de Jong proved that any variety X can be desingularized by an alteration f:X'-->X, i.e. a proper surjective generically finite morphism. This was strengthened by Gabber as follows: f can be chosen of degree prime to a fixed prime l invertible on X.In this talk, I'll tell about the most recent progress on the subject: if X is of finite type over a quasi-excellent threefold then one can desingularize X by an alteration whose degree is only divisible by primes non-invertible on X. We will also discuss finer results that […]

In positive characteristic the Base Point Free Theorem is still an open problem. 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.

Je donnerai une condition suffisante pour que l'anneau de cohomologie quantique soit semisimple et expliquerai comment l'appliquer à certaines variétés de Fano ayant un grand indice. Je considérerai tout particulièrement l'exemple des sections linéaires de grassmanniennes de droites.