Continuity of heights in families and complete intersections in toric varieties

Micha{\l} Szachniewicz; Nuno Hultberg; Pablo Destic

REVIEW 3 major objections 5 minor 2 references

Continuity of heights in families and complete intersections in toric varieties

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Pith's one-line read Heights of cycles vary continuously in flat families, yielding an exact toric limit-height formula.

desk verdict Proves the full Gualdi conjecture via a new GVF-analytification continuity theorem; the strategy is original and mostly solid, but two load-bearing steps are sketched rather than proved. read the letter →

arxiv 2412.15988 v1 pith:TFRT2ZE2 submitted 2024-12-20 math.NT math.AGmath.LO

Pablo Destic , Nuno Hultberg , Micha{l} Szachniewicz This is my paper · ORCID

classification math.NTmath.AGmath.LO MSC 14G4014M2552B2003C66

keywords toricvarietyArakelovgeometrygloballyvaluedfieldheightofcyclescompleteintersectionRonkinfunctionmixedintegrallimit

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 establishes that arithmetic heights of cycles vary continuously in flat projective families, once the base is viewed through its 'globally valued field' (GVF) analytification—a space whose points carry height functions satisfying a product formula rather than just valuations. From this continuity result it derives an exact limit formula: if Laurent polynomials define hypersurfaces in a proper toric variety, then the height of the intersection of their generically translated copies, in the limit of a generic small torus point, equals a sum over places of mixed integrals of the roof functions of the ambient divisors and the Legendre transforms of the Ronkin functions of the polynomials. This proves the conjecture posed in [Gua18a]. The payoff is that a transcendental-looking limit of heights becomes an explicit convex-geometric quantity, computable from Newton polytopes and concave functions; in one worked case it equals $2\zeta(3)/(3\zeta(2))$.

What carries the argument

The load-bearing device is the GVF analytification of a finite-type $K$-scheme $S$: points are pairs of a scheme point and a height function on its residue field extending the height of $K$, topologized so that every tuple of regular functions has continuous height. Over this space the paper defines globally integrable line bundles by uniform approximation from lattice line bundles pulled back from projective spaces with Weil or Fubini–Study metrics. The continuity theorem for $\widehat{\deg}$ on fibers is proved by expressing the intersection number through heights of resultants, which are manifestly continuous in the GVF topology. In the toric application the computation is carried by three further objects: Ronkin divisors, whose roof functions convert the height of a hypersurface into a height on the toric variety; the mixed integral, which polarizes integrals of concave functions just as mixed volume polarizes volume; and a non-Archimedean Fubini principle that lets integrals over the product torus be evaluated fiber by fiber, with the mutual vanishing of the Ronkin function and $\log|f|$ on each fiber doing the required cancellation.

What would settle it

Compute both sides of Theorem 4.5 independently in a new configuration, for example $K=\mathbb{Q}$, $T=\mathbb{P}^2$, $m=2$, $f_1=f_2=x_1+x_2+1$ with the same toric divisors used in the paper; the known value $2\zeta(3)/(3\zeta(2))$ provides a sharp target, so any mismatch would refute the theorem. Alternatively, test the Fubini step in Theorem 4.2 directly: over a non-Archimedean field, evaluate the double integral of $\log|g\,s|$ against $\pi^*c_1(\mathcal{O}(1))^n$ on the vanishing locus of $g$ and check whether the inner fiber integral vanishes identically; a nonzero boundary term would break the equality.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 4.5: for Laurent polynomials $f_1,\dots,f_m$ in $n$ variables over a number field $K$, a proper toric variety $T$ with torus $\mathbb{G}_m^n\subset T$, and semipositive toric Zhang divisors $D_0,\dots,D_{n-m}$, the limit $$\lim_{j\to\infty}\widehat{\deg}(D_0,\dots,D_{n-m}|_{\zeta_{1,j}V_1\cap\cdots\cap \zeta_{m,j}V_m})=\sum_{v\in M_K} n_v \operatorname{MI}(\theta_{0,v},\dots,\theta_{n-m,v},\rho_1^\vee,\dots,\rho_m^\vee)$$ holds for every generic small sequence $(\zeta_{1,j},\dots,\zeta_{m,j})$ in the torus, i.e. for torus points whose Weil height tends to zero and which are chosen generically. Here $V_i$ is the hypersurface of $f_i$, $\theta_{i,v}$ are the local roof functions of the toric divisors, and $\rho_i^\vee$ are Legendre transforms of the Ronkin functions. The proof combines a general continuity theorem for fiber heights in flat projective families over a GVF with a toric comparison between intersections on the family and intersections with Ronkin divisors; the core of that comparison is a non-Archimedean Fubini step.

Load-bearing premise

The load-bearing premise is the non-Archimedean Fubini principle used in Theorem 4.2: integrals of the forms defining the height of the intersection can be evaluated iteratively over the torus fibers with product measures and no boundary contributions, a step the paper sketches rather than proves in full.

Editorial extensions

If this is right

For a flat projective family over a number field, if a sequence of base points has convergent small-point heights, then all fiber intersection heights converge, with the limit read off from the GVF analytification and identified with an arithmetic intersection number by Proposition 3.26.
The height limit for translated complete intersections is independent of the chosen generic small sequence and depends only on Newton polytopes and concave roof functions of the polynomials and divisors.
Arithmetic heights of these intersections become explicit convex-geometric quantities; in the $m=2$, $T=\mathbb{P}^2$, $f_1=f_2=x_1+x_2+1$ case the limit equals $2\zeta(3)/(3\zeta(2))$.
Because every integrable Zhang line bundle on a projective variety over a number field is globally integrable, the continuity theorem applies to all semipositive toric divisors appearing in the conjecture.

Reading between the lines

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

A fully formal proof of the non-Archimedean Fubini step would make the same average-intersection method available for other families equipped with a torus fibration and product measures, not only complete intersections in toric varieties.
The continuity theorem suggests that the GVF analytification itself is a natural home for height-convergence statements: rather than fixing one polarisation and applying an equidistribution theorem, one could study convergence of all fiber heights directly on this space.
The mixed-integral right-hand side is an explicit computational target: for new polynomials and toric divisors, one can produce numerical predictions for generic small sequences and check them by direct height computation, which tests both the theorem and the Fubini step.

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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

Bottom line: this paper takes a real shot at Gualdi's conjecture for arbitrary m and, to my reading, hits it modulo two explicitly flagged technical gaps. The main new idea — prove a continuity/definability statement for heights in flat families over globally valued fields, then apply it to a family of translates of complete intersections and use Yuan's equidistribution — is original and well suited to the problem. The earlier special cases (P^2 with f1=f2=x1+x2+1, then m=2) used local logarithmic equidistribution; this paper instead goes through uniform convergence of resultant heights and a polarised GVF structure, which is a different and genuinely new route.

The resultants section is the strongest part: the intersection product on lattice line bundles is computed as a limit of heights of resultants with an explicit error bound (Remark 3.16), and the continuity of the fibre height then falls out in a clean epsilon argument. The appendix on mixed Mahler measures is self-contained and useful. The authors also honestly flag where they are sketching: Theorem 3.24 asserts without proof that arithmetically ample divisors are dense in semipositive Zhang divisors, and the non-Archimedean Fubini step in Theorem 4.2 is introduced with 'we sketch an argument' and 'the general case follows by approximation.' Both are load-bearing for the final theorem. I do not see an outright contradiction, and the specific worry that toric boundary strata carry positive Chambert-Loir mass does not land for the toric metrics used here — their supports sit in the open torus — but the iterative slicing argument on the complete-intersection cycle does need a real proof, not a sketch. The self-citation point is a red herring: the GVF framework comes from papers by two of the authors, but the central resultant argument is independent and the reliance is legitimate.

My overall take: this is a substantial advance, with a plausible and mostly explicit proof, but the two sketched steps mean the paper is not yet a finished reference. I'd want a referee to push hard on Theorem 4.2 before accepting the main theorem as proved. Who is it for: Arakelov geometers and model theorists working on heights, and toric geometers. It deserves serious peer review, and in the meantime I would cite the continuity theorem and the Gualdi result with a caveat.

Referee Report

3 major / 5 minor

Summary. The paper develops a theory of globally valued field (GVF) analytifications for finite type schemes and proves a continuity theorem for arithmetic intersection numbers in flat projective families over a GVF base. The main abstract result is Theorem 3.1/3.23: for a flat projective morphism X→S of relative dimension d and globally integrable line bundles L0,...,Ld on X/S, the fiber height s↦deg(L0,...,Ld|Xs) is continuous on S_GVF. The proof proceeds by expressing the intersection number as a limit of heights of resultants over the base, with a uniform error term (Remark 3.16). This continuity result is then applied to prove Gualdi's conjecture on limit heights of complete intersections in toric varieties (Theorem 4.5): for Laurent polynomials f1,...,fm over a number field and semipositive toric Zhang divisors D0,...,D_{n-m} on a proper toric variety T, the limit of the heights of ζ_{1,j}V1∩...∩ζ_{m,j}Vm along a generic small sequence equals a sum of mixed integrals of the roof functions of the Di and the Legendre transforms of the Ronkin functions of the fi. The proof combines the continuity theorem with a polarised GVF structure on the function field of the translation torus T^m, a resultant/Ronkin divisor computation (Theorem 4.2), and convex-geometric projection formulas (Lemmas 2.26 and 2.27).

Significance. If the missing technical arguments are supplied, this is a substantial contribution. It provides a general and apparently new continuity theorem for heights in flat families over GVF bases, with an explicit and quantitative resultant route rather than a purely abstract compactness argument. The application to Gualdi's conjecture is a concrete, checkable result: it gives the limit height as a sum of mixed integrals, generalizing the earlier special cases of Gualdi-Sombra. The paper also contains a self-contained Appendix A with an elementary Mahler-measure estimate needed for the resultant comparison. The GVF framework is imported from companion papers by the same authors, but the central uniform-convergence argument for resultant heights and the toric application appear original. However, two load-bearing steps are only sketched: the non-Archimedean Fubini/vanishing argument in Theorem 4.2 and the density of arithmetically ample divisors in semipositive Zhang divisors used in Theorem 3.24. These need to be proved or precisely referenced before the main theorems can be considered established.

major comments (3)

[Section 4, proof of Theorem 4.2] The identity deg(R1...Rm·π_h^*D0...π_h^*D_{n-m}·π_1^*O(1)^n...π_m^*O(1)^n|X) = deg(π_h^*D0...π_h^*D_{n-m}·π_1^*O(1)^n...π_m^*O(1)^n|V-tilde) is not proved. The paragraph beginning 'We need to be slightly careful when applying Fubini's theorem' explicitly says 'We sketch an argument' and concludes 'The general case follows by approximation.' The reduction to the open torus chart assumes that 'a Zariski closed subset with empty interior is a nullset with respect to a measure associated to differential forms,' and then invokes an adaptation of [Sto21, Proposition 3.4.21]. For Chambert-Loir measures attached to semipositive toric metrics on a toric compactification, this null-set statement is not automatic: the toric boundary can carry positive mass, and the residual measure obtained on div(g1)∩...∩div(g_{r-1}) can meet toric strata, for example when one of the gi is a monomial. This equality is load-bearing: Lemma 4.1 and Lemma 4.3 both feed into Theorem 4.5 through it, so Theorem 4.5 collapses if the Fubini factorization or the boundary vanishing fails. A complete proof of these analytic facts, or a reference covering exactly this situation, is required.
[Section 3.4, proof of Theorem 3.24] The statement 'Arithmetically ample divisors in turn are dense in semipositive Zhang divisors allowing us to finish the proof' is asserted without proof or reference. This density is not a formality; it is an arithmetic Demailly-type approximation statement, closely related to work of Charles and of Qu-Yin (see [QY23]). It is exactly the step that upgrades continuity from lattice line bundles to integrable Zhang divisors, and it is used in Corollary 3.2 and in the application in Section 4 (Lemma 4.1 says the construction 'makes sense by Theorem 3.24'). The proof given only addresses approximation by arithmetically ample divisors; the semipositive case is not supplied. Please provide a proof of the density assertion or a precise reference with a verification that its hypotheses are satisfied in the present setting.
[Section 4, proof of Theorem 4.5] The reduction 'We apply the projection formula to restrict to the case, where the NP(fi) define divisors on T' is not spelled out. Theorem 4.4 assumes NP(fi) define divisors on T, while Theorem 4.5 does not; the reduction must be checked on both sides of the limit identity, including the behavior of the translated hypersurfaces ζ_{i,j}Vi and of the Ronkin divisors under the toric modification. Please state the projection formula used and explain why the mixed-integral expression is unchanged under this reduction.

minor comments (5)

[Throughout, Theorem 1.1 and Section 4] The letter T is used both for the proper toric variety and for its open torus T=G^n; this is very confusing in statements such as 'T = G^n ⊂ T'. Please introduce a separate notation for the torus, for instance T0 or G^n.
[Definition 2.2] The notation ω(a):=−log|a|_ω and then h(x1,...,xn):=∫ −min_i(ω(x_i)) dν(ω) overloads ω for both a place and the associated valuation; please write val_ω(a) or |a|_ω consistently.
[Section 4, before Lemma 4.1] The map V→T^m is said to be flat over a dense Zariski open U⊆T^m and Theorem 3.1 is applied to V/U, but the limit point η_can of the generic small sequence is asserted to lie in U_GVF. Please justify this, or state explicitly that the construction of the limit point accounts for the flat locus.
[Theorem 4.2 and Lemma 4.3] The notation for the Ronkin divisors changes between the theorem and the lemma: in Theorem 4.2 the Ri are associated to gi=fi(z1w_{1,i}^{-1},...,znw_{n,i}^{-1}) on a toric blow-up of T×∏ P^n, while Lemma 4.3 assumes 'Ri denote the Ronkin line bundle associated to fi and suppose it is already defined on T'. Please clarify the relationship between these two sets of divisors and the toric blow-up X.
[Appendix A, Proposition A.1] The statement of Proposition A.1 is correct in substance, but the notation m_{S_{m1}}×...×S_{mn}(P) is easy to misread as a product of measures rather than an integral over a product of spheres; a short sentence defining the measure would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the continuity theorem and the toric limit formula rest on independent external results, and the self-citations supply framework rather than reducing the claim to its own inputs.

full rationale

The derivation chain is not circular. Theorem 3.1 is proved by expressing fiber degrees as normalized heights of resultants (Lemma 3.17, Proposition 3.18), using the Mahler-measure estimate of Appendix A and Chen–Moriwaki's adelic intersection product [CM21]; none of these inputs is the conclusion. Theorem 4.5 is assembled from Theorem 3.1, the external toric Arakelov identities of Burgos Gil–Philippon–Sombra [BPS14, Theorem 5.2.5] and Gualdi's Ronkin-divisor theorem [Gua18b, Theorem 5.12], plus combinatorial Lemma 2.27 proved inside the paper; no 'prediction' is a fitted parameter renamed. The proof of Theorem 4.2 contains an unproved non-Archimedean Fubini step ('We sketch an argument' and 'The general case follows by approximation'), and the identification of the limit point in Lemma 4.1 uses the authors' prior work [Sza23] and [Ben+24] through Corollary 2.15; these are genuine dependence risks, but they are not circular reductions, because the cited results are separate theorems with assumptions not containing Gualdi's conjecture, and the central toric identity is not assumed in their statements. Hence no part of the claimed derivation is equivalent by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central proof imports substantial external machinery, but no free parameters are fitted to data. The main unstated premises are the GVF/adelic-curve dictionary, Chen-Moriwaki intersection theory, Yuan equidistribution, the BPS toric mixed-integral formula, and a sketched non-Archimedean Fubini principle.

assumptions (6)

domain assumption Countable GVF structures are represented by proper adelic curves, and Chen-Moriwaki intersection theory over adelic curves is available.
Used to define heights and intersection products on fibers in Section 3; specifically Definition 3.4 and Theorem 3.10 invoke [CM21].
domain assumption Yuan's equidistribution theorem [Yua08], in the form [Cha21, Lemma 8.2], provides generic small sequences and polarised GVF structures.
Used in Section 2.3 to identify limits of small points with GVF structures coming from polarisations, especially Corollaries 2.12 to 2.15.
domain assumption The Burgos Gil-Philippon-Sombra mixed-integral formula for arithmetic intersection numbers of semipositive toric Zhang divisors [BPS14, Theorem 5.2.5].
This is the bridge between arithmetic intersection numbers on toric varieties and sums of mixed integrals of roof functions, used in Theorem 4.5.
domain assumption Arithmetically ample Zhang divisors are dense among semipositive Zhang divisors, an arithmetic Demailly-type statement used in Theorem 3.24.
The proof of Theorem 3.24 asserts this density without a precise proof or theorem number, citing [Zha95a], [QY23], and [Cha17] only in passing.
domain assumption A non-Archimedean Fubini theorem for very affine charts, adapted from [Sto21, Proposition 3.4.21], with product measures and vanishing boundary contributions.
This premise is central to Theorem 4.2, where integrals over products of tori are evaluated fiberwise; the paper only sketches the adaptation.
domain assumption Existential closedness of Q as a GVF [Sza23, Theorem A] supplies generic rational points with prescribed height limits.
Used in the proof of Corollary 2.12 to justify uniqueness of the limiting GVF structure; self-cited by the third author but complemented by Yuan's theorem.

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Pith. "Pith review of Continuity of heights in families and complete intersections in toric varieties." pith.science (2026). https://pith.science/paper/TFRT2ZE2

@misc{pith2026241215988,
  author       = {Pith},
  title        = {Pith review of: Continuity of heights in families and complete intersections in toric varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TFRT2ZE2}},
  note         = {Machine review of arXiv:2412.15988}
}

read the original abstract

We study the variation of heights of cycles in flat families over number fields or, more generally, globally valued fields. To a finite type scheme over a GVF we associate a locally compact Hausdorff space which we refer to as its GVF analytification. For a flat projective family, we prove that the height of fibres is a continuous function on the GVF analytification of the base. As an application, we prove Roberto Gualdi's conjecture on limit heights of complete intersections in toric varieties.

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Works this paper leans on

2 extracted references · 2 canonical work pages

[255]

Arakelov Geometry, Heigh ts, Equidistribution, and the Bogomolov Conjecture

issn: 0022-314X. doi: https://doi.org/10.1006/jnth.1999.2490. [Cha21] Antoine Chambert-Loir. “Arakelov Geometry, Heigh ts, Equidistribution, and the Bogomolov Conjecture”. In: Springer International Publishing Arakelov Geometry and Diophantine Applications (2021). doi: 10.1007 /978-3-030-57559-5_8 . [Cha17] François Charles. “Arithmetic ampleness and an a...

work page arXiv 2021
[1297]

Arithmetic Demailly Approximation Theorem

doi: 10.1007/s002220050358. [PP24] Fabien Pazuki and Riccardo Pengo. “On the Northcott p roperty for spe- cial values of L-functions”. In: Revista Matemática Iberoamericana 40.1 (2024), pp. 1–42. [QY23] Binggang Qu and Hang Yin. Arithmetic Demailly Approximation Theo- rem. 2023. arXiv: 2208.13230 [math.NT] . [Rém01] Gaël Rémond. “Géométrie diophantienne m...

work page Pith review arXiv 2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [255]

Arakelov Geometry, Heigh ts, Equidistribution, and the Bogomolov Conjecture

issn: 0022-314X. doi: https://doi.org/10.1006/jnth.1999.2490. [Cha21] Antoine Chambert-Loir. “Arakelov Geometry, Heigh ts, Equidistribution, and the Bogomolov Conjecture”. In: Springer International Publishing Arakelov Geometry and Diophantine Applications (2021). doi: 10.1007 /978-3-030-57559-5_8 . [Cha17] François Charles. “Arithmetic ampleness and an a...

work page arXiv 2021

[2] [1297]

Arithmetic Demailly Approximation Theorem

doi: 10.1007/s002220050358. [PP24] Fabien Pazuki and Riccardo Pengo. “On the Northcott p roperty for spe- cial values of L-functions”. In: Revista Matemática Iberoamericana 40.1 (2024), pp. 1–42. [QY23] Binggang Qu and Hang Yin. Arithmetic Demailly Approximation Theo- rem. 2023. arXiv: 2208.13230 [math.NT] . [Rém01] Gaël Rémond. “Géométrie diophantienne m...

work page Pith review arXiv 2024