Fano varieties: stability and symmetries

Project description

Objects of study in algebraic geometry are called varieties. They are studied using various tools from both algebra and geometry. Typically two external behaviours of a variety are interesting: how they deform and what other avrities they are isomorphic to. The latter has given rise to birational classification and the former has appeared in several new lines of research. In this project, we will be looking at Fano varieties, their deformations and their birational classification. Roughly speaking, Fano avrieties are those defined by polynomials of low degree, or equivalently geometric shapes with positive curvature. New developments in the field, such as finiteness of families of Fano varieties by Birkar (Fields medal 2018) or the development of the theory of K-stability has opened up several questions to be answered.