
6th SwissFrench workshop on algebraic geometry
Charmey (near Gruyères, Fribourg, Switzerland), January 913, 2017
The workshop will be held in Charmey from January 9 to 13, 2017.
Minicourses
In the morning, there will be three minicourses of 5 hours (3 times one hour each day).
Daniele FAENZI (Dijon)

Vector bundles, construction and classification 


Gaël REMOND (Grenoble) 
Mordell's conjecture and some generalisations 


Ronan TERPEREAU (Dijon)

Structure and linear representations of algebraic groups 


In the afternoon, we will have research talks of 50 minutes.
Schedule Talks: In Charmey, ( "Centre Les Dents Vertes, VivaGruyère" )
Monday
January 9

Tuesday
January 10

Wednesday
January 11 
Thursday
January 12 
Friday
January 13 
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 Elek
18h3019h20 van Santen
dinner

time for discussion / enjoying the mountain
17h2018h10 Dill
18h3019h20 Canci
dinner

time for discussion / enjoying the mountain
17h2018h10 Vargas De León
18h3019h20 Poloni
dinner


Minicourses  titles and abstracts
Daniele FAENZI  Vector bundles, construction and classification 

The course is intended as an introduction to some topics
related to vector bundles on algebraic varieties, mainly complex
projective manifolds.
We will first review some basic notions such as
locally free sheaves, line bundles and the Picard group,
Grassmannians and operations on vector bundles.
Then we will focus shortly on characteristic classes (notably Chern
classes) and related topics.
Next, we will consider the problem of
classification of vector bundles, introduce some splitting results
and say some words on stable bundles.
Finally (and hopefully) we will focus on a special class of bundles,
namely Ulrich bundles, and give a short survey of their properties.

Gaël REMOND  Mordell's conjecture and some generalisations 

Mordell's conjecture (1922) is the statement, proven by Faltings in 1983,
that a curve of genus at least two over a number field has only finitely
many rational points. A second proof by Vojta (1991) has led to several
generalisations, the most famous being the socalled MordellLang theorem
on subvarieties of abelian varieties, also due to Faltings. The aim of
this course is to present these results and discuss the strategy of proofs.
The main tool is the notion of height, which plays a central role in
Diophantine geometry and is particularly useful for finiteness statements.
The lectures will include an introduction to heights at a basic level
allowing to formulate the key height inequality, called (generalised)
Vojta inequality, and see how it implies the result on rational points.

Ronan TERPEREAU  Structure and linear representations of algebraic groups 

In this lecture we will prove the Chevalley's structure theorem: any connected algebraic group over an algebraically closed field is an extension of an abelian variety by a connected linear algebraic group, and these are unique. Moreover, we will see applications of the Chevalley's structure theorem (e.g. any algebraic subgroup of the Cremona group is linear).
Along the way we will review important facts regarding the structure of algebraic groups, algebraic group actions on algebraic varieties, and linear representations of algebraic groups. We will also present examples of algebraic groups that appear naturally in algebraic geometry.

Talks  titles and abstracts
Jung Kyu CANCI  Periodic points for rational functions and integral points of varieties 

I will present a joint work with S. Vishkautsan where we provide an explicit bound on the number of rational periodic points of a rational function of degree at least 2, where everything is defined over a given number field. Our result is obtained by applying some theorems about integral points of certain varieties.

Gabriel DILL  The AndréPinkZannier conjecture for curves 

In the spirit of the MordellLang conjecture, we consider the intersection of a curve in a family of abelian varieties with the images of a finiterank subgroup of a fixed abelian variety A_{0} under all isogenies between A_{0} and some member of the family. After excluding certain degenerate cases, we can prove that this intersection is finite. We describe the strategy of the proof, due to Zannier, and present some of the techniques we used. If time permits, we give a glimpse of our current work on replacing the curve by a subvariety of arbitrary dimension.

Balazs ELEK  KazhdanLusztig atlases on Toric surfaces 

A Bruhat atlas, introduced by He, Knutson and Lu, on a stratified variety is a way of modeling the stratification locally on the stratification of Schubert cells by opposite Schubert varieties. He, Knutson and Lu described Bruhat atlases on many interesting varieties, including partial flag varieties and on wonderful compactifications of groups. We will discuss some results toward a classification of varieties with Bruhat atlases, focusing on the 2dimensional toric case. In this case, the answer may be stated in terms of the moment polygon of the toric surface, which one should first slice up, then put toppings on, much like one would do while preparing a pizza.

PierreMarie POLONI  The Jonquières subgroup is a Borel subgroup (joint work with JeanPhilippe Furter) 

The Jonquières subgroup is the group B_{n} of triangular polynomial automorphisms of the complex affine nspace. By analogy with linear groups, many authors refer to it as the triangular "Borel" subgroup of Aut(𝔸^{n}). In this talk, we will show that the quotation marks could be dropped. Indeed, we will prove that B_{n} is maximal among all (connected) solvable subgroups of Aut(𝔸^{n}), thus a Borel subgroup of Aut(𝔸^{n}), when the latter is viewed as an indgroup.
We will also consider the following question: Are Borel subgroups of Aut(𝔸^{n}) all conjugate?

Immanuel VAN SANTEN  Uniqueness of Embeddings of the Affine Line into Algebraic Groups 

This is joint work with Peter Feller (Max Planck Institute for Mathematics, Bonn).
We consider embeddings X ⟶ Y of affine varieties and
study them up to automorphisms of Y. After recalling some classical results
and examples in case Y is the affine space ℂ^{n}, we focus on the main
result we present in this talk: Let Y be the underlying variety of
a connected algebraic group G. If the character group of G is
trivial and dim(G) ≠3, then all embeddings ℂ ⟶ Y
are the same up to automorphisms of Y. We give
some techniques used in the proof of this result and describe the main strategy.

Alejandro José VARGAS DE LEON  Chipfiring and a RiemannRoch Formula for Graphs 

We present an interaction between algebraic geometry and combinatorics developed by Matthew Baker and Serguei Norine. Certain chipfiring game played on a graph is described, and the idea is to introduce a language inspired by linear divisors. Encoding the distribution of chips as a linear divisor and the firing of chips inducing a linear equivalence suggests several analogies and conjectures, leading up to a RiemannRoch formula for graphs. The exposition also describes algorithms of practical importance for the game, and at the end we state several open questions.

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 close to the bus stop ( map).
Participants
Rémi BignaletCazalet (Dijon)
Arthur Bik (Bern)
Jérémy Blanc (Basel)
Réda Boumasmoud (EPFL)
Alberto Calabri (Ferrara)
Jung Kyu Canci (Basel)
Julie Déserti (Paris)
Gabriel Dill (Basel)
Adrien Dubouloz (Dijon)
Balazs Elek (Cornell)
Daniele Faenzi (Dijon)
Andrea Fanelli (Düsseldorf)
Enrica Floris (Basel)
Linda Frey (Basel)
JeanPhilippe Furter (La Rochelle)
Philipp Habegger (Basel)
Mattias Hemmig (Basel)
Stéphane Lamy (Toulouse)
Anne Lonjou (Toulouse)
Lucy MoserJauslin (Dijon)
Karol Palka (Warsaw)
Tomasz Pełka (Warsaw)
PierreMarie Poloni (Bern)
Gaël Rémond (Grenoble)
Julia Schneider (Basel)
Ronan Terpereau (Dijon)
Christian Urech (Basel/Rennes)
Immanuel van Santen (Hamburg)
Alejandro José Vargas De León (Bern)
Francesco Veneziano (Basel)
Marius Vuille (EPFL)
Susanna Zimmermann (Toulouse)
Send an email to susannamaria zimmermann gmail com if you would like to participate.
Organiser:
Susanna Zimmermann (Toulouse) with the help of
Adrien Dubouloz (Dijon)
Philipp Habegger (Basel)
Jérémy Blanc (Basel)
