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Non-free sections of Fano fibrations

Brian Lehmann, Eric Riedl, Sho Tanimoto

TL;DR

This paper develops a geometric framework for Geometric Manin's Conjecture over the complex numbers for Fano fibrations π:𝒳→𝐵, focusing on non-relatively free sections of high anticanonical degree. The authors show that such sections arise from accumulating maps f:𝒴→𝒳 that preserve or increase the Fujita invariant a(𝒴_η, -f^*K_{𝒳/𝐵}|_{𝒴_η}) and that all such accumulating maps lie in a bounded family, leading to a finite, well-controlled set of contributions to the moduli of sections. They develop a robust toolkit—Fujita invariants, Harder–Narasimhan stability, descent theory for vector bundles, and Hurwitz-space twists—to capture the geometry of sections, classify non-nef and nef components, and connect local-global phenomena via twists and base changes. The results verify the first aspect of Batyrev’s heuristic for Geometric Manin’s Conjecture over ℂ and yield arithmetic consequences for function-field settings by enabling upper bounds after suitable reductions. Overall, the work provides a comprehensive bridge between birational geometry, stability theory, and arithmetic heuristics, clarifying how pathological families are bounded and how they contribute to the asymptotics of rational points on Fano fibrations.

Abstract

Let $B$ be a smooth projective curve and let $π: \mathcal{X} \to B$ be a smooth integral model of a geometrically integral Fano variety over $K(B)$. Geometric Manin's Conjecture predicts the structure of the irreducible components $M \subset \textrm{Sec}(\mathcal{X}/B)$ which parametrize non-relatively free sections of sufficiently large anticanonical degree. Over the complex numbers, we prove that for any such component $M$ the sections come from morphisms $f: \mathcal{Y} \to \mathcal{X}$ such that the generic fiber of $\mathcal{Y}$ has Fujita invariant $\geq 1$. Furthermore, we prove that there is a bounded family of morphisms $f$ which together account for all such components $M$. These results verify the first part of Batyrev's heuristics for Geometric Manin's Conjecture over $\mathbb{C}$. Our result has ramifications for Manin's Conjecture over global function fields: if we start with a Fano fibration over a number field and reduce mod $p$, we obtain upper bounds of the desired form by first letting the prime go to infinity, then the height.

Non-free sections of Fano fibrations

TL;DR

This paper develops a geometric framework for Geometric Manin's Conjecture over the complex numbers for Fano fibrations π:𝒳→𝐵, focusing on non-relatively free sections of high anticanonical degree. The authors show that such sections arise from accumulating maps f:𝒴→𝒳 that preserve or increase the Fujita invariant a(𝒴_η, -f^*K_{𝒳/𝐵}|_{𝒴_η}) and that all such accumulating maps lie in a bounded family, leading to a finite, well-controlled set of contributions to the moduli of sections. They develop a robust toolkit—Fujita invariants, Harder–Narasimhan stability, descent theory for vector bundles, and Hurwitz-space twists—to capture the geometry of sections, classify non-nef and nef components, and connect local-global phenomena via twists and base changes. The results verify the first aspect of Batyrev’s heuristic for Geometric Manin’s Conjecture over ℂ and yield arithmetic consequences for function-field settings by enabling upper bounds after suitable reductions. Overall, the work provides a comprehensive bridge between birational geometry, stability theory, and arithmetic heuristics, clarifying how pathological families are bounded and how they contribute to the asymptotics of rational points on Fano fibrations.

Abstract

Let be a smooth projective curve and let be a smooth integral model of a geometrically integral Fano variety over . Geometric Manin's Conjecture predicts the structure of the irreducible components which parametrize non-relatively free sections of sufficiently large anticanonical degree. Over the complex numbers, we prove that for any such component the sections come from morphisms such that the generic fiber of has Fujita invariant . Furthermore, we prove that there is a bounded family of morphisms which together account for all such components . These results verify the first part of Batyrev's heuristics for Geometric Manin's Conjecture over . Our result has ramifications for Manin's Conjecture over global function fields: if we start with a Fano fibration over a number field and reduce mod , we obtain upper bounds of the desired form by first letting the prime go to infinity, then the height.
Paper Structure (61 sections, 71 theorems, 230 equations)

This paper contains 61 sections, 71 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Fano fibrations
  3. Geometric Manin's Conjecture
  4. Main results
  5. A geometric application
  6. An arithmetic application
  7. Stability and restriction
  8. Strategy
  9. History
  10. Background
  11. Convex geometry
  12. Vector bundles on curves
  13. Cohomology bounds
  14. Slope stability for smooth projective varieties
  15. Fujita invariant
  16. ...and 46 more sections

Key Result

Theorem 1.3

Let $\pi: \mathcal{X} \to B$ be a Fano fibration. There is a constant $\xi = \xi(\pi)$ with the following properties. Let $M$ be an irreducible component of $\operatorname{Sec}(\mathcal{X}/B)$ parametrizing a family of non-relatively free sections $C$ which satisfy $-K_{\mathcal{X}/B} \cdot C \geq \

Theorems & Definitions (177)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Example 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 167 more