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

Sho Tanimoto

TL;DR

The paper advances Manin's conjecture for rational curves over function fields by extending the homological sieve framework to split quintic del Pezzo surfaces and, more broadly, to low-degree cases. It develops a comprehensive machinery—conic-bundle fibrations, configuration covers, and bar complexes—to count $\,\mathrm{Mor}(\mathbb P^1,S,\alpha)$ and relate these counts to Peyre's Tamagawa numbers, obtaining all-height asymptotics and uniform upper and lower bounds. A central outcome is the all-height formula for quintic del Pezzo surfaces, yielding explicit Tamagawa constants and showing that the Mor spaces are rational in many cases; analogous lower-bound results are obtained for degrees $e\le 3$. The work unifies geometric and cohomological techniques to produce precise asymptotics in the function-field setting, with concrete implications for the distribution of rational curves on del Pezzo surfaces and for the broader program surrounding Manin's conjecture over global function fields.

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.

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

TL;DR

The paper advances Manin's conjecture for rational curves over function fields by extending the homological sieve framework to split quintic del Pezzo surfaces and, more broadly, to low-degree cases. It develops a comprehensive machinery—conic-bundle fibrations, configuration covers, and bar complexes—to count and relate these counts to Peyre's Tamagawa numbers, obtaining all-height asymptotics and uniform upper and lower bounds. A central outcome is the all-height formula for quintic del Pezzo surfaces, yielding explicit Tamagawa constants and showing that the Mor spaces are rational in many cases; analogous lower-bound results are obtained for degrees . The work unifies geometric and cohomological techniques to produce precise asymptotics in the function-field setting, with concrete implications for the distribution of rational curves on del Pezzo surfaces and for the broader program surrounding Manin's conjecture over global function fields.

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 assuming 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 assuming is large.

Paper Structure

This paper contains 27 sections, 33 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. History
  4. Del Pezzo surfaces
  5. Notation
  6. Del Pezzo surfaces
  7. Geometry of conic bundles
  8. The space of rational curves on del Pezzo surfaces
  9. The transversality of the space of sections/the space of rational curves
  10. Fibrations structures induced by conic bundle structures
  11. Configuration covers
  12. Basic properties of configuration covers
  13. Further structures of $Z^{\circ}_{a, \bold k}$
  14. Fibration structures for the space of rational curves
  15. Rationality of the space of rational curves
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $S$ be a smooth split quintic del Pezzo surface over $k = \mathbb F_q$. Then there exist uniform constants $C \geq 1$ and $C_3 \geq 1$ independent of $S$ and a rational homogeneous continuous piecewise linear function $\ell(\alpha)$ on $\mathrm{Nef}_1(S)$ which is independent of $S$ and which ta where $\tau_{-K_S}(S)$ is the Tamagawa number introduced in Peyre. In this situation, it is given b

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 53 more