Courses

Abstract:
I will present basic properties of the Cremona group in 2 variables. For instance, the Noether-Castelnuovo theorem states that the group is generated by the group of automorphisms and the standard quadratic map. I will also make the link with conic bundles and del Pezzo surfaces, which appear naturally when studying finite subgroups or more generally algebraic subgroups of the group.

Abstract:
Weil's regularization theorem asserts that for any rational action of an algebraic group G on a variety X, there exists a variety on which G acts regularly and which is equivariantly birational to X. This theorem and its refinements are key ingredients for classifying algebraic subgroups of groups of birational transformations, in classical work of Enriques and recent work of Blanc, Zimmermann, Fong and others. The lectures will first discuss the notions occurring in the regularization theorem, and some of its applications. We will then present a proof of this theorem and further developments.

Abstract:
Dynamical degrees were introduced by Russakovskii and Shiffman in 1997. They measure the growth of preimages of subvarieties of a fixed dominant rational map of a projective variety, and control many dynamical features of this map. In this first set of lectures, we shall define dynamical degrees using positivity properties of divisor classes, and explain how to compute them in low dimension and in some significant examples.

Abstract:
We start by defining the Grothendieck ring of varieties and the Burnside ring, discussing their fundamental properties and open questions. Then we explain various types of motivic invariants of birational maps along with their basic properties, including the more refined horizontal and vertical motivic invariants for maps between fibrations.
We state vanishing results of motivic invariants for surfaces over perfect fields and prove vanishing for all threefolds over the field of complex numbers, using the intermediate Jacobians and MRC fibrations.
Finally, we explain the nonvanishing of motivic invariants for various Cremona groups, starting with P^3. For P^4 over the field of complex numbers, we explain the unboundedness of motivic invariants, via K3 and elliptic surfaces. Time permitting, we also introduce invariants of pairs that relate motivic invariants to the construction of Genevois–Lonjou–Urech.

Abstract:
Methods from geometric group theory have turned out to be essential tools to study Cremona groups. Especially isometric actions on spaces of negative or non-positive curvature have proven to be very useful. In this course, we will focus on isometric actions of Cremona groups on CAT(0) cube complexes, which allow us to better understand certain group theoretical and dynamical properties of groups of birational transformations. After a basic introduction to the geometry of CAT(0) cube complexes, we construct several complexes with actions of Cremona groups. In a first step, we look at the blow-up complex, which allows us to give uniform proofs for various results about groups of birational transformations in dimension two. In a second step, we construct cube complexes that allow isometric actions for groups of birational transformations in arbitrary dimension.

Abstract:
Two main themes : how do dynamical degrees relate to entropy, and how do these notions of complexity vary as parameters change?
The recent work of Junyi Xie on semicontinuity of dynamical degrees, a priori only in codimension 1. This will be a continuation of the first mini-course on dynamical degree by Favre.
Relation of dynamical degrees to topological entropy: Gromov upper bound on entropy for endomorphisms, and a few words on Yomdin’s lower bound (for instance proving the lower bound in terms of topological degree).

Abstract:
Two main goals : Skolem-Mahler-Lech theorem and constraints on groups acting by birational transformations (for instance \(\mathrm{SL}(m,\mathbf Z)\) does not act faithfully by birational transformations on Xif \(\dim(X)<m-1\)). A tentative plan is

1. p-adic numbers, p-adic analytic functions, principle of isolated zeroes (Strassman)
2. Skolem-Mahler-Lech and its non-linear version (theorems of J. Bell, of Bell, Ghioca Tucker)
3. \(\mathrm{SL}(m,\mathbf Z)\) does not embed in \(\mathrm{Aut}(A^n)\) if \(n<m\).
4. \(\mathrm{SL}(m,\mathbf Z)\) does not embed in \(\mathrm{Bir}(A^n)\) if \(n<m-1\).

Abstract:
I shall discuss the action of the Cremona group on the infinite dimensional space of b-divisors (where b stands for birational), obtained as a limit of all Néron-Severi spaces for all possible blowups dominating the projective plane. This space is naturally endowed with a quadratic form of Minkowski type, and so contains an infinite dimensional hyperboloid. One application of this construction is to produce many normal subgroups in the Cremona group, via a general result in small cancellation theory. More generally, the action allows to classify elements or subgroups or the Cremona group along the familiar elliptic/parabolic/loxodromic subtypes in hyperbolic geometry. I plan to discuss a few classification results along these lines, for instance subgroups that contain a positive dimensional algebraic normal subgroup, or subgroups of elliptic elements that do not preserve a pencil of rational curves.

Abstract:
The primary objective of this course is to outline key guidelines for proving the classification of maximal connected algebraic subgroups of the Cremona group in dimension 3 over an algebraically closed field. The five lectures will address the following topics

• Lecture 1: Aims and scope, a warm-up: tori and additive groups in the Cremona groups
• Lecture 2: Standard conic bundles over rational surfaces
• Lecture 3: Automorphism groups of standard conic bundles whose generic fiber is not \(\mathbf P^1\)
• Lecture 4: Mori del Pezzo fibrations of degree at most 6 over \(\mathbf P^1\)
• Lecture 5: Umemura quadric fibrations over \(\mathbf P^1\)

Résumé :
The first step of the course is to see the following: any birational map of the plane can be decomposed into the blow-up of a point, elementary transformations between Hirzebruch surfaces and the exchange of the two fibrations of \(\mathbf{P}^1\times\mathbf{P}^1\). So, the Fano surface \(\mathbf P^2\) and the Hirzebruch surfaces, and elementary birational maps between them, are the base of birational maps of the plane.
The second step of the course is to generalise this concept to non-closed fields. The base-stones will be minimal del Pezzo surface and minimal conic fibrations over the projective line, and elementary birational maps between them.

Résumé :
A fundamental invariant for the dynamics of a Cremona transformation \(f:\mathbf P^n\to\mathbf P^n\) is its "first dynamical degree", an asymptotic measure of how quickly the degree of a hypersurface grows when pulled back by the iterates of f. In many particular situations first dynamical degrees can be realized and effectively computed as eigenvalues of integer matrices. When the dimension n is two, this is always the case. I will discuss a joint work with Jason Bell, Mattias Jonsson and Holly Krieger in which we construct examples of Cremona transformations in dimensions three and higher with first dynamical degrees that are transcendental numbers. Analyzing the examples involves, among other things, interesting toric geometry and substantial results about diophantine approximation.

Abstract:
Regularizable transformations of the Cremona group are the ones that admit a birational conjugate that is an automorphism of a projective surface. In this mini course, we will focus on the following question: Consider a finitely generated subgroup G of the Cremona group such that all its elements are regularizable. Is G regularizable? The key tool to start to tackle this question is the construction of an action on the blow-up complex, which is a CAT(0) cube complex. Note that for now, the question is still open in general.

Résumé :
Which quotients do Cremona groups admit? The answer depends on the field and on the rank. For example, the plane Cremona group over an algebraically closed field has no abelian quotient and no finite quotient. This is no longer true for non-closed fields, and it is also not true for Cremona groups of higher ranks. We will discuss several constructions of quotients in these cases, focusing on those using the Sarkisov program.