and weak approximation." pith.science (2026). https://pith.science/paper/EIMTWMT2

@misc{pith2026250911021,
  author       = {Pith},
  title        = {Pith review of: Homological stability and weak approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EIMTWMT2}},
  note         = {Machine review of arXiv:2509.11021}
}
read the original abstract

We investigate homological stability for the space of sections of Fano fibrations over curves in the context of weak approximation, and establish it for projective bundles, as well as for conic and quadric surface bundles over curves.

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Works this paper leans on

37 extracted references · 5 linked inside Pith

  1. [1]

    Aumonier

    A. Aumonier. An h-principle for complements of discriminants. Geom. Topol. , 29(3):1441--1488, 2025

    2025

  2. [2]

    V. V. Batyrev. Distribution of rational points of bounded height. a lecture at Math. Inst. Berlin, Thu 21th Jul, 1988

    1988

  3. [3]

    Burke and E

    A. Burke and E. Jovinelly. Geometric M anin's C onjecture for F ano 3 -folds. arXiv:2209.05517, 2022

    arXiv 2022

  4. [4]

    Beheshti, B

    R. Beheshti, B. Lehmann, E. Riedl, and S. Tanimoto. Moduli spaces of rational curves on F ano threefolds. Adv. Math. , 408(Paper No. 108557):60 pp., 2022

    2022

  5. [5]

    V. V. Batyrev and Yu. I. Manin. Sur le nombre des points rationnels de hauteur born\'e des vari\'et\'es alg\'ebriques. Math. Ann. , 286(1-3):27--43, 1990

    1990

  6. [6]

    V. V. Batyrev and Y. Tschinkel. Tamagawa numbers of polarized algebraic varieties. Ast\'erisque , (251):299--340, 1998. Nombre et r\'epartition de points de hauteur born\'ee (Paris, 1996)

    1998

  7. [7]

    R. L. Cohen, J. D. S. Jones, and G. B. Segal. Stability for holomorphic spheres and M orse theory. In Geometry and topology: A arhus (1998) , volume 258 of Contemp. Math. , pages 87--106. Amer. Math. Soc., Providence, RI, 2000

    1998

  8. [8]

    R. Das, B. Lehmann, S. Tanimoto, and P. Tosteson. Homological stability and M anin's conjecture for rational curves on quartic del P ezzo surfaces. arXiv:2506.17071, submitted, 2025

    arXiv 2025

Show all 37 references
  1. [9]

    Das and P

    R. Das and P. Tosteson. Homology of spaces of curves on blowups. arXiv:2405.12968, 2024

    2024 arXiv

  2. [10]

    J. S. Ellenberg, T. Tran, and C. Westerland. Fox- N euwirth- F uks cells, quantum shuffle algebras, and M alle's conjecture for function fields. arXiv:1701.04541, 2023

    2023 arXiv

  3. [11]

    J. S. Ellenberg, A. Venkatesh, and C. Westerland. Homological stability for H urwitz spaces and the C ohen- L enstra conjecture over function fields. Ann. of Math. (2) , 183(3):729--786, 2016

    2016

  4. [12]

    Franke, Yu

    J. Franke, Yu. I. Manin, and Y. Tschinkel. Rational points of bounded height on F ano varieties. Invent. Math. , 95(2):421--435, 1989

    1989

  5. [13]

    Graber, J

    T. Graber, J. Harris, and J. Starr. Families of rationally connected varieties. J. Amer. Math. Soc. , 16(1):57--67, 2003

    2003

  6. [14]

    M. A. Guest. The topology of the space of rational curves on a toric variety. Acta Math. , 174(1):119--145, 1995

    1995

  7. [15]

    Harris, M

    J. Harris, M. Roth, and J. Starr. Rational curves on hypersurfaces of low degree. J. Reine Angew. Math. , 571:73--106, 2004

    2004

  8. [16]

    Hassett and Yu

    B. Hassett and Yu. Tschinkel. Weak approximation over function fields. Invent. Math. , 163(1):171--190, 2006

    2006

  9. [17]

    Hassett and Yu

    B. Hassett and Yu. Tschinkel. Weak approximation for hypersurfaces of low degree. In Algebraic geometry--- S eattle 2005. P art 2 , volume 80, Part 2 of Proc. Sympos. Pure Math. , pages 937--955. Amer. Math. Soc., Providence, RI, 2009

    2005

  10. [18]

    Hassett and Y

    B. Hassett and Y. Tschinkel. Spaces of sections of quadric surface fibrations over curves. In Compact moduli spaces and vector bundles , volume 564 of Contemp. Math. , pages 227--249. Amer. Math. Soc., Providence, RI, 2012

    2012

  11. [19]

    Landesman and I

    A. Landesman and I. Levy. An alternate computation of the stable homology of dihedral group H urwitz spaces. arXiv:2410.22222, 2024

    2024 arXiv

  12. [20]

    Landesman and I

    A. Landesman and I. Levy. The C ohen-- L enstra moments over function fields via the stable homology of non-splitting H urwitz spaces. arXiv:2410.22210, 2024

    2024

  13. [21]

    Landesman and I

    A. Landesman and I. Levy. Homological stability for H urwitz spaces and applications. arXiv:2503.03861, 2025

    2025

  14. [22]

    Lehmann, E

    B. Lehmann, E. Riedl, and S. Tanimoto. Non-free sections of F ano fibrations. Mem. Amer. Math. Soc. , 2025. to appear

    2025

  15. [23]

    Lehmann, A

    B. Lehmann, A. K. Sengupta, and S. Tanimoto. Geometric consistency of M anin's C onjecture. Compos. Math. , 158(6):1375--1427, 2022

    2022

  16. [24]

    Lehmann and S

    B. Lehmann and S. Tanimoto. Geometric M anin's conjecture and rational curves. Compos. Math. , 155(5):833--862, 2019

    2019

  17. [25]

    Lehmann and S

    B. Lehmann and S. Tanimoto. Rational curves on prime F ano threefolds of index 1. J. Algebraic Geom. , 30(1):151--188, 2021

    2021

  18. [26]

    Lehmann and S

    B. Lehmann and S. Tanimoto. Classifying sections of del P ezzo fibrations, II . Geom. Topol. , 26(6):2565--2647, 2022

    2022

  19. [27]

    Lehmann and S

    B. Lehmann and S. Tanimoto. Classifying sections of del P ezzo fibrations, I . J. Eur. Math. Soc. (JEMS) , 26(1):289--354, 2024

    2024

  20. [28]

    F. Okamura. The irreducibility of the spaces of rational curves on del P ezzo manifolds. Int. Math. Res. Not. IMRN , (12):9893--9909, 2024

    2024

  21. [29]

    F. Okamura. Rational curves on F ano threefolds with G orenstein terminal singularities. Eur. J. Math. , 11(2):Paper No. 38, 25, 2025

    2025

  22. [30]

    E. Peyre. Hauteurs et mesures de T amagawa sur les vari\'et\'es de F ano. Duke Math. J. , 79(1):101--218, 1995

    1995

  23. [31]

    E. Peyre. Points de hauteur born\'ee, topologie ad\'elique et mesures de T amagawa. volume 15, pages 319--349. 2003. Les XXII\`emes Journ\'ees Arithmetiques (Lille, 2001)

    2003

  24. [32]

    E. Peyre. Libert\'e et accumulation. Doc. Math. , 22:1615--1659, 2017

    2017

  25. [33]

    Riedl and D

    E. Riedl and D. Yang. Kontsevich spaces of rational curves on F ano hypersurfaces. J. Reine Angew. Math. , 748:207--225, 2019

    2019

  26. [34]

    G. Segal. The topology of spaces of rational functions. Acta Math. , 143(1-2):39--72, 1979

    1979

  27. [35]

    Zh. Tian. Weak approximation for cubic hypersurfaces. Duke Math. J. , 164(7):1401--1435, 2015

    2015

  28. [36]

    C. Voisin. Hodge theory and complex algebraic geometry. I , volume 76 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2002. Translated from the French original by Leila Schneps

    2002

  29. [37]

    Ch. Xu. Weak approximation for low degree del P ezzo surfaces. J. Algebraic Geom. , 21(4):753--767, 2012

    2012

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