GROUPES en ACTIONS

1. Real forms of toric varieties

Intervenants : Giancarlo Lucchini-Arteche (UChile Santiago) et Ronan Terpereau (Lille)

Toric varieties are central objects in algebraic geometry, whose combinatorial description makes them particularly tractable. In this mini-course, we will first review their classification over the complex numbers. We will then explain how Galois descent can be used to obtain a classification over the real numbers.
* Lecture 1 (1h30): Complex toric varieties
* Lecture 2 (1h30): Real forms, Galois cohomology, and the classification of real tori and torsors under the action of a real torus
* Lecture 3 (1h): Real forms of toric varieties: main result and examples

Notes de cours

2. $K3$ surfaces and their automorphisms groups

Intervenants : Yulieth Prieto (PUC Santiago) et Alessandra Sarti (Poitiers)

$K3$ surfaces are important algebraic surfaces due to their remarkable properties. The name was given by A. Weil in 1958 in honour of the mountain K2 in Kashmir and the three renowned mathematicians Kummer, Kähler, and Kodaira. The aim of these lectures is to introduce the basic definitions and properties of K3 surfaces, while also providing many examples. In particular, we will consider quartic surfaces, complete intersections, Kummer surfaces, and elliptic fibrations. We will also discuss the role of their automorphisms group, which is a fundamental tool for understanding their geometry. Furthermore, we will study $K3$ surfaces endowed with an elliptic fibration and an automorphism acting on it, present some general results, and give several applications.

3. Topics in polynomial derivations and their automorphism groups

Intervenants : Ivan Pan et Alvaro Rittatore (CMAT Montevideo)

Let $K$ be an algebraically closed field. It is well known that, if $n\geq 2$, one cannot endow the so-called \emph{Affine Cremona Group} ${\rm ACr}(n)$, consisting of the automorphisms of the affine space ${\mathbb A}^n_K$, with a structure of variety (or scheme), but ${\rm ACr}(n)$ can be seen as an ind-variety. Under this point of view, a lot of work has been done. For example, one may characterize when a closed subgroup $G\subset {\rm ACr}(n)$ inherits a structure of algebraic group --- in this case we have a faithfull action of $G$ on the affine space. Analogously, one may ask for subgroup schemes of ${\rm ACr}(n)$. On the other hand, by thinking of the elements in ${\rm ACr}(n)$ as $K$-algebra automorphisms of $K[x_1,\ldots,x_n]$, we see that ${\rm ACr}(n)$ acts by conjugation on $\mathrm{Der}(K[x_1,\ldots,x_n])$, the vector space of the $K$-derivations $D:K[x_1,\ldots,x_n]\to K[x_1,\ldots,x_n]$ --- a $K$-derivation is a linear map $D$ satisfying the Leibnitz rule $D(fg)=fD(g)+gD(f)$. The isotropy ${\rm Aut}(D)$ of such a $D$, which necessarily codifies important properties of its conjugation class, inherits a structure of closed ind-subvariety of ${\rm ACr}(n)$. It is natural to ask in which mesure (the equivalence class of) $D$ is determined by $\mathrm{Aut}(D)$ in the case where that subgroup acts on ${\mathbb A}^n_K$ as an algebraic group (or group scheme). This mini-course focuses in introducing some tools to lead with this type of problem. More precisely: first we explain how to endow ${\rm Aut}(D)$ with a structure of closed ind-subgroup of ${\rm ACr}(n)$ and propose some problems relating to that. Subsequently, we describe the situation in dimension 2, where all is quite known. Finally, we give some results in higher dimension and propose some possible themes for studying.

Notes de cours   Slides

4. Birational geometry of rational surfaces

Intervenants : Pedro Montero (UTFSM Valparaiso) et Enrica Floris (Toulouse)

A fundamental notion in algebraic geometry is that of a rational variety, which is a geometric object that looks like projective space. In this mini-course we will present basic definitions and examples of rational surfaces alongside with some fundamental classification theorems.

5. Groupes arithmétiques et amalgames : le Théorème de Nagao

Intervenants : Claudio Bravo (Talca) et Benoit Loisel (Poitiers)

Présentation d'un groupe et groupes libres. Produits libres, amalgames et énoncé du Théorème de Nagao. Le $p$-arbre de Bruhat-Tits de $\mathrm{(P)SL}_2(\mathbb{Z})$. Interprétation via les fibrés en droites, Théorème de Grothendieck-Birkhoff et preuve du Théorème de Nagao. Généralisation à $\mathrm{SL}_n$: immeubles et le théorème de Soulé

Notes de cours