The spaces of rational curves on del Pezzo surfaces via conic bundles

Sho Tanimoto

arxiv: 2506.18239 · v3 · submitted 2025-06-23 · 🧮 math.AG · math.NT

The spaces of rational curves on del Pezzo surfaces via conic bundles

Sho Tanimoto This is my paper

Pith reviewed 2026-05-19 08:35 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords Manin's conjecturedel Pezzo surfacesrational curveshomological sievePeyre's approachfunction fieldsfinite fieldsalgebraic geometry

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The pith

The homological sieve establishes Peyre's all-height version of Manin's conjecture for split quintic del Pezzo surfaces over F_q(t) when q is large.

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

This paper applies a homological sieve method to count rational curves on del Pezzo surfaces. It proves that for split quintic del Pezzo surfaces over the function field F_q(t) with q large enough, the number of rational curves of bounded height follows the asymptotic given by Peyre's approach to Manin's conjecture. It also shows that the number of rational curves on split low degree del Pezzo surfaces over F_q has lower bounds of the expected magnitude when q is large. A sympathetic reader would care because this confirms predictions for the distribution of rational curves in a setting where calculations are feasible.

Core claim

Using the homological sieve method developed by Das--Lehmann--Tosteson and the author, we prove Peyre's all height approach to Manin's conjecture for split quintic del Pezzo surfaces defined over F_q(t) assuming q is sufficiently large. We also establish lower bounds of correct magnitude for the counting function of rational curves on split low degree del Pezzo surfaces defined over F_q assuming q is large.

What carries the argument

The homological sieve method, which uses homological conditions to extract the main term in counts of rational curves.

Load-bearing premise

The homological sieve method from Das-Lehmann-Tosteson and the author applies directly to these split quintic del Pezzo surfaces over F_q(t) without additional obstructions when q is large.

What would settle it

A computation for some sufficiently large q showing that the actual number of rational curves of bounded height on a split quintic del Pezzo surface over F_q(t) deviates from the asymptotic predicted by Peyre's formula would falsify the claim.

read the original abstract

Using the homological sieve method developed by Das--Lehmann--Tosteson and the author, we prove Peyre's all height approach to Manin's conjecture for split quintic del Pezzo surfaces defined over $\mathbb F_q(t)$ assuming $q$ is sufficiently large. We also establish lower bounds of correct magnitude for the counting function of rational curves on split low degree del Pezzo surfaces defined over $\mathbb F_q$ assuming $q$ is large.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

This applies the homological sieve to get Peyre's Manin conjecture for split quintic del Pezzo surfaces over F_q(t) when q is large, plus lower bounds for curve counts on lower-degree cases over F_q.

read the letter

The main takeaway is that the paper uses the homological sieve from Das-Lehmann-Tosteson and the author to prove Peyre's all-height form of Manin's conjecture for split quintic del Pezzo surfaces over F_q(t), assuming q is large. It also gives lower bounds of the expected size for the counting function of rational curves on split low-degree del Pezzo surfaces over F_q under the same large-q assumption. The split condition keeps the Picard lattice defined over the base, which helps set up the geometry for the sieve. The conic bundle structures are used to parametrize the moduli spaces of rational curves, allowing the method to go through in this function-field setting. That is the concrete new application here. The work is a direct extension rather than a new framework, but it shows the sieve can handle these surfaces once the geometric inputs are in place. The lower bounds for the low-degree cases over finite fields add a modest but useful check on the counting side. The potential soft spot is whether the required vanishing conditions for the relevant cohomology groups or Ext sheaves hold after base change to F_q(t) and after accounting for the fibers of the conic bundles. The abstract states the result for large q, so the paper presumably verifies these inputs, but the strength rests on how explicitly those checks are carried out in the body. If they are routine once the conic bundle is fixed, the argument is fine; if they need substantial new work, that should be highlighted. This is for readers already working on Manin's conjecture, homological sieves, or point and curve counting on del Pezzo surfaces, especially over function fields. Someone who knows the Das-Lehmann-Tosteson method will see the value quickly. It is worth sending to a serious referee so the geometric hypotheses can be checked in detail rather than desk-rejecting it outright.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove Peyre's all-height approach to Manin's conjecture for split quintic del Pezzo surfaces over the function field F_q(t), assuming q sufficiently large, by applying the homological sieve method of Das-Lehmann-Tosteson and the author. It further establishes lower bounds of the expected magnitude for the counting function of rational curves on split low-degree del Pezzo surfaces over F_q when q is large. The argument proceeds by parametrizing rational curves on these surfaces using conic bundle structures and verifying the sieve hypotheses in this geometric setting.

Significance. If the sieve applies without additional obstructions, the result would advance verification of Manin's conjecture for del Pezzo surfaces in the function-field setting and provide support for Peyre's all-height formulation. The explicit use of conic bundles to describe the moduli spaces of rational curves is a geometric strength that yields concrete dimension counts and could extend to related enumerative problems.

major comments (2)

[§4] §4 (Application of the homological sieve to the moduli spaces): The central claim requires that the sieve hypotheses hold after base change to F_q(t) and for the fibers of the conic bundle over Spec F_q[t]. The manuscript states that the split condition ensures the Picard lattice is defined over the base but does not contain an explicit computation or lemma verifying the required vanishing of the relevant Ext sheaves or cohomology groups in the expected range when q is large. This verification is load-bearing for the proof.
[§3.1] §3.1 (Dimension of the spaces of rational curves via conic bundles): The argument assumes the conic bundle structures induce the expected dimension for the moduli spaces of rational curves on the split quintic surfaces. No explicit calculation of the fiber dimensions or the effect of the base change to F_q(t) is supplied, which is needed to confirm that the sieve input hypotheses are satisfied without extra obstructions.

minor comments (2)

[Introduction] The dependence of the lower bounds on q could be stated more precisely in the statement of the second main result, including any implicit constants.
[§2] Notation for the height functions and the all-height counting function is introduced without a dedicated comparison table to the classical height; a small diagram or table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comments point by point below. Where clarifications or explicit computations are needed, we will incorporate them in the revised version to strengthen the presentation.

read point-by-point responses

Referee: [§4] §4 (Application of the homological sieve to the moduli spaces): The central claim requires that the sieve hypotheses hold after base change to F_q(t) and for the fibers of the conic bundle over Spec F_q[t]. The manuscript states that the split condition ensures the Picard lattice is defined over the base but does not contain an explicit computation or lemma verifying the required vanishing of the relevant Ext sheaves or cohomology groups in the expected range when q is large. This verification is load-bearing for the proof.

Authors: We agree that making the verification explicit will improve clarity. In the revision, we will insert a new lemma in §4 that directly computes the relevant Ext sheaves and cohomology groups after base change to F_q(t). The argument relies on the split hypothesis, which ensures that the Picard lattice is constant over the base, combined with the fact that for sufficiently large q the fibers of the conic bundle are smooth rational curves whose cohomology vanishes in the required degrees by standard vanishing theorems for coherent sheaves on del Pezzo surfaces. This computation confirms that the homological sieve hypotheses of Das–Lehmann–Tosteson are satisfied without additional obstructions. revision: yes
Referee: [§3.1] §3.1 (Dimension of the spaces of rational curves via conic bundles): The argument assumes the conic bundle structures induce the expected dimension for the moduli spaces of rational curves on the split quintic surfaces. No explicit calculation of the fiber dimensions or the effect of the base change to F_q(t) is supplied, which is needed to confirm that the sieve input hypotheses are satisfied without extra obstructions.

Authors: We will add an explicit dimension calculation to §3.1 in the revision. Using the conic bundle structure on the split quintic del Pezzo surface, the moduli space of rational curves is identified with a space of sections of the bundle over the base curve. The fiber dimension is computed via the Riemann–Roch theorem applied to the relative tangent bundle after base change to F_q(t); the split condition guarantees that the Picard group remains constant, so no extra components or obstructions arise for large q. This yields the expected dimension matching the sieve input requirements. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to homological sieve method from prior collaborative work

specific steps

self citation load bearing [Abstract]
"Using the homological sieve method developed by Das--Lehmann--Tosteson and the author, we prove Peyre's all height approach to Manin's conjecture for split quintic del Pezzo surfaces defined over F_q(t) assuming q is sufficiently large."

The proof invokes the homological sieve from prior work that includes the present author as a coauthor. The central claim therefore depends on the direct applicability of that method to the conic bundle geometry and moduli spaces arising here, with the large-q assumption standing in for explicit verification of vanishing conditions or base-change hypotheses within this manuscript.

full rationale

The paper applies the homological sieve method from prior work coauthored by the present author to prove the stated case of Manin's conjecture for split quintic del Pezzo surfaces over F_q(t). This is a standard citation to an established technique rather than a reduction of the target result to a tautology or to fitted quantities within the present paper. The assumption that q is sufficiently large supplies an independent parameter, and the derivation chain remains self-contained against external benchmarks from the cited method. No equations or steps reduce the claimed prediction to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the result rests on applicability of the cited homological sieve and largeness of q.

pith-pipeline@v0.9.0 · 5593 in / 1002 out tokens · 24799 ms · 2026-05-19T08:35:16.142034+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

Using the homological sieve method developed by Das–Lehmann–Tosteson … configuration covers Za,k … bar complex B(PU,ZPU) … virtual height zeta function … Tamagawa number τ−KS(S)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

We prove Peyre's all height approach to Manin's conjecture for split quintic del Pezzo surfaces

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Homological sieve and Manin's conjecture
math.AG 2026-05 unverdicted novelty 2.0

Survey of the homological sieve and its applications to Manin's conjecture.