Seminar

The group of birational transformation of an irreducible variety has a partially defined action on the set of hypersurfaces: for a given transformation, this action is defined on all hypersurfaces but finitely many. This can be extended to an action by a canonical procedure; the resulting action can also be made explicit using basic algebraic geometry: it acts on the set of all hypersurfaces of all birational models of X, suitably defined.  This action "commensurates" the subset of those hypersurfaces that are in the original variety. This is known to induce an action on a CAT(0) cube complex. This construction allows, when a group is known to have restrictions on its actions on CAT(0) cube complex (such as Property FW: every action on any CAT(0) cube complex has a fixed point), to provide restrictions on its birational actions on varieties. The most optimistic conclusion is to prove that the action is by variety automorphisms on some model; we can reach this conclusion for birational actions on surfaces of groups with Proprety FW.



I will carefully define the basic definitions, both on the side of CAT(0) cube complexes, and especially on the algebraic geometry side (where I will essentially only consider the projective plane), so that the talk requires no prerequisite in algebraic geometry.