A reconstruction theorem for varieties
arXiv:1902.04668
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.
This has been superseded by a corrected and vastly expanded paper that also includes additional coauthors, available at arXiv:2003.04847