Klt del Pezzo surfaces which are not globally F-split

We construct a klt del Pezzo surface which is not globally F-split, over any algebraically closed field of positive characteristic.Comment: 18 page

On log del Pezzo surfaces in large characteristic

We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally F-regular or it admits a log resolution which is liftable to characteristic zero. As a consequence, we prove the Kawamata-Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.Comment: v2: final version. To appear in Compositio Mathematic

On the anti-canonical ring and varieties of Fano type

We study the anti-canonical ring of a projective variety and we characterise varieties of log Fano type depending on the singularities of these models.Comment: 13 pages, v2: remove Example 5.2 from v1, v3: Reference adde

New outlook on the minimal model program, I

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.Comment: to appear in Duke Math.

The Minimal Model Program Revisited

We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.Comment: to appear in Contributions to Algebraic Geometry, EMS Series of Congress Report