
SwissFrench workshop on algebraic geometry
Enney (near Gruyères, Fribourg, Switzerland), February 2024, 2012
The workshop was held in Enney from February 20 to 24, 2012.
Minicourses
In the morning, there were three minicourses of 5 hours (3 times one hour each day).
Frédéric MANGOLTE (Angers) 
Course on real algebraic geometry



Lucy MOSERJAUSLIN (Dijon)
PierreMarie POLONI (Basel) 
Course on locally nilpotent derivations and extensions of automorphisms



Adrien DUBOULOZ (Dijon)
JeanPhilippe FURTER (La Rochelle)
Immanuel STAMPFLI (Basel) 
Course on indvarieties (algebraic varieties of infinite dimension) 
In the afternoon, we had research talks of 50 minutes.
Schedule Talks: In Enney, ( "Centre l'Ondine, VivaGruyère" )
Monday
February 20

Tuesday
February 21

Wednesday
February 22 
Thursday
February 23 
Friday
February 24 

breakfast
9h10h minicourse 1
10h1511h15 minicourse 2
11h3012h30 minicourse 3

breakfast
9h10h minicourse 1
10h1511h15 minicourse 2
11h3012h30 minicourse 3

breakfast
9h10h minicourse 1
10h1511h15 minicourse 2
11h3012h30 minicourse 3

breakfast
9h10h minicourse 1
10h1511h15 minicourse 2
11h3012h30 minicourse 3


lunch 
lunch 
lunch 
lunch 
13h3014h30 welcome
14h3015h30 minicourse 1
15h4516h45 minicourse 2
17h0018h00 minicourse 3
dinner

time for discussion / enjoying the mountain
17h3018h20 talk  A. Perepechko
18h3019h20 talk  M. Robayo
dinner

time for discussion / enjoying the mountain
17h3018h20 talk  H. Kraft
18h3019h20 talk  S. Maubach
dinner

time for discussion / enjoying the mountain
17h3018h20 talk  K. Kuyumzhiyan
18h3019h20 talk  J. Blanc
dinner


Minicourses  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 MOSERJAUSLIN, PierreMarie 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 KorasRussell 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, JeanPhilippe FURTER, Immanuel STAMPFLI  Indvarieties and Indschemes 

We will first introduce the concepts of indvariety and indgroups 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 indvarieties 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 proaffine algebra after Kambayashi (1996) and explain in which sense these topological algebras give the appropriate algebraic local counterpart of indvarieties. Then the third lecture will be devoted to an overview of a general formalism which unifies these two notions. We will reinterpret 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 minicourses of the week. There will be more questions that results.
The group of polynomial automorphisms of the affine plane has a wellknown 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 nSpace 

We will discuss conjugacy classes in the group Aut(A^n) of automorphisms of affine nspace 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 mtuples 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 socalled 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 infinitetransitivity for nongenerate affine toric varieties of dimension > 1, normal affine cones over flag varieties G/P and the socalled 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 infinitelytransitive action of the group of special automorphisms is unirational. If time permits, we will discuss the relation between infinitetransitive 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)
JungKyu Canci (Basel)
Adrien Dubouloz (Dijon)
Emilie Dufresne (Basel)
JeanPhilippe 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 MoserJauslin (Dijon)
Shameek Paul (Dijon)
Charlie Petitjean (Dijon)
Alexander Perepechko (Grenoble)
PierreMarie Poloni (Basel)
Alexandre RamosPeon (Bern)
Maria Fernanda Robayo (Basel)
Immanuel Stampfli (Basel)
Anne Christina Wald (Bochum)
Tommy Wuxing Cai (Basel)
Susanna Zimmermann (Basel)
Organisers
Adrien Dubouloz (Dijon)
Jérémy Blanc (Basel)
