Manin's conjecture for quintic del Pezzo surfaces with a conic bundle structure

Daniel Loughran; D. R. Heath-Brown

Manin's conjecture for quintic del Pezzo surfaces with a conic bundle structure

T0 review · 0 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Manin's conjecture is proved for infinitely many quintic del Pezzo surfaces with a conic bundle structure.

desk verdict First asymptotic for infinitely many non-split degree-five del Pezzo surfaces, with Peyre's constant genuinely computed from geometry; long, technical, and worth refereeing. read the letter →

arxiv 2506.02829 v1 pith:BOOI5GNI submitted 2025-06-03 math.NT

D. R. Heath-Brown , Daniel Loughran This is my paper

classification math.NT MSC 11D4514G0514J26

keywords Manin'sconjecturedelPezzosurfacesdegreefiveconicbundlerationalpointsofboundedheightPeyre'sconstantequidistributioncirclemethod

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 targets Manin's conjecture for smooth del Pezzo surfaces of degree five that admit a conic bundle structure, embedded as $y_0Q_0(x)+y_1Q_1(x)=0$ in $\mathbb{P}^2\times\mathbb{P}^1$. It establishes matching upper and lower bounds of order $B(\log B)^{\rho-1}$ for the number of rational points of anticanonical height at most $B$ on all such surfaces, where $\rho$ is the Picard rank. In the Galois-generic case, where the base locus $Q_0=Q_1=0$ has no rational point and no nontrivial rational linear combination of the two quadrics is singular, it proves the full asymptotic $N(U,B)\sim c_S B\log B$ with $c_S$ exactly Peyre's constant. These are the first infinitely many nonsplit del Pezzo surfaces of degree at most five for which the conjecture is verified. The proof also yields equidistribution of the rational points with respect to Peyre's Tamagawa measure.

What carries the argument

The argument is built on a hyperbola split of the counting function into $N_1$ where $H(x)\le H(y)$ and $N_2$ where $H(x)>H(y)$. For $N_1$, each $x$ determines a unique $\pm y$ unless both quadrics vanish, reducing the count to lattice-point sums; Poisson summation, a dyadic decomposition in the gcd $d=\gcd(Q_0(x),Q_1(x))$, and bounds for the real density produce the asymptotic. For $N_2$, the count is organized by the conics $Q_y=0$; a uniform circle-method estimate bounds the error term for each conic in terms of the minimal zero $z(y)$ of the conic, and a sieve argument averages these errors over $y$. The main terms are summed over the sparse set of $y$ for which the conic has a rational point; the Hilbert-symbol product formula and a multiplicative-function analysis of the local densities convert this sum into Peyre's constant. The geometry of the effective cone, generated by the exceptional divisor $E$ and a conic-bundle fibre $F$ with $2L=E+F$, gives the factor $2/3$ and explains the asymmetric $1:3$ contributions of $N_1$ and $N_2$.

What would settle it

Take a concrete surface satisfying the hypotheses, for instance the blow-up of $\mathbb{P}^2$ in a closed point of degree $4$ whose splitting field has Galois group $A_4$, compute $N(U,B)$ by enumerating primitive $(x,y)$ with $H(x)H(y)\le B$ up to large $B$, and check whether $N(U,B)/(B\log B)$ tends to $\frac{2}{3}\tau_\infty\prod_p\tau_p$; a deviation would refute the theorem.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.2: for a smooth surface $S\subset\mathbb{P}^2\times\mathbb{P}^1$ of bidegree $(2,1)$ with $Q_0,Q_1$ as above, if the intersection $Q_0(x)=Q_1(x)=0$ has no $\mathbb{Q}$-points and no non-trivial $\mathbb{Q}$-linear combination of $Q_0$ and $Q_1$ is singular, then $\rho=2$ and \[N(U,B)\sim c_S B\log B,\qquad c_S=\frac{2}{3}\tau_\infty\prod_p\tau_p,\] as $B\to\infty$, where $U$ is the complement of all lines on the surface and $\tau_\infty,\tau_p$ are Peyre's local densities. The geometric conditions are equivalent to the surface being the blow-up of $\mathbb{P}^2$ in a single closed point of degree $4$ whose splitting field has Galois group $A_4$ or $S_4$. The constant $\frac{2}{3}$ is Peyre's effective-cone constant for these surfaces, and the paper verifies that the full Manin–Peyre prediction holds in these cases.

Load-bearing premise

The proof's asymptotic for the conic-counting half requires that the two quadrics $Q_0=Q_1=0$ share no rational point; if such a point existed, the minimal integer zero of some isotropic conic in the pencil would be uniformly small, and the averaged error-term bound would lose its force.

Editorial extensions

If this is right

Manin's conjecture holds for infinitely many smooth del Pezzo surfaces of degree $5$ that are far from split, namely those whose blow-up centres have Galois group $A_4$ or $S_4$.
The asymptotic constant is exactly Peyre's constant, including the effective-cone factor $2/3$, so the full Manin–Peyre prediction is verified in these cases.
The hyperbola decomposition is not symmetric: $N_1$ contributes one quarter and $N_2$ three quarters of the main term, a ratio predicted by the geometry of the effective cone.
Rational points on these surfaces are equidistributed with respect to Peyre's Tamagawa measure, so the asymptotic is independent of the chosen adelic metric on the anticanonical bundle.
The new uniform conic-counting estimate gives an alternative proof of the lower bound $N(U,B)\gg B(\log B)^{\rho-1}$ for all such surfaces.

Reading between the lines

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

The assumption that $Q_0=Q_1=0$ has no rational point is probably essential to the method: it prevents any conic in the pencil from having an abnormally small rational point, which is exactly the regime where the uniform conic-counting bound is favourable. A surface with such a point would require a genuinely different treatment.
The $1:3$ ratio between the two hyperbola regions is a testable prediction for other conic-bundle del Pezzo surfaces: the ratio should be computable from the nef cone, as in the quartic case studied by Browning and Sofos.
The equidistribution result suggests that these surfaces have no Brauer–Manin obstruction to weak approximation and that rational points are spread according to the local Tamagawa measures, which is consistent with the general expectation for rationally connected varieties.
The uniform conic-counting estimate may be applicable to other sparse families of varieties where the fibres are conics and the base has few rational points, provided a minimal-zero bound is available.

Share X Bluesky LinkedIn Reddit HN

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 is the real thing—a proof of Manin's conjecture for infinitely many non-split quintic del Pezzo surfaces, with the Peyre constant computed from geometry rather than fitted to the count. That is a genuine step beyond the split and two-point-conjugate cases, and the first such family in degree five. The paper deserves a serious referee.

What is actually new: the asymptotic for the H(x)≤H(y) half (Theorem 1.3), the asymptotic for the other half (Theorem 1.4), and the resulting Theorem 1.2 with constant (2/3) τ∞ ∏ τp. The equidistribution theorem (1.5) is a bonus. The unequal 1/4 and 3/4 contributions from the hyperbola split are surprising at first, and the explanation via the effective cone (Section 2.2) is convincing. The paper also gives a clean geometric characterization (A4 or S4) of the surfaces covered and shows why the assumption M(Q)=∅ is not a hidden technicality.

Where I would be careful: the proof is long and rests on imported analytic estimates—Heath-Brown's uniform conic theorem, the sieve bound from Iwaniec–Kowalski, and the authors' own earlier work with Frei and Sofos. I did not check every line of Proposition 4.5 or Lemma 6.16, but the delicate points I would have flagged (the theta(p) formula (4.17), the Hilbert-product step, the dyadic summation range) are handled correctly. The restriction to M(Q)=∅ is a stated hypothesis, not an unstated assumption, and the paper says plainly that without it the minimal zero z(y) can be too small for the error bound. The only real criticism is readability: the authors themselves note that some symbols have different meanings in different parts of the paper. That is annoying but not a mathematical defect.

I would send this to peer review. It is a well-posed, honest paper with a major new theorem and a sensible proof strategy. It will be valuable to the Manin-conjecture community and to anyone working on conic-bundle counting. I would cite it.

Referee Report

0 major / 5 minor

Summary. The paper proves matching upper and lower bounds and, under genericity hypotheses, an asymptotic formula for rational points of bounded height on smooth quintic del Pezzo surfaces with a conic bundle structure. Such a surface is written as y0Q0(x)+y1Q1(x)=0 in P^2 x P^1, and the open set U is the complement of the ten (-1)-curves. Theorem 1.1 gives N(U,B) = B(log B)^{rho-1} up to constants. Theorems 1.2-1.4 give, when Q0=Q1=0 has no Q-point and no nontrivial Q-linear combination of Q0,Q1 is singular, rho=2 and N(U,B) ~ c_S B log B with the Peyre constant c_S=(2/3) tau_infty prod_p tau_p; the proof splits the count by the hyperbola condition H(x)<=H(y) versus H(x)>H(y), using lattice-point asymptotics in Section 5 and an averaged conic-counting argument in Section 6. Theorem 1.5 derives equidistribution of rational points with respect to Peyre's measure. A geometric lemma identifies the hypotheses with blowing up P^2 in a closed point of degree 4 with splitting field A4 or S4, so the family is infinite and far from split.

Significance. If correct, this is a substantial advance: it gives the first asymptotic verification of Manin's conjecture for an infinite family of non-split del Pezzo surfaces of degree 5, in a regime where the conic bundle has very few fibres with rational points. The paper's strengths are the explicit derivation of the Peyre constant from the effective cone and local densities rather than from the counting function, the transparent use of the hyperbola method, and the careful averaging of Heath-Brown's uniform conic estimate over the sparse set of soluble fibres. I checked the principal algebraic identities and constant bookkeeping: the decomposition (1.3), the formula (4.17), the Hilbert-symbol step in Lemma 6.16, and the matching of the products in Section 5.2 with prod_p tau_p; I found no circularity or internal inconsistency. The main limitation, that the conic-counting error term is only controllable under the hypothesis M(Q)=empty (and the main-term averaging also needs C(Q)=empty), is explicitly stated as a hypothesis of Theorem 1.2 and is not an unstated assumption.

minor comments (5)

[1.1, Theorem 1.1] The displayed surface has bidegree (2,1), not (1,2); the text should be corrected, and the phrase 'split del quintic' should read 'split del Pezzo'.
[1.1] The claim that Theorem 1.2 'verifies for the first time that Manin's conjecture holds for infinitely many smooth del Pezzo surfaces of degree at most 5' is not supported by the cited literature, since references [4] and [5] already provide infinite families of degree-5 del Pezzo surfaces (split and near-split). Please restrict the novelty claim to the non-split conic-bundle setting and adjust the wording accordingly.
[2.1] 'Peterson graph' is a typo for 'Petersen graph'.
[2.3] The remark immediately before Lemma 2.10 says 'the second integral in Lemma 2.6'; it should refer to Lemma 2.9, which contains the integral in question.
[2.2.2] The phrase 'adelic metrics on the basis' is imprecise; the metrics should be placed on the corresponding line bundles.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the asymptotic constant is computed from geometry and local densities, not fitted, and the main theorem is derived from independent analytic estimates.

full rationale

The derivation chain is self-contained with respect to the claimed theorem. The constant in Theorem 1.2 is not fitted to the counting function: the effective cone contribution 2/3 is computed directly from Lemma 2.7 and equation (2.4), the real density is evaluated explicitly in Lemma 2.9, and the product of p-adic densities is built from the local analysis in Sections 2.4 and 2.5 with convergence established in Lemma 2.19. The counting arguments are separate: upper bounds via lattices and conics in Sections 3 and 4, asymptotics via lattices in Section 5, and asymptotics via conics in Section 6. The hypothesis that M(Q) is empty is a stated assumption of Theorem 1.2, and it is used in Lemma 6.12 only to ensure that small pairs (y,z) lie in the open set U, so that the already-proved upper bound of Theorem 1.1 applies; this is a limitation of the method, not a circular reduction. The uniform conic estimate in Proposition 6.9 is quoted from the published work [18] and is not used to define Peyre's constant. Citations to [14] supply geometric embedding, Picard-rank facts, and an alternative lower bound, but these are external published results and do not force the final asymptotic formula. No equation in the paper sets the target constant equal to a fitting parameter, and no input is defined in terms of the output. The claim is therefore not circular.

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

The proof introduces no fitted parameters and no speculative entities. Everything the central claim rests on is either a theorem hypothesis, a standard analytic number theory input, or a geometric fact about the effective cone and the Galois action. The largest external inputs are Heath-Brown's uniform conic-counting theorem and the sieve bound, both quoted with precise references.

assumptions (5)

domain assumption The surface S defined by y0Q0(x)+y1Q1(x)=0 is smooth, equivalently C(y)=det(y0Q0+y1Q1) is separable.
Smoothness is a hypothesis in all theorems and is used to ensure singular fibres are rank-2 conics and that M is finite etale.
standard math The biprojective height H(x)H(y) is an anticanonical height for S.
This identifies the counting function with Manin's height setting and justifies the shape of the conjectured asymptotic.
standard math The external analytic bounds used in the proof, namely Heath-Brown's uniform conic estimate, the circle method for quadratic forms, Perron's formula and the Iwaniec-Kowalski sieve bound, are valid.
These imported results supply the error terms and averaging bounds on which Propositions 6.2 and 6.9 depend.
domain assumption In the Galois-generic case, the scheme M=Q0=Q1=0 is a closed point of degree 4 with splitting field A4 or S4, giving rho=2.
This is the geometric content of Lemma 2.5 and is used explicitly in Section 6 to justify the convergence and averaging assumptions.
standard math The global Hilbert-symbol product formula, together with the fact that a smooth conic over a finite field has q+1 points and lifts by Hensel, is used to identify the local invariants chi_p(Q_y).
This is the key identity in Lemma 6.16 relating local densities to the divisor sums Sigma_1 and Sigma_2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Manin's conjecture for quintic del Pezzo surfaces with a conic bundle structure." pith.science (2026). https://pith.science/paper/BOOI5GNI

@misc{pith2026250602829,
  author       = {Pith},
  title        = {Pith review of: Manin's conjecture for quintic del Pezzo surfaces with a conic bundle structure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BOOI5GNI}},
  note         = {Machine review of arXiv:2506.02829}
}

read the original abstract

We investigate Manin's conjecture for del Pezzo surfaces of degree five with a conic bundle structure, proving matching upper and lower bounds, and the full conjecture in the Galois general case.

Figures

Figures reproduced from arXiv: 2506.02829 by the authors.

**Figure 2.1.** Blow-up model for S¯ Lemma 2.4. We have the following formulas for rank Pic S. (1) Let #M be the number of closed points of the subscheme M from (2.2). Then rank Pic S = #M + 1. (2) We have rank Pic S = 2 + #{closed points P ∈ P 1 : π −1 1 (P) is singular and split}. Proof. The surface S is obtained by blowing up P 2 in the irreducible components of M. Each blow up increases the rank of the Picard group by 1. The re… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

Bernert, U

C. Bernert, U. Derenthal, Points of bounded height on quintic del Pezzo surfaces over number fields,arxiv:2405.20293

work page arXiv
[2]

Points of bounded height on quintic del Pezzo surfaces over the rational numbers

C. Bernert, U. Derenthal, Points of bounded height on quintic del Pezzo surfaces over the rational numbers,arxiv:2505.06065

work page Pith review arXiv
[3]

Bonolis, T.D

D. Bonolis, T.D. Browning, and Z. Huang, Density of rational points on some quadric bundle threefolds,Math. Annalen390(2024), 4123–4207

work page 2024
[4]

de la Bretèche, Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5,Duke Math

R. de la Bretèche, Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5,Duke Math. J.,113(2002), no. 3, 421–464

work page 2002
[5]

de la Bretèche, E

R. de la Bretèche, E. Fouvry, L’éclaté du plan projectif en quatre points dont deux con- jugués,J. reine angew. Math.,576(2004), 63–122

work page 2004
[6]

T. D. Browning, Revisiting the Manin-Peyre conjecture for the split del Pezzo surface of degree 5,New York J. Math.,28(2022), 1193–1229

work page 2022
[7]

Browning and D.R

T.D. Browning and D.R. Heath-Brown, Counting rational points on quadric surfaces, Discrete Analysis,15(2018), 29 pp

work page 2018
[8]

Browning and D.R

T.D. Browning and D.R. Heath-Brown, Density of rational points on a quadric bundle in P3 ×P 3,Duke Math. J.,169(2020), no. 16, 3099–3165

work page 2020

Show all 29 references

[9]

T. D. Browning and E. Sofos, Counting rational points on quartic del Pezzo surfaces with a rational conic.Math. Ann.373(2019), no. 3-4, 977–1016

2019
[10]

Cassels,An introduction to the geometry of numbers, Classics in Mathematics

J.W.S. Cassels,An introduction to the geometry of numbers, Classics in Mathematics. (Springer-Verlag, Berlin, 1971)

1971
[11]

Derenthal, M

U. Derenthal, M. Joyce, Z. Teitler, The nef cone volume of generalized Del Pezzo surfaces, Algebra Number Theory,2(2008), no. 2, 157–182

2008
[12]

Erdős, On the sumPx k=1 d(f(k)),J

P. Erdős, On the sumPx k=1 d(f(k)),J. London Math. Soc.,27(1952), 7–15

1952
[13]

Franke, Y

J. Franke, Y. I. Manin, Y. Tschinkel, Rational points of bounded height on Fano varieties. Invent. Math.95(1989), no. 2, 421–435

1989
[14]

C. Frei, D. Loughran and E. Sofos, Rational points of bounded height on general conic bundle surfaces,Proceedings of the London Mathematical Society, (3)117(2018), no. 2, 407–440

2018
[15]

Heath-Brown, Diophantine approximation with square-free numbers,Math

D.R. Heath-Brown, Diophantine approximation with square-free numbers,Math. Zeit., 187(1984), no. 3, 335–344

1984
[16]

Heath-Brown, A new form of the circle method, and its application to quadratic forms,J

D.R. Heath-Brown, A new form of the circle method, and its application to quadratic forms,J. reine angew. Math.,481(1996), 149–206

1996
[17]

Heath-Brown, The density of rational points on cubic surfaces,Acta Arith.,79 (1997), 17–30

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

1997
[18]

Heath-Brown, The distribution of rational points on conics,Acta Arith.,209(2023), 91–128

D.R. Heath-Brown, The distribution of rational points on conics,Acta Arith.,209(2023), 91–128

2023
[19]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic number theory, American Mathematical Society Colloquium Publications, 53. (American Mathematical Society, Providence, RI, 2004). 98 D.R. HEATH-BROWN AND D. LOUGHRAN

2004
[20]

Landau,Vorlesungen über Zahlentheorie, Vol II, (Chelsea, New York, 1969)

E. Landau,Vorlesungen über Zahlentheorie, Vol II, (Chelsea, New York, 1969)

1969
[21]

Lewis and K

D.J. Lewis and K. Mahler, On the representation of integers by binary forms,Acta Arith., 6(1960/61), 333–363

1960
[22]

Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano.,Duke Math

E. Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano.,Duke Math. J.,79 (1995), no. 1, 101–218

1995
[23]

Peyre, Beyond heights: slopes and distribution of rational points.Arakelov geometry and Diophantine applications, 215–279, Lecture Notes in Math., 2276, Springer, 2021

E. Peyre, Beyond heights: slopes and distribution of rational points.Arakelov geometry and Diophantine applications, 215–279, Lecture Notes in Math., 2276, Springer, 2021

2021
[24]

Serre,A course in arithmetic, Graduate Texts in Mathematics,7, (Springer-Verlag, New York-Heidelberg, 1973)

J.-P. Serre,A course in arithmetic, Graduate Texts in Mathematics,7, (Springer-Verlag, New York-Heidelberg, 1973)

1973
[25]

Serre Spécialisation des éléments deBr2(Q(T1,

J.-P. Serre Spécialisation des éléments deBr2(Q(T1, . . . , Tn)).C. R. Acad. Sci. Paris Sér. I Math.311(1990), no. 7, 397–402

1990
[26]

Shiu, A Brun-Titchmarsh theorem for multiplicative functions,J

P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions,J. Reine Angew. Math., 313(1980), 161–170

1980
[27]

Uchida, On Artin L-functions,Tohoku Math

K. Uchida, On Artin L-functions,Tohoku Math. J. (2),27(1975), 75–81

1975
[28]

van der Waall, On a conjecture of Dedekind on zeta-functions,Nederl

R.W. van der Waall, On a conjecture of Dedekind on zeta-functions,Nederl. Akad. Wetensch. Proc. Ser. A, 78.Indag. Math.,37(1975), 83–86

1975
[29]

Wittenberg,Intersections de deux quadriques et pinceaux de courbes de genre1, Lecture Notes in Mathematics,1901

O. Wittenberg,Intersections de deux quadriques et pinceaux de courbes de genre1, Lecture Notes in Mathematics,1901. (Springer, Berlin, 2007). Roger Heath-Brown, Mathematical Institute, Radcliffe Obser v atory Quarter, Woodstock Road, Oxford, OX2 6GG, UK. Daniel Loughran, Depar...

1901

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Bernert, U

C. Bernert, U. Derenthal, Points of bounded height on quintic del Pezzo surfaces over number fields,arxiv:2405.20293

work page arXiv

[2] [2]

Points of bounded height on quintic del Pezzo surfaces over the rational numbers

C. Bernert, U. Derenthal, Points of bounded height on quintic del Pezzo surfaces over the rational numbers,arxiv:2505.06065

work page Pith review arXiv

[3] [3]

Bonolis, T.D

D. Bonolis, T.D. Browning, and Z. Huang, Density of rational points on some quadric bundle threefolds,Math. Annalen390(2024), 4123–4207

work page 2024

[4] [4]

de la Bretèche, Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5,Duke Math

R. de la Bretèche, Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5,Duke Math. J.,113(2002), no. 3, 421–464

work page 2002

[5] [5]

de la Bretèche, E

R. de la Bretèche, E. Fouvry, L’éclaté du plan projectif en quatre points dont deux con- jugués,J. reine angew. Math.,576(2004), 63–122

work page 2004

[6] [6]

T. D. Browning, Revisiting the Manin-Peyre conjecture for the split del Pezzo surface of degree 5,New York J. Math.,28(2022), 1193–1229

work page 2022

[7] [7]

Browning and D.R

T.D. Browning and D.R. Heath-Brown, Counting rational points on quadric surfaces, Discrete Analysis,15(2018), 29 pp

work page 2018

[8] [8]

Browning and D.R

T.D. Browning and D.R. Heath-Brown, Density of rational points on a quadric bundle in P3 ×P 3,Duke Math. J.,169(2020), no. 16, 3099–3165

work page 2020