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be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on
using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on
. We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.
We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a Kawamata log terminal pair (X,Δ) can be detected on a birational model of the base of the (KX+Δ)-trivial reduction map. We then interpret the main conjectures of the minimal model program as a natural statement about the existence of curves on X.
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