Moduli spaces of rational curves on Fano threefolds

TL;DR

This paper classifies components of the moduli space of rational curves on smooth Fano threefolds, confirming a conjecture about their growth and using Geometric Manin's Conjecture to analyze free curves.

Contribution

It proves classification results for moduli space components and confirms Batyrev's conjecture on their growth, advancing understanding of rational curves on Fano threefolds.

Findings

Confirmed Batyrev's conjecture on component growth

Classified moduli space components for rational curves

Applied Geometric Manin's Conjecture to free curves

Abstract

We prove several classification results for the components of the moduli space of rational curves on a smooth Fano threefold. In particular, we prove a conjecture of Batyrev on the growth of the number of components as the degree increases. The key to our approach is Geometric Manin's Conjecture which predicts the number of components parameterizing free curves.