8th swiss-french workshop in Algebraic Geometry

7 - 11 January 2019, in Charmey
(near Gruyères, Fribourg, Switzerland)

Mini-courses

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
  

Schedule

Monday  
January 7
Tuesday  
January 8
Wednesday  
January 9
Thursday  
January 10
Friday  
January 11
 






12h30 welcome
 breakfast

8h45-9h45 
 mini-course 1

10h15-11h15 
 mini-course 2

11h45-12h45 
 mini-course 3

 breakfast

8h45-9h45 
 mini-course 1

10h15-11h15 
 mini-course 2

11h45-12h45 
 mini-course 3

 breakfast

8h45-9h45 
 mini-course 1

10h15-11h15 
 mini-course 2

11h45-12h45 
 mini-course 3

 breakfast

8h45-9h45 
 mini-course 1

10h-11h 
 mini-course 2

11h15-12h15 
 mini-course 3

 lunch  lunch  lunch  lunch  bus at 12h42


14h30-15h30 
 mini-course 1

16h00-17h00 
 mini-course 2

17h30-18h30 
 mini-course 3

 dinner

time for discussion / enjoying the mountain side

  


17h20-18h10 
 Frey

18h30-19h20 
 Paemurru

 dinner

time for discussion / enjoying the mountain side

  


17h20-18h10 
 Cifani

18h30-19h20 
 Urech

 dinner

time for discussion / enjoying the mountain side

  


17h20-18h10 
 Durighetto

18h30-19h20 
 Dill

 dinner

Location

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.

Titles and abstracts

Mini-courses

Olivier Benoist - Algebraic cycles on real algebraic varieties
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.

 
Dan Loughran - Rational points on varieties (Lecture and Exercises)
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, local-global 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.

 
Etienne Mann - Algebraic stacks
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.
 

Research talks

Maria Gioia Cifani- Monodromy group of projections of hypersurfaces
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).

 
Gabriel Dill - Unlikely Intersections with Isogeny Orbits in Fibered Powers of Elliptic Schemes
In the spirit of the Mordell-Lang conjecture, we consider the intersection of a closed irreducible algebraic subvariety $V$ of a non-isotrivial family of abelian varieties over a base curve with the images of a finite-rank 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 André-Pink-Zannier 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.
 
Sara Durighetto - Cremona contractibility
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 con figurations of lines.

 
Linda Frey - Denominators of Igusa Invariants of Genus 2 Curves with CM Jacobian (work in progress)
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 Bilu-Habegger-Kühne. I will talk about several approaches to prove the conjecture and will be happy to talk about ideas of the audience after the talk.

 
Erik Paemurru - Birational models of terminal sextic double solids
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.

 
Christian Urech - Representation dimension of finite subgroups of Cremona groups
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 $n=2$.

 

Participants

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 Jauslin-Moser (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.

Organisers

Philipp Habegger (University of Basel)
Ronan Terpereau (University of Burgundy)
Susanna Zimmermann (University of Angers)
Logistic support: Adrien Dubouloz (University of Dijon)

The swiss-french workshops in Algebraic Geometry

Here are the previous ones:

1st, 2nd , 3rd, 4th, 5th, 6th, 7th swiss-french workshop in Algebraic Geometry

Financial support

We gratefully acknowledge support from:
ANR FIBALGA
Institut de Mathématiques de Bourgogne
GDR Singularités (CNRS)
ISITE-BFC 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)