7  11 January 2019, in Charmey
(near Gruyères, Fribourg, Switzerland)
Olivier Benoist (ENS Paris)

Algebraic cycles on real algebraic varieties 
Dan Loughran (University of Manchester)  Rational points on varieties 
Etienne Mann (University of Angers)

Algebraic stacks 
Monday January 7 
Tuesday January 8 
Wednesday January 9 
Thursday January 10 
Friday January 11 
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  bus at 12h42 
14h3015h30 minicourse 1 16h0017h00 minicourse 2 17h3018h30 minicourse 3 dinner 
time for discussion / enjoying the mountain side 17h2018h10 Frey 18h3019h20 Paemurru dinner 
time for discussion / enjoying the mountain side 17h2018h10 Cifani 18h3019h20 Urech dinner 
time for discussion / enjoying the mountain side 17h2018h10 Durighetto 18h3019h20 Dill dinner 
VIVA GRUYERE Charmey, Rte des Arses 4, 1637 Charmey
The journey to Charmey is 2h10 from Geneva, 2h30 from Basel/Zürich, 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.
 
The real locus of a smooth projective real algebraic variety is a differentiable manifold. How much of its topology can be seen by algebraic geometers? For instance, are all differentiable submanifolds close to the real locus of an algebraic subvariety? Homologous to the real locus of an algebraic subvariety? We will explain classical results, progress in collaboration with Olivier Wittenberg, and open questions.
 
 
The topic of rational points on varieties is a venerable one, and is the modern way to study solutions to Diophantine equations. The fundamental problems are to determine whether a given variety has a rational point, and if so, "how many" rational points it has. We shall begin with the study of rational points on simple classes of varieties over the field of rational numbers, and introduce tools such as heights, localglobal principles, and techniques from analytic number theory. With these tools in hand, towards the end of the course we shall study the problem of existence of rational points in families of varieties.
 
 
In these lectures, we will first explain the motivation of stacks, then explain the notion fibered categories in groupoids. After, we will give the definition of stacks and algebraic stacks. We will give a lot of easy examples where we can compute everything. At the end, we will define the orbifold cohomology and may be we will explain the root construction and how to define a toric stack. 
 
Let X be a irreducible, reduced, non developable, projective hypersurface over the complex numbers; take a point p not in X and consider the linear projection of X from p, that is a finite map of degree d=deg(X). To this map we can associate the monodromy group, that is a transitive subgroup of the symmetric group S_d.
It is known that the general point is uniform, i.e. the monodromy group associated to the projection from a general point is S_d. We proved that X admits at most a finite number of non uniform points (this is a j.w.w A.Cuzzucoli and R.Moschetti).
 
 
In the spirit of the MordellLang conjecture, we consider the intersection of a closed irreducible algebraic subvariety $V$ of a nonisotrivial family of abelian varieties over a base curve with the images of a finiterank subgroup $\Gamma$ of a fixed abelian variety $A_0$ under all isogenies between $A_0$ and some member of the family, where everything is defined over the field of algebraic numbers. After excluding certain degenerate cases, the AndréPinkZannier conjecture predicts that this intersection is not Zariski dense in $V$. Known results in this direction have so far been confined to the cases where $V$ is contained in a fiber of the family, $V$ is a curve or $\Gamma$ is a torsion group. We could prove the statement for arbitrary $\Gamma$ and arbitrary $V$ under certain technical restrictions on $A_0$ and the family of abelian varieties. They are satisfied for example for any fibered power of the Legendre family of elliptic curves if $A_0$ is equal to a power of an elliptic curve without complex multiplication. In the proof, a height bound is crucial that is obtained via a generalization of Rémond's generalized Vojta inequality.
 
 
The plane Cremona group Cr2 is the group of birational automorphisms of P^2.
From the half of the XIXth century the plane Cremona group has been studied
under many aspects. We say that a curve is contractible if it can be contracted to
a finite set of points. Starting from Castelnuovo and Enriques untill Calabri and
Ciliberto, the problem has been investigated and related to adjoint linear systems.
In this talk I will present the state of the art and give some general results about
the contractibility of a curve. Then we will focus on the study of contractible configurations of
lines.
 
 
In this talk I will give a short introduction into Igusa invariants of genus 2 hyperelliptic curves. As with the genus 1 curves (elliptic curves) there is a modular view of the world and an algebraic one. We will learn about important results on the denominators of the Igusa invariants and I will state conjecture which is a work in progress and a generalization of a result of BiluHabeggerKühne. I will talk about several approaches to prove the conjecture and will be happy to talk about ideas of the audience after the talk.
 
 
A sextic double solid is a double cover of ℙ^3 branched along a sextic surface. Terminal threefold hypersurface singularities are compound du Val singularities cA_n, cD_n and cE_n, that is, where the general hyperplane section is an A_n, D_n or E_n singularity. My aim is to construct birational models for sextic double solids with a cA_n singularity for n ≥ 4. I will introduce birational rigidity, analytic singularities, and discuss progress so far.
 
 
I will talk about some joint work in progress with Alexander Duncan, in which we look at the representation dimension of finite subgroups of Cremona groups. More precisely, for a given integer $n$ and a field $k$, we look at the question whether there exists an integer $m$ such that all finite subgroups of the Cremona group in $n$ variables over the field $k$ can be embedded into the general linear group $GL_m(k)$. I will explain when such an integer exists and give the precise values for $n=2$.

Olivier Benoist (Paris)
Cinzia Bisi (Ferrara)
Anna Bot (ETH)
Jérémy Blanc (Basel)
Alberto Calabri (Ferrara)
Jung Kyu Canci (Basel)
Maria Gioia Cifani (Pavia)
Benoît Dejoncheere (Lyon)
Gabriel Dill (Basel)
Adrien Dubouloz (Dijon)
Sara Durighetto (Ferrara)
Daniele Faenzi (Dijon)
Andrea Fanelli (Versailles)
Linda Frey (Copenhagen)
Pascal Fong (Basel)
Richard Griffon (Basel)
Philipp Habegger (Basel)
Isac Hedén (Warwick)
Lucy JauslinMoser (Dijon)
Stéphane Lamy (Toulouse)
Pierre Le Boudec (Basel)
Anne Lonjou (Basel)
Dan Loughran (Manchester)
Frédéric Mangolte (Angers)
Etienne Mann (Angers)
Giao Nguyen (Ferrara)
Erik Paemurru (Loughborough)
Joachim Petit (Basel)
Quentin Posva (EPFL)
Sarah Scherotzke (Münster)
Julia Schneider (Basel)
Ursina Schweizer (EPFS)
Samuel Streeter (Manchester)
Ronan Terpereau (Dijon)
Ettore Turatti (Dijon)
Christian Urech (Imperial College)
Immanuel van Santen (Basel)
Christian Urech (London)
Francesco Veneziano (Pisa)
Egor Yasinsky (Basel)
Sokratis Zikas (Basel)
Susanna Zimmermann (Angers)
The registration is closed.
Philipp Habegger (University of Basel)
Ronan Terpereau (University of Burgundy)
Susanna Zimmermann (University of Angers)
Logistic support: Adrien Dubouloz (University of Dijon)
Here are the previous ones:
1st, 2nd , 3rd, 4th, 5th, 6th, 7th swissfrench workshop in Algebraic Geometry
We gratefully acknowledge support from:
ANR FIBALGA
Institut de Mathématiques de Bourgogne
GDR Singularités (CNRS)
ISITEBFC Project Motivic Invariants of Algebraic Varieties
LAREMA
PEPS (CNRS)
Swiss Academy of Sciences
Swiss doctoral program (cuso)
Swiss mathematical society
University of Angers
University of Basel
University of Rennes (CNRS)