Jérémy Blanc - Universität Basel - Mathematik
Swiss-French workshop on algebraic geometry
Enney (near Gruyères, Fribourg, Switzerland), February 20-24, 2012
The workshop was held in Enney from February 20 to 24, 2012.

Mini-courses
In the morning, there were three mini-courses of 5 hours (3 times one hour each day).
 Frédéric MANGOLTE (Angers) Course on real algebraic geometry Lucy MOSER-JAUSLIN (Dijon) Pierre-Marie POLONI (Basel) Course on locally nilpotent derivations and extensions of automorphisms Adrien DUBOULOZ (Dijon) Jean-Philippe FURTER (La Rochelle) Immanuel STAMPFLI (Basel) Course on ind-varieties (algebraic varieties of infinite dimension)

In the afternoon, we had research talks of 50 minutes.

Schedule
Talks: In Enney, ("Centre l'Ondine, Viva-Gruyère" )
 Monday   February 20 Tuesday   February 21 Wednesday   February 22 Thursday   February 23 Friday   February 24 breakfast 9h-10h  mini-course 1 10h15-11h15  mini-course 2 11h30-12h30  mini-course 3 breakfast 9h-10h  mini-course 1 10h15-11h15  mini-course 2 11h30-12h30  mini-course 3 breakfast 9h-10h  mini-course 1 10h15-11h15  mini-course 2 11h30-12h30  mini-course 3 breakfast 9h-10h  mini-course 1 10h15-11h15  mini-course 2 11h30-12h30  mini-course 3 lunch lunch lunch lunch 13h30-14h30 welcome 14h30-15h30  mini-course 1 15h45-16h45  mini-course 2 17h00-18h00  mini-course 3  dinner time for discussion / enjoying the mountain 17h30-18h20  talk - A. Perepechko 18h30-19h20  talk - M. Robayo  dinner time for discussion / enjoying the mountain 17h30-18h20  talk - H. Kraft 18h30-19h20  talk - S. Maubach  dinner time for discussion / enjoying the mountain 17h30-18h20  talk - K. Kuyumzhiyan 18h30-19h20  talk - J. Blanc  dinner

Mini-courses - titles and abstracts
 Frédéric MANGOLTE - Real algebraic geometry This series of lectures is dedicated to a general audience with basic knowledge in algebraic geometry. We will explore the topology of real algebraic manifolds by the study of fundamental examples. Our goal is to state and give sketch of proofs of some old and new results. Nash's Theorem, topological manifolds in dimension 2 and 3. Real surgeries by blowups in dimension 2 and 3, rational and nearly rational algebraic varieties defined over R, Nash's conjecture. Comessatti's Theorem on rational surfaces, Kollar's Theorem on uniruled threefolds. Regular morphisms in algebraic real geometry vs. real algebraic geometry (sic). Algebraic varieties with large automorphism groups. Lucy MOSER-JAUSLIN, Pierre-Marie POLONI - Locally nilpotent derivations and extensions of automorphisms We will give an introduction to the study of locally nilpotent derivations and automorphism groups of certain hypersurfaces of affine complex space. Locally nilpotent derivations on the coordinate ring of an affine variety correspond to actions of the additive group (C,+). We will describe techniques of how to use these actions to determine properties of automorphism groups and isomorphism classes of Danielewski hypersurfaces and Koras-Russell threefolds. We will also investigate derivations with zero divergence and how to use them to construct polynomial automorphisms of affine space. This process involves finding automorphisms of formal power series, truncating, and then, lifting them to automorphisms of a polynomial ring. Finally, we will develop some techniques based on the isomorphisms of hypersurfaces to determine when automorphisms extend to the ambient space. Adrien DUBOULOZ, Jean-Philippe FURTER, Immanuel STAMPFLI - Ind-varieties and Ind-schemes We will first introduce the concepts of ind-variety and ind-groups following Shafarevich (1960): these are defined from a point set theoretical viewpoint as limits of increasing chains $X_1 \subseteq X_2 \subseteq X_3 \subseteq \ldots$ of varieties $X_n$, each one closed in the next. Many examples illustrating the difference between these ind-varieties and the ordinary ones will be given, two prominent ones ones on which we will focus during all the course being the group of polynomial automorphisms $Aut(\mathbb{C}^n)$ and the group of one parameter families of linear automorphisms $GL_2(\mathbb{C}[t])$. As a second step, we will introduce and discuss the notion of pro-affine algebra after Kambayashi (1996) and explain in which sense these topological algebras give the appropriate algebraic local counter-part of ind-varieties. Then the third lecture will be devoted to an overview of a general formalism which unifies these two notions. We will re-interpret the previous examples in this framework. The last two lectures will focus on more concrete questions about the group $Aut(\mathbb{C}^n)$ of polynomial automorphisms of $\mathbb{C}^n$ for $n=2$ and $3$. For $n= 2$, the length of an automorphism is defined as the minimum number of triangular automorphisms we need to express it as a composition of triangular and affine automorphisms. Then, we show that the length function is lower semicontinuous on $Aut(\mathbb{C}^2)$. For $n=3$, let $T_z$ be the subgroup of $Aut(\mathbb{C}^3)$ whose elements are the tame automorphisms fixing the last coordinate. Then, we show that $T_z$ is closed in $Aut(\mathbb{C}^3)$. We also look at the group $GL_2(R)$, where $R= \mathbb{C}[X_1, \ldots,X_N]$. By definition, the subgroup $GE_2 (R)$ is the one generated by elementary matrices. If $n>1$, it is a classical result from Cohn, that $GE_2(R)$ is a strict subgroup of $GL_2(R)$ and we show that $GE_2(R)$ is closed in $GL_2(R)$.

Talks - titles and abstracts
 Jérémy BLANC - What are the automorphisms of the plane? This talk is a recreative talk on a subject close to the topics of the three mini-courses of the week. There will be more questions that results. The group of polynomial automorphisms of the affine plane has a well-known structure of an amalgamated product. In particular, it is generated by affine and de Jonquières transformations, which are easy to understand. Working over the field of real numbers, we can say that a morphism is a rational map defined over all real points, and look for automorphisms of the affine real plane (or the Euclidean plane). I will show some natural automorphisms, and explain why they DO NOT generate the group. There is up to now no natural set of generators for the group. Hanspeter KRAFT - Conjugacy Classes in the Automorphism Group of affine n-Space We will discuss conjugacy classes in the group Aut(A^n) of automorphisms of affine n-space A^n. A lot is known in case n=2 due to the amalgamated product structure of the group, and almost nothing for n>2. Using basic properties of families of automorphisms we can give short proofs of several known results, get some new ones, partially extend these in different directions and apply them to the linearization problem. It will also become clear what obstructions we do have to overcome in higher dimension. Karine KUYUMZHIYAN - Varieties with infinitely transitive action of the group of Special Automorphisms Let X be an affine algebraic variety, and let Aut(X) be the group of its algebraic automorphisms. We say that the action of Aut(X) on X is infinitely transitive if for every integer m this action is transitive on m-tuples of pairwise distinct smooth points of the variety. The class of such varieties X is rather poor. The simplest example of such X is the affine space A^n for n>1. Since it is not easy to work with Aut(X), in our proofs we use only the so-called special automorphism group, i.e. the group of automorphisms which can be described in terms of locally nilpotent derivations of the algebra of functions k[X]. In the talk, we will discuss different examples of varieties with this property, constructed in the joint work with Arzhantsev and Zaidenberg. We show infinite-transitivity for non-generate affine toric varieties of dimension > 1, normal affine cones over flag varieties G/P and the so-called suspensions over varieties, already having this property. As it was shown in the joint work with F. Mangolte, the last series of examples works also over the ground field R. A recent result of Arzhantsev, Flenner, Kaliman, Kutzschebauch and Zaidenberg shows that every variety with the infinitely-transitive action of the group of special automorphisms is unirational. If time permits, we will discuss the relation between infinite-transitive varieties and unirational varieties. Stefan MAUBACH - Recent results on polynomial maps over finite fields. In this talk I will discuss some of the recent results on polynomial maps over finite fields. Alexander PEREPECHKO - Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5. We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. Maria Fernanda ROBAYO - On birational diffeomorphisms of the sphere We present the first steps in the classification of the conjugacy classes of elements of finite order of the group of birational diffeomorphisms of the sphere S, the smooth real projective surface S=\{ (w:x:y:z)\in P^3_R | w^2=x^2+y^2+z^2 \}.

How to come
The journey to Enney is 2 hours from Geneva, 2h30 from Basel/Zurich, 1h30 from Lausanne.
See timetables on www.cff.ch.
The train station of Enney is at 10 minutes by foot from the center. See the map, with the blue path given by google (pay attention to the snow!)

Participants
Bachar Al Hajjar (Dijon)
Jérémy Blanc (Basel)
Jung-Kyu Canci (Basel)
Emilie Dufresne (Basel)
Jean-Philippe Furter (La Rochelle)
Isaac Heden (Uppsala)
Nikita Kalinin (Geneva)
Hanspeter Kraft (Basel)
Karine Kuyumzhiyan (Moscow)
Kevin Langlois (Grenoble)
Stefan Maubach (Bremen)
Frédéric Mangolte (Angers)
Lucy Moser-Jauslin (Dijon)
Shameek Paul (Dijon)
Charlie Petitjean (Dijon)
Alexander Perepechko (Grenoble)
Pierre-Marie Poloni (Basel)
Alexandre Ramos-Peon (Bern)
Maria Fernanda Robayo (Basel)
Immanuel Stampfli (Basel)
Anne Christina Wald (Bochum)
Tommy Wuxing Cai (Basel)
Susanna Zimmermann (Basel)

Organisers