
3rd SwissFrench workshop on algebraic geometry
Enney (near Gruyères, Fribourg, Switzerland), January 2731, 2014
The workshop was held in Enney from January 27 to 31, 2014.
Minicourses
In the morning, there were three minicourses of 5 hours (3 times one hour each day).
Michel BRION (Grenoble) 
Local properties of actions of linear algebraic groups



Serge CANTAT (Rennes) 
Dynamical Degrees



Yuri PROKHOROV (Moscow)

Finite subgroups of the space Cremona group and Fano threefolds. 
In the afternoon, we had research talks of 50 minutes.
Schedule Talks: In Enney, ( "Centre l'Ondine, VivaGruyère" )
Monday
January 27

Tuesday
January 28

Wednesday
January 29 
Thursday
January 30 
Friday
January 31 
12h30 welcome

breakfast
8h459h45 minicourse 1
10h1511h15 minicourse 2
11h4512h45 minicourse 3

breakfast
8h459h45 minicourse 1
10h1511h15 minicourse 2
11h4512h45 minicourse 3

breakfast
8h459h45 minicourse 1
10h1511h15 minicourse 2
11h4512h45 minicourse 3

breakfast
8h459h45 minicourse 1
10h1511h15 minicourse 2
11h4512h45 minicourse 3

lunch 
lunch 
lunch 
lunch 

14h3015h30 minicourse 1
16h0017h00 minicourse 2
17h3018h30 minicourse 3
dinner

time for discussion / enjoying the mountain
17h2018h10 talk  S. Vishkautsan
18h3019h20 talk  M. Robayo
dinner

time for discussion / enjoying the mountain
17h2018h10 talk  C. Petitjean
18h3019h20 talk  J. Furter
dinner

time for discussion / enjoying the mountain
17h2018h10 talk  M. Shkolnikov
18h3019h20 talk  S. Lamy
dinner


Minicourses  titles and abstracts
Michel BRION  Local properties of actions of linear algebraic groups 

The minicourse will present some fundamental properties
of actions of linear algebraic groups (especially tori) on algebraic
varieties: linearization of invertible sheaves; existence of invariant
affine neighborhoods; the BialynickiBirula decomposition. The setting
will be algebraic geometry over an arbitrary field.

Serge CANTAT  Dynamical Degrees 

Let V be a complex projective variety and let f be a rational transformation
of V. The graph G(f) is an algebraic subset of VxV ; taking iterates, one gets a sequence
of graphs G(f^n) in VxV. The dynamical degrees of f are nonnegative real numbers
that describe how the sequence G(f^n) grows with n. For instance, they capture
the exponential growth rate of the volume of G(f^n) as n goes to infinity.
I shall explain the definition of these numbers, their dynamical meaning, and some
of the very nice arithmetical properties that they satisfy (at least conjecturally).

Yuri PROKHOROV  Finite subgroups of the space Cremona group and Fano threefolds 

The aim of this course is to introduce a techniques
 based on the minimal model program 
that can be used
to classify finite subgroups of Cremona groups.
I concentrate on the threedimensional case.
Lecture 1.
Introduction. GMMP. FanoMori models. Examples.
Lecture 2.
Gorenstein Fano threefolds. Projective models. NonGorenstein case.
Lecture 3.
Special kinds of Fano threefolds: del Pezzo varieties.
Lecture 4.
MoriMukai classification.
Sarkisov links and Fano threefolds of rank 2.
Some invariants of conjugacy classes.
Lecture 5.
Prime Fano threefolds.
Iskovskikh and Mukai classification. Applications.

Talks  titles and abstracts
JeanPhilippe FURTER  Generalities on the groups SL_{2}( [x_{1},...,x_{n}] ) 

The group SL_{2}() is often considered as a good
prototype of a linear algebraic group. Analogously, the theory of indgroups
should probably begin with groups such as SL_{2}( [x_{1},...,x_{n}] ).
In my lecture, I will address different questions related with these groups,
especially considered as indgroups. In particular, we will have a look
at the essentially different cases n=1 and n ≥2.

Stéphane LAMY  Blowup of smooth curves 

I will blowup some smooth curves in threefolds and decide when the variety obtained is weakFano. Joint work with JB.

Charlie PETITJEAN  A combinatorial description of Tvarieties 

The coordinate ring A of a normal affine variety endowed with an effective action of the torus T=(*)n admits a natural grading indexed by the lattice of characters of T. We will review a method due to Altmann and Hausen to describe the graded pieces of A in terms of pairs (Y,D) consisting of a variety Y and a certain divisor D on Y with "combinatorial" coefficients.

Maria Fernanda ROBAYO  Finite order birational diffeomorphisms of the sphere for the conic bundle case 

Let S be the real algebraic projective surface defined by the equation w2=x2+y2+z2 in 2_{}. The group of birational diffeomorphisms of S is denoted by Diffbir(S) or Aut(S()) and corresponds to those birational transformations f such that f and f 1 are defined at every real point of S.
For g∈Diffbir(S) of finite order it is known that if (X,g') is the minimal resolution of (S,g'), where g'∈Aut(X) is the lift of g, then
(1) rank Pic(X)g'=1 and X is a Del Pezzo surface, or
(2) rank Pic(X)g'=2 and Π: X→ 1 is a conic bundle.
We present some algebraic and geometric results about the conjugacy classes of birational diffeomorphisms of finite order for the case (2).

Mikhail SHKOLNIKOV  Counting curves on surfaces over nonalgebraically closed fields 

Consider a very ample line bundle on a surface over an algebraically closed field. There is a classical enumerative problem of counting curves in the corresponding linear system of given genus and passing through a generic configuration of points. This numbers are usually called the Severi Degrees, in many cases they are known and well understood. One can try to formulate the same problem over an arbitrary field. But the corresponding number of curves will depend on the configuration of points even if it is generic. In the real case Welschinger managed to define a weighted invariant count of rational curves on a surface. Both Severi Degrees and Welschinger invariants for toric surfaces can be computied in a quite similar way via tropical geometry. Surprisingly, it appears that there exist a one parametric family of such refined invariant tropical counts with so called BlochGoettsche multiplicities which in a sense connects real and complex cases. It is expected that they provide answers to some unknown enumerative problems. In my talk I will prose some candidates. 
Soli VISHKAUTSAN  Arithmetic dynamics and residual periodicity of birational automorphisms of cubic surfaces 

We present "residual periodicity", a relatively new concept in arithmetic dynamics, as defined by Bandman, Grunewald and Kunyavskii. A rational selfmap of a quasiprojective variety defined over a number field is strongly residually periodic if its minimal periods are bounded modulo almost every prime (where the bound is uniform, i.e. not depending on the prime). We discuss some interesting examples, and present results about residual periodicity of birational automorphisms of cubic surfaces.

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
Ivan Bazhov (Geneva)
Mohammed Benzerga (Angers)
Cinzia Bisi (Ferrara)
Michel Brion (Grenoble)
Jérémy Blanc (Basel)
Serge Cantat (Rennes)
Jung Kyu Canci (Basel)
Adrien Dubouloz (Dijon)
JeanPhilippe Furter (La Rochelle)
Isac Héden (Uppsala)
Johannes Josi (Geneva)
Natalia Kolokolnikova (Geneva)
Nikon Kurnosov (Moscow)
Kevin Langlois (Grenoble)
Stéphane Lamy (Toulouse)
Matthias Leuenberger (Bern)
Lucy MoserJauslin (Dijon)
Frédéric Mangolte (Angers)
Charlie Petitjean (Dijon)
PierreMarie Poloni (Basel)
Yuri Prokhorov (Moscow)
Abdul Rauf (Bremen)
Andriy Regeta (Basel)
Maria Fernanda Robayo (Basel)
Christoph Schiessl (ETHZ)
Junliang Shen (ETHZ)
Mikhail Shkolnikov (Geneva)
Ronan Terpereau (Mainz)
Artur Tomberg (Moscow)
Christian Urech (Basel/Rennes)
Soli Vishkautsan (BenGurion)
Susanna Zimmermann (Basel)
Send an email to Jeremy Blanc unibas ch if you would like to participate.
Organisers
Adrien Dubouloz (Dijon)
Jérémy Blanc (Basel)
