Geometric Manin's Conjecture for Fano 3-Folds

TL;DR

This paper classifies families of free rational curves on smooth Fano threefolds, proving Geometric Manin's Conjecture in dimension three and enabling explicit counts of moduli space components.

Contribution

It provides a complete classification of free rational curve families on Fano threefolds and confirms the conjecture in this dimension, with explicit enumeration results.

Findings

Family of very free rational curves is either irreducible or empty.

Proved Geometric Manin's Conjecture for Fano threefolds.

Explicitly counted moduli space components for various classes.

Abstract

We classify families of free rational curves on all smooth Fano threefolds over the complex numbers. In particular, we prove the family of very free rational curves representing any fixed numerical curve class is either irreducible or empty. This proves Geometric Manin's Conjecture in dimension three. For general Fano threefolds of each deformation type, our results allow us to explicitly count the number of components of the moduli space of irreducible, geometrically rational curves, which may not be free, representing any numerical class.