Atiyah flip: birational geometry VS GIT

Venue: The event will take place online through the ZOOM platform. To get the access codes please contact the secretary office (phd.maths [at] unitn.it)

Time: 11.00 a.m.

• Alberto Franceschini - PhD in Mathematics, University of Trento

Abstract:
The aim of Minimal Model Program (MMP) is to classify projective varieties up to birational equivalence. Starting with a smooth projective variety X, one can associate a cone, called the Mori cone, and birational maps from X to a projective variety Y correspond to the contraction of certain extremal rays in the negative part, w.r.t. the canonical divisor, of this cone. It can happen that contracting a ray in the Mori cone of X produces a small contraction, that is a birational map where the exceptional locus has codimension smaller than 2. Small contractions produce, in general, a variety Y with too bad singularities to go on with the MMP, flips arise to solve this problem. A flip is a birational map form X to a projective variety (with not too bad singularities) X+ such that the birational map from X+ to Y is given by the contraction of a ray in the positive side of the Mori cone of X+. As long as we can find such a variety, we can go on with MMP.
In this seminar, we present an example of a flip (on which Atiyah worked on first) and we produce its construction  using techniques from both birational geometry and GIT.

Contact person: Eduardo Luis Sola Conde

The seminar corresponds to the final exam of Algebraic Geometry II, a planned course within  Franceschini's first year PHD study programme