Abstract:

Rational homogeneous varieties G/P generalize projective spaces and Grassmannians, and they are very simple objects from the birational point of view: they carry a transitive action of a semisimple group G, hence they are smooth rational Fano varieties, and in fact Mori dream spaces. Even so, their quotients by natural torus actions, such as the action of a maximal torus of G, turn out to be hard to understand. In this talk I will introduce the necessary background, starting from algebraic groups and rational homogeneous varieties and moving on to GIT and Chow quotients. In the last part I will present the main results of recent joint work with L. Barban and L. E. Solá Conde on maximal geometric quotients of complete flag varieties.