**Luogo:** Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari**Ore** 14:00

**Relatore:**

Imperial College**Diletta Martinelli**(**)**

**Abstract: **

I will start by giving some motivations as to why we are interested in doing birational geometry in characteristic p. I will give an overview of the main differences from the complex case, such as the failure of Kodaira vanishing theorem and the lack of a proof of resolution of singularities. Then I will explain how these have been replaced with new methods, such as the Frobeniuos morphism. In particular, I will focus on the theory of F-singularities and their relations in characteristic zero.

In the second part of the talk, I will address a specific open problem in positive characteristic, the Base Point Free Theorem. In a joint work with Jakub Witaszek and Yusuke Nakamura, we proved the Base Point Free theorem for varieties of dimension three with log canonical singularities defined over the algebraic closure of a finite field. I will give an introduction to the problem and describe the tools that we used in the proof.

**Referente: **Claudio Fontanari