Geometric Manin's Conjecture for Fano Threefolds

This paper proves Geometric Manin’s Conjecture for smooth Fano threefolds over the complex numbers and of Picard rank at least two. Geometric Manin’s Conjecture translates Batyrev’s heuristic for Manin’s Conjecture to a statement about rational curves on Fano varieties. For a smooth Fano variety X, the conjecture predicts there is a constant bound on the number of Manin components of the morphism scheme of rational curves of each numeric class. On a Fano threefold and for numeric classes of anticanonical degree at least four, Manin components are precisely those which generically parameterize very free curves. We prove there is at most one such component for each numeric class.

First, we describe how a recent, movable version of Mori’s Bend-and-Break for free rational curves on Fano threefolds reduces Geometric Manin’s Conjecture to a study of low degree free curves. We then establish monodromy properties of Mori fiber space structures on Fano threefolds that are general in their deformation class. This allows us to describe families of low degree free rational curves explicitly on such threefolds. Where necessary, we then prove the number of Manin components of curves remains constant under specialization of X. We conclude with a case-by-case analysis of the 88 deformation types of smooth Fano threefolds of Picard rank at least two.