
5th SwissFrench workshop on algebraic geometry
Charmey (near Gruyères, Fribourg, Switzerland), January 1822, 2016
The workshop was held in Charmey from January 18 to 22, 2016.
Minicourses
In the morning, there will be three minicourses of 5 hours (3 times one hour each day).
Ivan CHELTSOV (Edinburgh)

Gbirationally rigid rational Fano threefolds 


Robin DE JONG (Leiden) 
Intersection theory on arithmetic surfaces 


Jean FASEL (Grenoble)

Unstable classification of vector bundles over smooth affine schemes 


In the afternoon, we will have research talks of 50 minutes.
Schedule Talks: In Charmey, ( "Centre Les Dents Vertes, VivaGruyère" )
Monday
January 18

Tuesday
January 19

Wednesday
January 20 
Thursday
January 21 
Friday
January 22 
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
10h11h minicourse 2
11h1512h15 minicourse 3

lunch 
lunch 
lunch 
lunch 

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

time for discussion / enjoying the mountain
17h2018h10 L. Frey
18h3019h20 T. Pełka
dinner

time for discussion / enjoying the mountain
17h2018h10 L. Kühne
18h3019h20 N. Kalinin
dinner

time for discussion / enjoying the mountain
16h1017h00 F. Veneziano
17h2018h10 I. Krylov
18h3019h20 R. Terpereau
dinner


Minicourses  titles and abstracts
Ivan CHELTSOV 
Gbirationally rigid rational Fano threefolds 

For a rational Fano threefold X faithfully acted by a finite group G, being Gbirationally rigid means
that X is the unique output of a Gequivariant Minimal Model Program, i.e. X is a GMori fiber space over a point,
and X is not Gbirational to any other GMori fiber space.
We describe existing methods of proving Gbirational rigidity of Fano threefolds,
and apply them to construct many explicit examples of rational Gbirationally rigid Fano threefolds.
As an application we obtain many examples of nonconjugate embeddings of finite groups into the Cremona group of rank 3.

Robin DE JONG  Intersection theory on arithmetic surfaces 

Arithmetic surfaces are certain 2dimensional models of curves defined over a number field. One can develop an intersection theory on arithmetic surfaces in analogy with the classical intersection theory on projective surfaces over a field, involving such results as adjunction formula, RiemannRoch and a Noether formula. Arithmetic intersection numbers involve both nonarchimedean data (coming from the finite primes of the number field) and archimedean data (coming from the embeddings of the number field into the complex numbers). These archimedean data are given by complex analytic Green's functions, which make arithmetic intersection numbers both hard to compute and interesting to study. Recently it has become clear that these Green's functions, when studied in degenerating families of compact Riemann surfaces, themselves display a certain asymptotic behavior whose leading terms can be explained by the nonarchimedean data in arithmetic intersection theory.

Jean FASEL  Unstable classification of vector bundles over smooth affine schemes 

In these lectures, we will explain how to classify vector bundles on smooth affine schemes using the machinery of the A^1homotopy category. We will start with a quick introduction on the A^1homotopy category, and then explain the results of Morel on the classification of vector bundles. We will then explain how to extend these results using Postnikov towers, together with the known computations of the second non trivial A^1homotopy sheaf of the affine punctured space of a given dimension. Time permitting, we wil also explain how to classify symplectic or orthogonal bundles.

Talks  titles and abstracts
Linda FREY  Heights and elliptic curves 

After shortly introducing heights and elliptic curves we will look at the torsion points E_{tor} of an elliptic curve E over and adjoin their coordinates to . What we get is (E_{tor}) which is a Bogomolov field. In a Bogomolov field, the height of a nonzero number which is not a root of unity is bounded from below by a positive constant. We will compute this constant explicitly and give an overview of the proof.

Nikita KALININ 
Tropical curves and sandpile models 

We will discuss sandpile models. They concern moving chips on the vertices of some graph and emerge in many different places: physics, combinatorics, and algebraic geometry. In a particular situation, when we perturb the maximal stable state by adding a few chips, we will see emergence of tropical curves, aka planar graphs with straight edges with rational slopes, and balancing condition at vertices. If time permits, we will discuss the initial motivation for defining such models as well as speculations about the future development.

Lars KÜHNE  A glimpse of anabelian geometry 


Igor KRYLOV  Classification and birational rigidity of del Pezzo fibrations with an action of the Klein simple group 

Study of embeddings of a finite group G into the Cremona group is equivalent to study of Gbirational geometry of rational GMori fiber spaces. A good place to start study finite subgroups is a study of simple subgroup. We prove that any del Pezzo fibration over projective line with an action of the Klein simple group is either a direct product or a certain singular del Pezzo fibration X_{n} of degree 2. It is known that del Pezzo fibrations of degree 2 satisfying the K^2condition are birationally superrigid. I extend this result to singular del Pezzo fibrations and prove that X_{n} are superrigid, in particular not rational, for n>2.

TOMASZ PEŁKA  Planar cuspidal curves with **fibered complements 

To classify complex rational cuspidal curves E⊂2 it remains to classify these whose complements are of log general type, i.e. these for which κ(K_{X}+D)=2, where (X,D) is a log resolution of (2,E). It is conjectured that κ(K_{X}+1/2D)=&infty; and hence 2\E is **fibered, where **=\{0,1}, or (K_{X}+\frac{1}{2}D) is ample on some minimal model of (X,1/2D). A vast majority of known examples turns out to be of the first type. We discuss our recent progress on their classification. This is a joint project with Karol Palka.

Ronan TERPEREAU  Maximal connected subgroups of the Cremona groupk 

This talk is about a work in progress with Jérémy Blanc and Andrea Fanelli. The socalled Cremona group is the group of birational transformations of the ndimensional complex projective space. This group is not an algebraic group for n>1, but we can hope (at least in small dimension) classify its maximal connected algebraic subgroups. In dimension 2, the classification is old and quite easy (F. Enriques, 1893). In dimension 3, the first rigorous treatment was done by H. Umemura in the 1980's in a series of six (quite long and technical) papers. In this talk I will explain how we can hope to recover his results in a much simpler way using the now welldeveloped Mori theory and discuss several possible generalizations.

Francesco VENEZIANO  Rational points on explicit families of curves


I will present a method, of easy application, to compute all the
rational points on a fairly general family of curves in products of elliptic
curves, proving in particular the explicit MordellLang Conjecture for
these curves. We prove some explicit and very sharp estimates for the height
of such rational points. The bounds are so good that we can implement a
computer search. I will present several explicit examples in which this has
been done.
All results are in collaboration with Sara Checcoli and Evelina Viada.

How to come
The address is VIVA GRUYERE Charmey Rte des Arses 4 1637 Charmey
The journey to Charmey is 2h10 from Geneva, 2h30 from Basel/Zurich, 1h30 from Lausanne. See timetables on www.cff.ch, the bus stop is "Charmey (Gruyère), Le Chêne". The place is very near the bus stop. See the map
Participants
Raphaël Achet (Grenoble)
Rémi BignaletCazalet (Dijon)
Cinzia Bisi (Ferrara)
Jérémy Blanc (Basel)
JungKyu Canci (Basel)
Ivan Cheltsov (Edinburgh)
Robin De Jong (Leiden)
Adrien Dubouloz (Dijon)
Andrea Fanelli (Basel)
Jean Fasel (Grenoble)
Linda Frey (Basel)
JeanPhilippe Furter (Basel)
Philipp Habegger (Basel)
Isac Hedén (Kyoto)
Mattias Hemmig (Basel)
Johannes Josi (Genève)
Nikita Kalinin (Genève)
Lars Kühne (Bonn)
Igor Krylov (Edinburgh)
Stéphane Lamy (Toulouse)
Bruno Laurent (Grenoble)
Anne Lonjou (Toulouse)
Lucy MoserJauslin (Dijon)
Jan Nagel (Dijon)
Karol Palka (Warsaw)
Tomasz Pełka (Warsaw)
PierreMarie Poloni (Bern)
Lukas Pottmeyer (Basel)
Maria Fernanda Robayo (Basel)
Andriy Regeta (Basel)
Immanuel Stampfli (Hamburg)
Stefan Schmid (Basel)
Ronan Terpereau (Bonn)
Christian Urech (Basel/Rennes)
Francesco Veneziano (Basel)
Susanna Zimmermann (Basel)
Send an email to Jeremy Blanc unibas ch if you would like to participate.
Organisers
Adrien Dubouloz (Dijon)
Philipp Habegger (Basel)
Jérémy Blanc (Basel)
