A reconstruction theorem for varieties (1902.04668v2)

Abstract: We show that varieties of dimension at least 2 over infinite fields are determined as abstract schemes by their Zariski topological spaces together with the rational equivalence relation on the set of effective divisors. This gives a universal Torelli theorem in the sense of Bogomolov and Tschinkel. The proof relies heavily on a rational version of the classical Fundamental Theorem of Projective Geometry.