Birational Geometry

I will survey some old and new developments of the Minimal Model Program. After an elementary introduction, I will describe how the program has been recently generalised to several contexts. The talk will be accessible to Master and young PhD students interested in algebraic geometry.

I will discuss recent joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia, in which we construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral (i.e., not finitely generated) cone of effective divisors, both in characteristic 0 and in prime characteristic. As a consequence, the Grothendieck-Knudsen moduli space of stable rational curves with n>=10 markings has a non polyhedral cone of effective divisors and, in particular, it is not a Mori Dream Space. The talk will aim at explaining the motivation behind questions about the birational geometry of moduli spaces, as well as the techniques needed to reduce such questions to blow-ups of toric surfaces. To settle the question in the case of surfaces, we will use some basic toric geometry and computations with elliptic curves. The talk will be accessible to master and PhD students interested in algebraic geometry.

The relation among birational geometry, non-archimedean geometry and mirror symmetry is a relatively new, rich and fascinating research subject in algebraic geometry. In this talk I will try to describe some aspects of this connection. In this perspective I will present the notion of dual complex. I will then construct retractions of non-archimedean spaces onto dual complexes, and hint at their connection with SYZ mirror symmetry. The talk will be accessible to PhD students interested in algebraic geometry.

Toric varieties are algebraic varieties which are constructed following combinatorial recipes. There is a very powerful dictionary between the algebro-geometric properties of toric varieties and the properties of their combinatorial avatars (polytopes and cones). In this talk I will try to explain some constructions and show some examples which are relevant for the birational study of algebraic varieties. The talk will be accessible to master and PhD students interested in algebraic geometry.