Arithmetic finiteness results for Fano varieties and hypersurfaces   
-- Ariyan Javanpeykar, April 13, 2018

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Faltings proved that there are only finitely many abelian schemes of dimension g over a fixed "arithmetic curve". Similar finiteness results hold (for instance) for smooth proper curves of fixed genus g >1 (Faltings), and smooth cubic surfaces (Scholl). In this talk, we will investigate similar finiteness results for Fano varieties and hypersurfaces, and explain the "relation" with hyperbolic moduli spaces. For instance, we will explain how the Lang-Vojta conjecture implies vast generalizations of Faltings's finiteness result. This is joint work with Daniel Loughran.