Pith. sign in

REVIEW 2 major objections 7 minor 33 references

Linear growth and moduli spaces of rational curves

T0 review · 2 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For del Pezzo surfaces over $\mathbb{F}_q(t)$, rational-point counts off the exceptional curves come arbitrarily close to linear growth when $q$ is large.

desk verdict A clean moduli-counting trick yields near-linear function-field bounds for del Pezzo surfaces, but the del Pezzo half rests on unverified characteristic assumptions and the counting lemma has a false proof. read the letter →

arxiv 2505.07645 v1 pith:SRZOIP75 submitted 2025-05-12 math.NT math.AG

Jakob Glas This is my paper · ORCID
classification math.NTmath.AG MSC 14H1011D4511P5514G0514J70
keywords rationalcurvesmodulispacesdelPezzosurfacesBatyrev–ManinconjecturepositivecharacteristicF_q(t)-pointscubichypersurfacesintersectionsoftwoquadrics
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 proves strong upper bounds for the number of $\mathbb{F}_q(t)$-points of bounded height on several Fano varieties over finite fields. The main result is that on a smooth del Pezzo surface of degree $1 \le d \le 5$, after deleting all rational curves whose anticanonical degree is at most $1$, the number of points of height $q^e$ is $O(C_d^e q^e)$, with constants $C_5=1024$, $C_4=16$, $C_3=27$, $C_2=4^4$, $C_1=6^6$ independent of $e$. Since $C_d$ is fixed, for large $q$ this is arbitrarily close to the linear growth $q^{e(1+\varepsilon)}$ predicted by the Batyrev--Manin conjecture. The method also shows that the moduli spaces of rational curves on smooth cubic hypersurfaces and on smooth intersections of two quadrics of dimension at least $3$ have the expected dimension in positive characteristic, yielding analogous point-counting bounds.

What carries the argument

The engine is an elementary finite-field point count: any locally closed subvariety $Y\subset \mathbb{A}^N$ over $\mathbb{F}_q$ satisfies $\#Y(\mathbb{F}_q) \le \deg(Y)q^{\dim Y}$, proved by slicing with hyperplanes and applying B\'ezout's inequality. Because $\mathbb{F}_q(t)$-points of height $q^e$ are exactly the $\mathbb{F}_q$-points of the morphism scheme $\operatorname{Mor}(\mathbb{P}^1,X,e)$, this turns counting into a dimension estimate for those moduli spaces. For del Pezzo surfaces, the dimension estimate is imported: the only components of $\operatorname{Mor}(\mathbb{P}^1,X,e)$ of larger than expected dimension are multiple covers of rational curves $C$ with $-K_X\cdot C \le 1$, so deleting those curves leaves components of dimension $e+2$. For cubic hypersurfaces and intersections of two quadrics, the paper proves the expected-dimension statement directly, using a bend-and-break reduction to lines together with the geometry of Fano schemes of lines and a positive-characteristic finiteness result for Eckardt points.

What would settle it

Check the hypotheses of the cited theorem in the source: if the classification of non-expected-dimensional components of the del Pezzo morphism space is proved only away from characteristic $2$ or $3$, then Theorem 1.1 is overclaimed for the remaining characteristics. Alternatively, compute $\operatorname{Mor}(\mathbb{P}^1,X,e)$ for a smooth del Pezzo surface of degree $4$ or $5$ over an algebraically closed field of characteristic $2$ and look for an irreducible component of dimension strictly above $e+2$ whose general member is not a multiple cover of a line; finding one would break the key input.

Watch

Extended reading notes

Core claim

On the author's own terms, the central discovery is that point counting over $\mathbb{F}_q(t)$ can be reduced to dimension bounds for moduli spaces of rational curves, and that those bounds are available in positive characteristic for del Pezzo surfaces of degree at most $5$. The precise form is $N_U(q^e) = O(C_d^e q^e)$ for $U$ the complement of all rational curves $C$ with $-K_X \cdot C \le 1$, with the implied constant depending only on $q$. Because $C_d$ is a fixed constant, taking $q$ large forces the exponent of $q^e$ arbitrarily close to $1$, matching the growth predicted by Batyrev--Manin. In the same framework, smooth cubic hypersurfaces of dimension at least $3$ satisfy $N_X(e)=O(27^e q^{e(n-3)})$ and smooth intersections of two quadrics satisfy $N_X(e)=O(16^e q^{e(n-4)})$.

Load-bearing premise

Everything rests on two imported facts that the paper does not verify in the full characteristic range: that every component of the morphism space has dimension at least the expected one, and that on del Pezzo surfaces the only components exceeding it are multiple covers of curves with $-K_X\cdot C\le 1$; if either fails over $\mathbb{F}_2$ or $\mathbb{F}_3$, the main estimate is not established as stated.

Editorial extensions

If this is right

  • For any smooth del Pezzo surface of degree $1\le d\le 5$ over $\mathbb{F}_q$, the bound $N_U(q^e)\ll (C_d q)^e$ gives exponents $1+\log_q C_d$, which tend to $1$ as $q\to\infty$; this locates the dominant contribution in the deleted low-degree curves.
  • For smooth cubic surfaces over $\mathbb{F}_q$ with $q$ sufficiently large, the theorem yields $N_U(B)\ll B^{4/3-\theta}$ for any fixed $\theta<1/3$, settling the $\mathbb{F}_q(t)$-version of a problem raised by Browning.
  • For every $e\ge 1$, the moduli spaces of degree-$e$ rational curves on smooth cubic hypersurfaces (in characteristic $>3$) and on smooth intersections of two quadrics of dimension at least $3$ have the expected dimension, extending characteristic-zero results.
  • The general counting theorem shows that any Fano variety over $\mathbb{F}_q$ whose morphism spaces of rational curves have the expected dimension for all large $e$ automatically has near-linear growth; the bottleneck is purely the moduli-space dimension theory.

Reading between the lines

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

  • If the imported del Pezzo classification carries an unstated characteristic restriction, Theorem 1.1 would need to be narrowed to the range where that theorem holds; checking the hypotheses of the cited source is the cheapest way to test the main result.
  • The same counting strategy should transfer to any Fano variety once a positive-characteristic expected-dimension statement is known; the constants in the bound would come from the degrees of the defining equations, so improving those constants is a separate combinatorial problem.
  • Because the argument identifies height-$q^e$ points with morphisms, it is particular to $\mathbb{F}_q(t)$; moving to number fields would require an arithmetic analogue of the moduli-space dimension bounds, which is not provided here.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. The paper proposes an elementary counting method over finite fields and applies it to moduli spaces of rational curves on Fano varieties over F_q. Its main result (Theorem 1.1) states that for a smooth del Pezzo surface X of degree 1 ≤ d ≤ 5 over F_q, with U the complement of all rational curves C with -K_X·C ≤ 1, the number of F_q(t)-points of anticanonical height q^e is O(C_d^e q^e), with explicit constants. Theorem 1.2 gives a general bound for Fano complete intersections in terms of the dimension of the relevant moduli space, and Theorem 1.3 establishes that spaces of rational curves on smooth cubic hypersurfaces (char > 3) and smooth intersections of two quadrics (dim ≥ 3) have the expected dimension, leading to Theorem 1.4 bounds. The proofs use a Bezout-type point-counting lemma (Proposition 2.1), degree bounds for affine cones of moduli spaces, and imported results: [3, Theorem 1.1] for del Pezzo surfaces and bend-and-break arguments following [22], [13], [29]. The paper is clearly written and the numerical constants are explicit.

Significance. If the hypotheses of the imported results are satisfied, Theorem 1.1 is a substantial step: for large fixed q it gives upper bounds arbitrarily close to linear growth, improves on known bounds over F_q(t) for large q, and gives the first F_q(t)-version of Browning's problem on cubic surfaces. The general counting lemma and the positive-characteristic expected-dimension results for complete intersections are of independent interest. However, the main del Pezzo theorem rests entirely on the unstated characteristic scope of [3], and the self-contained proof of Proposition 2.1 is flawed as written; both issues must be addressed before the results can be considered established.

major comments (2)
  1. [2, proof of Proposition 2.1] The proof asserts that one can find an index i0 such that the hyperplane x_i0 = a intersects Z2 properly for every a in F_q. This is false in general: for Z2 = V(x1) union V(x2) in A^2, the hyperplane x1 = 0 contains the component V(x1) and the hyperplane x2 = 0 contains V(x2), so no coordinate hyperplane has the required property for all a. Consequently, the displayed summation over a in F_q of #(Z2 ∩ Ha)(F_q) is not justified. The proposition itself is true, as the author notes by citing [12], [18], [16], [10], so this is a flaw in the self-contained proof rather than in the statement; the proof should be repaired or replaced by a citation to one of the existing proofs.
  2. [3, Theorem 3.2 and Remark 3.3] The characteristic hypotheses of [3, Theorem 1.1] are not stated. Remark 3.3 claims that the weak del Pezzo condition 'a general member of |-K_X| is smooth' is automatic for smooth del Pezzo surfaces, citing [3, Section 3]. This is not correct in characteristic 2: a smooth del Pezzo surface of degree 2 given by y^2 = f_4(u,v,w) in P(2,1,1,1) has anticanonical members that are non-reduced double covers of lines, so the general member of |-K_X| is not smooth. Unless [3, Theorem 1.1] nevertheless applies to such surfaces, Theorem 1.1 is overclaimed in all characteristics. Please state the exact hypotheses of [3, Theorem 1.1], confirm they hold for all smooth del Pezzo surfaces over F_q, or add the necessary characteristic restriction to Theorem 1.1.
minor comments (7)
  1. [1, Theorem 1.1 display] The constants C_2 and C_1 are printed as '44' and '66', which is easy to misread; they should be typeset as 4^4 and 6^6.
  2. [3, d = 2 paragraph] The text says 'f in F_q[u,v,w] is a binary quartic form'; since f has three variables, this should be 'ternary quartic form'.
  3. [4, Corollary 4.10 and Lemma 4.9] Corollary 4.10 is stated without the characteristic assumption, but its proof relies on Lemma 4.9, which requires char(K) != 2,3. The corollary should either be restricted to char != 2,3 or an argument covering characteristics 2 and 3 should be supplied.
  4. [4, proof of Corollary 4.10] In the incidence argument, the fiber whose dimension is n-3 by Lemma 4.2 is the fiber of p1 over x, not 'p2(x)^{-1}'; the notation should be corrected.
  5. [5, Proposition 5.3] In the base case of the induction, the phrase 'since we assume n + d − 1 ≥ 2' appears to be a typo for 'n + 1 − d ≥ 2', which is the actual hypothesis used in the paper.
  6. [5, notation for Kontsevich spaces] The open substack of stable maps with irreducible domain is denoted by the same symbol as the full Kontsevich stack M_{0,k}(X,e); a distinct notation (for example M^0_{0,k}(X,e)) would avoid confusion. Also, in the definition of M_{0,k}(X,e), the marked points are listed as p1,...,pn but should be p1,...,pk.
  7. [1, dimension lower bound] The lower bound 'dim every non-empty irreducible component of Mor(P1,X,e) is at least e + dim(X)' is cited to Debarre [14, Theorem 2.6], which is a standard characteristic-0 reference; since the paper works over possibly positive characteristic, a reference covering positive characteristic (e.g., Kollár, Rational Curves on Algebraic Varieties, or Harris-Roth-Starr) or a brief justification would be more appropriate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: main geometric inputs are external theorems, counting constants are derived from defining degrees, and the author's own prior work appears only in comparative remarks.

full rationale

The paper's derivation chain is not circular. Theorem 1.1 rests on Theorem 3.2, which is quoted verbatim from Beheshti–Lehmann–Riedl–Tanimoto [3, Theorem 1.1], a published theorem by different authors; the exclusion of components parameterising multiple covers of curves with -K_X·C ≤ 1 is an external geometric input, not a re-statement of the counting conclusion. The constants C_d = 1024, 16, 27, 4^4, 6^6 arise from the degrees of the defining equations (two quadrics repeated as needed, the cubic, and the weighted equations for d = 1, 2) via Bezout estimates, not from fitting the point counts. Proposition 2.1 is proved in the paper from Bezout, and Proposition 3.1 applies it to cones over moduli spaces. Theorem 1.3 is proved by an independent bend-and-break induction using Harris–Roth–Starr [22], Riedl–Yang [30], and classical facts about Fano schemes of lines, with no parameter fitted to the final bound. The author's self-citations ([20], [21], [9]) occur only in the introduction for comparison with number-field and circle-method bounds and are not load-bearing. The possible concern about unstated characteristic hypotheses in [3] or about the positive-characteristic validity of [22, Lemma 5.1] is a correctness risk, not a circularity; no equation of the paper reduces to its own input by definition.

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

The paper introduces no new entities or fitted parameters. Its axiomatic load consists of standard moduli-space theorems (some requiring positive-characteristic care) and the imported BLRT expected-dimension theorem for del Pezzo surfaces. The most delicate assumption is the unstated characteristic scope of [3, Theorem 1.1].

assumptions (6)
  • domain assumption Theorem 3.2 (BLRT): for a smooth del Pezzo surface over F_q, the only irreducible components of Mor(P^1,X,e) not of the expected dimension are multiple covers of curves C with -K_X·C ≤ 1.
    Imported from [3, Theorem 1.1], used directly in the proof of Theorem 1.1. No characteristic hypothesis is restated in the paper.
  • domain assumption Every non-empty irreducible component of Mor_e(P^1,X) has dimension at least e(n+1-d)+dim(X) in positive characteristic.
    Cited to Debarre [14, Theorem 2.6], a characteristic-0 reference; used to establish equality in Theorem 1.3.
  • standard math The Kontsevich spaces M_{0,k}(X,e) exist as proper algebraic stacks of finite type and have projective coarse moduli spaces in positive characteristic.
    Invoked in Section 5, Proposition 5.3, relying on [28, Theorem 4.7].
  • domain assumption Lemma 5.1 (no complete curve in a fiber of the evaluation map) is valid in all characteristics.
    The paper checks that the argument of Harris-Roth-Starr [22, Lemma 5.1] does not use characteristic 0. This is used in Corollary 5.2 and Proposition 5.3.
  • domain assumption Finiteness of Eckardt points on a smooth cubic hypersurface in characteristic not 2,3.
    Corollary 4.10 is proved using Lemma 4.9 from [27], which assumes char(K) ≠ 2,3. This is consistent with the char>3 hypothesis in Theorem 1.3.
  • standard math Lefschetz hyperplane theorem via étale cohomology in positive characteristic.
    Used in Proposition 4.1 to rule out linear subspaces of dimension > dim(X)/2 in smooth intersections of two quadrics.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Linear growth and moduli spaces of rational curves." pith.science (2026). https://pith.science/paper/SRZOIP75

@misc{pith2026250507645,
  author       = {Pith},
  title        = {Pith review of: Linear growth and moduli spaces of rational curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SRZOIP75}},
  note         = {Machine review of arXiv:2505.07645}
}
abstract

Working in positive characteristic, we show how one can use information about the dimension of moduli spaces of rational curves on a Fano variety $X$ over $\mathbb{F}_q$ to obtain strong estimates for the number of $\mathbb{F}_q(t)$-points of bounded height on $X$. Building on work of Beheshti, Lehmann, Riedl and Tanimoto~\cite{BeheshtiLehmannRiedlTanimoto.dP}, we apply our strategy to del Pezzo surfaces of degree at most 5. In addition, we also treat the case of smooth cubic hypersurfaces and smooth intersections of two quadrics of dimension at least 3 by showing that the moduli spaces of rational curves of fixed degree are of the expected dimension. For large but fixed $q$, the bounds obtained come arbitrarily close to the linear growth predicted by the Batyrev--Manin conjecture.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 29 canonical work pages

  1. [22]

    Harris, M

    J. Harris, M. Roth, and J. Starr. Rational curves on hypersu rfaces of low degree. J. Reine Angew. Math. 571 (2004), pp. 73–106

    work page 2004

  2. [13]

    Coskun and J

    I. Coskun and J. Starr. Rational curves on smooth cubic hype rsurfaces. Int. Math. Res. Not. IMRN (2009), pp. 4626–4641

    work page 2009

  3. [29]

    F. Okamura. The Irreducibility of the Spaces of Rational Curve s on del Pezzo Manifolds. Int. Math. Res. Not. IMRN (2024), pp. 9893–9909

    work page 2024

  4. [3]

    Beheshti, B

    R. Beheshti, B. Lehmann, E. Riedl, and S. Tanimoto. Rational cur ves on del Pezzo surfaces in positive characteristic. Trans. Amer. Math. Soc. Ser. B 10 (2023), pp. 407– 451. REFERENCES 15

    work page 2023

  5. [12]

    Cafure and G

    A. Cafure and G. Matera. Improved explicit estimates on the nu mber of solutions of equations over a finite field. Finite Fields Appl. 12 (2006), pp. 155–185

    work page 2006

  6. [18]

    J. S. Ellenberg, R. Oberlin, and T. Tao. The Kakeya set and maxim al conjectures for algebraic varieties over finite fields. Mathematika 56 (2010), pp. 1–25

    work page 2010

  7. [16]

    Z. Dvir, J. Koll´ ar, and S. Lovett. Variety evasive sets. Comput. Complexity 23 (2014), pp. 509–529

    work page 2014

  8. [10]

    Browning, P

    T. Browning, P. Vishe, and S. Yamagishi. Rational curves on com plete intersections and the circle method, Preprint (2024), arXiv: 2404.11123

    arXiv 2024

Show all 33 references
  1. [1]

    A. B. Altman and S. L. Kleiman. Foundations of the theory of Fano schemes. Compositio Math. 34 (1977), pp. 3–47

    1977

  2. [2]

    V. V. Batyrev and Y. I. Manin. Sur le nombre des points rationnels de hauteur borne des varietes algebriques. Mathematische Annalen 286 (1990), pp. 27–44

    1990

  3. [4]

    Behrend and Y

    K. Behrend and Y. Manin. Stacks of stable maps and Gromov-Witt en invariants. Duke Math. J. 85 (1996), pp. 1–60

    1996

  4. [5]

    Bhargava, A

    M. Bhargava, A. Shankar, T. Taniguchi, F. Thorne, J. Tsimerma n, and Y. Zhao. Bounds on 2-torsion in class groups of number fields and integral points on e lliptic curves. J. Amer. Math. Soc. 33 (2020), pp. 1087–1099

    2020

  5. [6]

    Browning

    T. Browning. An overview of Manin’s conjecture for del Pezzo su rfaces. In: Analytic number theory. 7. Clay Math. Proc. Amer. Math. Soc., Providence, RI, 2007, pp. 39 –55

    2007

  6. [7]

    Browning and D

    T. Browning and D. R. Heath-Brown. The density of rational poin ts on non-singular hypersurfaces. II. Proc. London Math. Soc. (3) 93 (2006). With an appendix by J. M. Starr, pp. 273–303

    2006

  7. [8]

    Browning and P

    T. Browning and P. Vishe. Rational points on cubic hypersurface s over Fq(t). Geometric and functional analysis. 25 (June 2015), pp. 671–732

    2015

  8. [9]

    Browning, J

    T. Browning, J. Glas, and V. Y. Wang. Sums of three cubes over a function field, Preprint (2024), arXiv: 2402.07146

    2024 arXiv

  9. [11]

    A. Brumer. The average rank of elliptic curves. I. Invent. Math. 109 (1992), pp. 445– 472

    1992

  10. [14]

    O. Debarre. Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, New York, 2001, pp. xiv+233

    2001

  11. [15]

    Dolgachev and G

    I. Dolgachev and G. Martin. Automorphisms of del Pezzo surfa ces in odd characteristic. J. Lond. Math. Soc. (2) 109 (2024), Paper No. e12905, 40

    2024

  12. [17]

    Edidin, B

    D. Edidin, B. Hassett, A. Kresch, and A. Vistoli. Brauer groups and quotient stacks. Amer. J. Math. 123 (2001), pp. 761–777

    2001

  13. [19]

    Fukasawa, K

    S. Fukasawa, K. Furukawa, and H. Kaji. Projective varieties a dmitting an embedding with Gauss map of rank zero. Adv. Math. 224 (2010), pp. 2645–2661

    2010

  14. [20]

    Glas and L

    J. Glas and L. Hochfilzer. Rational points on del Pezzo surface s of low degree, Preprint (2024), arXiv: 2401.04759

    2024 arXiv

  15. [21]

    Glas and L

    J. Glas and L. Hochfilzer. On a question of Davenport and diagon al cubic forms over Fq(t). Math. Ann. 391 (2025), pp. 5485–5533

    2025

  16. [23]

    D. R. Heath-Brown. The density of rational points on cubic sur faces. Acta Arith. 79 (1997), pp. 17–30

    1997

  17. [24]

    D. R. Heath-Brown. Counting rational points on cubic surface s. Ast´ erisque251 (1998), pp. 13–30

    1998

  18. [25]

    J. Heintz. Definability and fast quantifier elimination in algebraically closed fields. The- oret. Comput. Sci. 24 (1983), pp. 239–277. 16 REFERENCES

    1983

  19. [26]

    H. A. Helfgott and A. Venkatesh. Integral points on elliptic cur ves and 3-torsion in class groups. J. Amer. Math. Soc. 19 (2006), pp. 527–550

    2006

  20. [27]

    Huybrechts

    D. Huybrechts. The geometry of cubic hypersurfaces . Vol. 206. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2 023, pp. xvii+441

  21. [28]

    A. J. de Jong, X. He, and J. M. Starr. Families of rationally simply c onnected varieties over surfaces and torsors for semisimple groups. Publ. Math. Inst. Hautes ´Etudes Sci. (2011), pp. 1–85

    2011

  22. [30]

    Riedl and D

    E. Riedl and D. Yang. Kontsevich spaces of rational curves on Fano hypersurfaces. J. Reine Angew. Math. 748 (2019), pp. 207–225

    2019

  23. [31]

    Salberger

    P. Salberger. Uniform bounds for rational points on cubic hype rsurfaces. In: Arithmetic and geometry. 420. London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Ca m- bridge, 2015, pp. 401–421

    2015

  24. [32]

    Salberger

    P. Salberger. Counting rational points on projective varieties . Proc. Lond. Math. Soc. (3) 126 (2023), pp. 1092–1133

    2023

  25. [33]

    V. Y. Wang. Sums of cubes and the Ratios Conjectures, Preprint (2023), arXiv: 2108.03398. Leibniz Universit ¨at Hannover, Welfengarten 1, 30167 Hannover, Germany Email address : glas@math.uni-hannover.de

    2023 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.