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We describe the moduli space of rational curves on smooth Fano varieties of coindex 3. For varieties of dimension 5 or greater, we prove the moduli space has a single irreducible component for each effective numerical class of curves. For varieties of dimension 4, we describe families of rational curves in terms of Fujita's a-invariant. Our results verify Lehmann and Tanimoto's Geometric Manin's Conjecture for all smooth coindex 3 Fano varieties over the complex numbers.
Jovinelly et al. (Sun,) studied this question.