Reading seminar on the Geometric Manin Conjecture, Fall 2025The Geometric Manin's conjecture concerns the dimension and number of components of the space of rational curves on a projective variety. The aim of this seminar is to understand the paper Geometric Manin's conjecture and rational curves by Lehmann and Tanimoto. The starting points are Batyrev's heuristics and Manin's conjecture for rational points on Fano varieties over number fields. The authors observe that in order to have a reasonable control on the number of components, one must discount an exceptional set. They thus define the so-called Manin components as those that parametrize families of rational curves that capture the geometry of the variety, rather than the geometry of a subvariety or of a cover of. The Geometric Manin's Conjecture predicts an asymptotic formula for a generating function encoding the number and dimension of the Manin components of the space of morphisms of bounded degree from the projective line. The paper shows that the Geometric Manin's Conjecture holds for Fano threefolds of Picard rank 1 and index 2. ProgramSee the detailed program for the literature and a description of each talk. Each talk lasts 2 hours including breaks.
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