Non-Free Sections of Fano Fibrations

Let B B be a smooth projective curve and let π : X → B \pi : \mathcal {X} \to B be a smooth integral model of a geometrically integral Fano variety over K ( B ) K(B) . Geometric Manin’s Conjecture predicts the structure of the irreducible components M ⊂ S e c ( X / B ) M \subset 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 M the sections come from morphisms f : Y → X f: \mathcal {Y} \to \mathcal {X} such that the generic fiber of Y \mathcal {Y} has Fujita invariant ≥ 1 \geq 1 . Furthermore, we prove that there is a bounded family of morphisms f f which together account for all such components M M . These results verify the first part of Batyrev’s heuristics for Geometric Manin’s Conjecture over C \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 p , we obtain upper bounds of the desired form by first letting the prime go to infinity, then the height.