Geometric Manin’s conjecture and rational curves

Let X be a smooth projective Fano variety over the complex numbers. We study the moduli space 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.