# Geometric Manin's Conjecture and rational curves

## TL;DR

This paper explores the structure of rational curves on smooth projective Fano varieties, proposing a geometric version of Manin's Conjecture to predict the growth of related moduli spaces.

## Contribution

It introduces a Geometric Manin's Conjecture and provides bounds on the dimension and components of rational curve moduli spaces on Fano varieties.

## Key findings

- Bounded the dimension of rational curve moduli spaces
- Estimated the number of irreducible components
- Proposed a growth rate conjecture for these components

## Abstract

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli spaces of rational curves on $X$ using the perspective of Manin's Conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on $X$. We propose a Geometric Manin's Conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.