The objects of the talk are the projective algebraic varieties over a field that are homogeneous under a semisimple algebraic group G. Such varieties are well understood in characteristic 0: they are the flag varieties G/P (where P is a parabolic subgroup of G) and their forms, for example the Severi-Brauer varieties. The situation is more complicated in positive characteristic, since parabolic subgroups may be non-smooth.

The talk will present the classification of projective homogeneous varieties over an algebraically closed field of characteristic p > 0, due to Wenzel (1989) for p > 3 and Matilde Maccan (2024-2025) for p = 2,3, and its generalization to an arbitrary field based on work in progress with Maccan and Srimathy Srinivasan. In particular, varieties of Picard rank 1 are "almost as expected" and many new phenomena arise in higher Picard rank.

Let X be a complex projective variety. The group of automorphisms Aut(X) acts on the cohomology of X, for instance on the second cohomology group H^2(X;\Z). This gives a linear representation of Aut(X) in GL(H^*(X;\Z)). The problem I will discuss is the following : how can we measure the complexity of the image of Aut(X) in GL(H^*(X;\Z)) ? How does it depend on the geometry of X, or on the dimension of X ?

I will talk about the work of my PhD student Sebastian Fuentes who classified almost simple finite subgroups in the space Cremona group.

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Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona modification sending one onto the other. In the last decades Cremona equivalence has been investigated widely and we have now a complete theory for non divisorial reduced schemes. The case of irreducible divisors is completely different and not much is known beside the case of plane curves and few classes of surfaces.

In this talk I will provide a gentle introduction to the state of the art, recall the, somewhat surprising, result of reduced schemes in codimension 2 and study in detail the class of smooth surfaces generically projected in P^3, where an unexpected relation with sectional genus seems to appear.

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I will talk about the birational geometry of del Pezzo surfaces of degree 4, that is, smooth intersections of two quadrics. In particular, we will discuss a classification of their birational models and properties of birational automorphism groups.

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I will discuss new invariants in equivariant birational geometry (joint with A. Kresch, F. Scavia, and Zh. Zhang)

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