Non-free sections of Fano fibrations

TL;DR

This paper investigates the structure of non-free sections in Fano fibrations over complex curves, confirming parts of Geometric Manin's Conjecture and providing bounds relevant to Manin's Conjecture over function fields.

Contribution

It proves that non-free sections originate from morphisms with fibers of Fujita invariant at least one and establishes a bounded family of such morphisms, verifying a key aspect of Batyrev's heuristics.

Findings

Non-free sections come from morphisms with fibers of Fujita invariant ≥ 1.

There exists a bounded family of morphisms accounting for all such sections.

Results support the first part of Batyrev's heuristics for Geometric Manin's Conjecture.

Abstract

Let $B$ be a smooth projective curve and let $\pi: \mathcal{X} \to B$ 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.