Talks

Abstract:
We discuss the following result. If G is a finitely generated nilpotent group acting faithfully on a complex algebraic variety X, then the dimension of X is larger than the virtual derived length of G, i.e the minimum of the derived length of the finite index subgroup of G. The proof uses p-adic analysis and p-adic Lie groups method. We show that up to finite index there exists a prime p such that G acts analytically on a p-adic manifold related to X. The study of this action gives the desired bound.

Abstract:
In this talk, I will discuss the following question, attributed to Gizatullin: "Which automorphisms of a smooth quartic surface in projective 3-space are restrictions of Cremona transformations of the ambient space?'' I will present a general framework that can be used to study this problem, namely a special "volume preserving" version of the Sarkisov program, and report on recent progress obtained in collaboration with A. Corti and A. Massarenti and D. Paiva and S. Zikas.

Abstract:
Del Pezzo surfaces and their automorphism groups play a key role in the classification up to conjugacy of subgroups of the Cremona group of the plane. Over an algebraically closed field, they are completely classified together with their automorphism groups. In this talk, we will first focus on del Pezzo surfaces of degree 5 defined over a perfect field. In this case, the classification as well as the description of all their possible automorphism groups is reduced to understanding the possible actions of the Galois group on the graph of (-1)-curves. We will next focus on rational del Pezzo surfaces of degree 4 defined over the real numbers, and see how studying the action of the Galois group on the conic bundle structures enables us to give a complete description of their automorphism groups by generators in terms of automorphisms and birational automorphisms.

Abstract:
Equivariant completions with an open orbit of abelian unipotent groups into smooth projective varieties have been studied in relation to several applications ranging from the study of projective completions of complex affines spaces to arithmetic questions concerning the asymptotic distribution of rational points of bounded height on varieties defined over number fields. A central tool in this context is the so-called Hasset-Tschinkel correspondences which relates conjugacy classes of equivariant completions with an open orbit of abelian unipotent groups into projective spaces to isomorphism classes of local Artinian commutative algebras. Building on this correspondence and on the classification of certain completions of the affine 3-space into smooth Fano threefolds of Picard rank 2 obtained by Kishimoto, the classification of equivariant completions of \(\mathbf{G}_a^3\) into smooth Fano threefolds was achieved by Huang and Montero. In this talk, I will explain recent progress towards a similar classification for completions of the Heisenberg group —the unique 3-dimensional non-abelian unipotent group over a field of characteristic zero— into total spaces of smooth threefolds Mori fiber spaces, based on a non-commutative (and non-associative) generalization of the Hasset-Tschinkel correspondence. [Joint work in progress with Takashi Kishimoto and Pedro Montero].

Abstract:
In this talk I will present joint projects with Fabio Bernasconi, Julia Schneider, Stefan Schröer and Susanna Zimmermann aimed to study some aspects of the birational geometry of regular algebraic surfaces over imperfect fields. These objects naturally appear when one works with fibrations over algebraically closed fields in positive characteristics. I will also discuss applications to the Cremona group.

Abstract:
An important invariant of an automorphism f of an algebraic variety X is its growth, i.e., how does the norm of the inverse image operator on the cohomology group \(H^2(X,\mathbf C)\) of the n'th power \(f^n\) depend on n. The growth can be exponential, polynomial, or bounded. There are many beautiful results describing constraints on the geometry of X admitting an automorphism with exponential or polynomial growth. We are going to discuss the geometric properties of a variety X which admits an infinite order automorphism with bounded growth. It appears there are only two sources of these automorphisms; namely, automorphisms of projective spaces and translations on abelian varieties. In particular, a rationally connected threefold whose automorphism group contains an element f of infinite order with bounded growth is necessarily a rational variety, and an iterate of the automorphism f is birationally conjugate to a regular automorphism of the projective space. I am going to explain how to work with these automorphisms in any dimension and to prove this result.

Abstract:
This is joint work with Andriy Regeta (University of Jena/Padova) and Christian Urech (ETH Zürich). In this talk, I will highlight our recent progress in understanding Borel subgroups of the group Bir\((X)\) of birational transformations and the group Aut\((X)\) of automorphisms of an irreducible variety X. We prove that any Borel subgroup of Bir\((X)\) has derived length at most twice the dimension of X, with equality holding if and only if X is birationally equivalent to the affine space \(\mathbf A^n\), and the Borel subgroup is conjugate to the standard Borel subgroup in the Cremona group Bir\((\mathbf A^n)\). Furthermore, we provide examples of non-standard Borel subgroups of the Cremona group Bir\((\mathbf A^n)\) for \(n\geqslant 2\) and of the affine Cremona group Aut\((\mathbf A^n)\) for \(n\geqslant 3\), thereby resolving conjectures by Popov and Furter-Poloni.

Abstract:
I will talk about some recent results and open questions in the birational geometry of algebraic surfaces with a group action (a Galois group or a finite group of automorphisms). This will complement the mini-courses of Blanc, Schneider and Zimmermann.