Abstract: A famous conjecture of Manin predicts an asymptotic formula for counting the number of rational points of bounded height on a Fano variety defined over a number field. In the 1990s, Batyrev developed a heuristic argument for a version of Manin's Conjecture over finite fields that assumes irreducibility of certain spaces of embedded rational curves. Though Batyrev's heuristics and Manin's initial conjecture are false in general, Geometric Manin’s Conjecture (GMC) translates Batyrev’s heuristic for Manin’s Conjecture to statements about free rational curves on Fano varieties. In this talk, I will first review this translation and motivate the framework of GMC with concrete examples. I will then describe a recent proof of GMC for smooth Fano threefolds over the complex numbers by appealing to relationships between this framework and the Mori structures of Fano threefolds.
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