Noncommutative Geometry Seminar

In a lecture course given at the end of the 1990s in Brussels, Tits explained his axiomatic approach to "Cremona planes": he defined an incidence structure consisting of "points" and "figures," and endowed this structure with a large set of synthetic axioms based on the geometry of (smooth, projective) models of C(X,Y). One of the main features of his approach was that the automorphism group of this incidence geometry precisely is the group of automorphisms of C(X,Y) fixing C elementwise, that is, the Cremona group. In the paper which was based on these lectures, and published in his Evres, Tits posed the question as to whether a "thin" Cremona plane exists. In September 2014, Deligne, after reading Tits's paper, answered the question in a letter to Tits, by providing a description of a thin Cremona plane (defined over F_1). In this lecture, we mention some aspects of this theory.