9th swiss-french workshop in Algebraic Geometry

13 - 17 January 2020, in Charmey
(near Gruyères, Fribourg, Switzerland)

Mini-courses

Cinzia Casagrande
(University of Torino)
Fano manifolds and birational geometry
  
Pierre Le Boudec
(University of Basel)
The Hasse principle and random Fano hypersurfaces
  
Bernd Sturmfels
(University of California at Berkeley)
Applications of Algebraic Geometry
  

Schedule

Monday  
January 13
Tuesday  
January 14
Wednesday  
January 15
Thursday  
January 16
Friday  
January 17
 






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 
  tba

18h30-19h20 
  tba

 dinner

time for discussion / enjoying the mountain side

  


 19h dinner

20-20h50 
  tba

21h10-22h 
  tba

time for discussion / enjoying the mountain side

  


17h20-18h10 
  tba

18h30-19h20 
  tba

 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

Cinzia Casagrande - Fano manifolds and birational geometry
We will illustrate some of the techniques in birational geometry and the Minimal Model Program in the framework of Mori dream spaces, and their applications to the study of (smooth, complex) Fano manifolds, with a particular focus to dimension 4. A tentative schedule:
• Mori dream spaces and birational geometry;examples
• Fano varieties and their properties as Mori dream spaces
• the Lefschetz defect of Fano varieties, properties and study via birational geometry
• geometry of Fano 4-folds with large second Betti number.

 
Pierre Le Boudec - The Hasse principle and random Fano hypersurfaces
A projective variety defined over a number field is said to fail the Hasse principle if it has points everywhere locally but no global point. Determining which classes of varieties satisfy the Hasse principle is a central topic in number theory, in particular because checking if a variety has points everywhere locally can be done in a finite number of steps. We will start by reviewing classical results in this area, starting with the celebrated Hasse-Minkowski theorem. Then, the main goal of the lectures will be to investigate the following question: in the family of all projective hypersurfaces of fixed degree and dimension and defined over the field of rational numbers, what is the probability for a hypersurface to satisfy the Hasse principle? Poonen and Voloch have conjectured that for Fano hypersurfaces this probability is equal to 1, and I shall report on recent work (joint with Tim Browning and Will Sawin) which comes close to establishing this conjecture.

 
Bernd Sturmfels - Applications of Algebraic Geometry
This mini-course consists of five independent lectures that offer a panorama of current themes in applied algebraic geometry. The topics to be discussed are 3264 conics in a second, sextics in the real plane, nonegative polynomials versus sums of squares, Gaussian mixtures, and signature tensors. The presentations will be aimed at non-experts and illustrated with many pictures.
• Monday: 3264 Conics in a Second (link to paper)
• Tuesday: Sixty-four Curves of Degree Six (link to paper)
• Wednesday: Nonnegative Polynomials versus Sums of Squares (link to paper)
• Thursday: Gaussian Mixtures and their Tensors (link to paper)
• Friday: Varieties of Signature Tensors (link to paper)

 

Research talks

Arthur Bik - Semi-algebraic properties of Minkowski sums of a twisted cubic segement
Consider the following problem: given the first k ≤ n moments tr(A), tr(A^2), ... , tr(A^k) of a real orthogonal (2n+1)x(2n+1) matrix A, determine the sets of possible eigenvalues of A. This problem was encountered by Michael Rubinstein and Peter Sarnak when trying to compute zeros of L-functions. In order to tackle this problem, they replaced the eigenvalues of A by their real parts. This transforms the problem into computing all possible -1 ≤ t_1, ... ,t_n ≤ 1 given their first k power sums. This can be done recursively, provided that we have membership tests for the Minkowski sums of set of vectors (t, t^2 ... ,t^k) with -1 ≤ t ≤ 1. In this talk, I will describe how to get such membership tests for k=3 using semi-algebraic descriptions of the Minkowski sums. This is joint work with Adam Czapliński and Markus Wageringel.

 
Myrto Mavraki - tba
tba
 
Türkü Özlüm - Algebraic Computations of Theta Constants
David Mumford showed that a principally polarized abelian variety can be written as an intersection of quadrics in a projective space. The coefficients of these quadrics are determined by certain constants, called theta constants, which are the values of transcendental functions, namely theta functions, at zero. We will present an algebraic way to compute the constants associated with a non-hyperelliptic curve. The method is implemented in the mathematical software package Magma. We will finalize the talk with a demonstration of the implementation.

 
Tomasz Pelka - Some "new" affine surfaces of Kodaira dimension zero
Smooth complex affine surfaces which are not of log general type are considered understood, mostly by means of powerful theory of almost MMP developed by Miyanishi, Fujita and others. Among these surfaces, the ones with Kodaira dimension zero are rather peculiar (just like Calabi-Yau varieties in the projective world). Those whose coordinate ring is factorial and has trivial units were classified by Gurjar and Miyanishi ('88). However, recently Freudenburg, Kojima and Nagamine discovered a series of new examples not contained in that list. In my talk, I will explain how to fix that classification: it is a simple adjustment, which I will use as an excuse to provide a gentle introduction into the theory of affine surfaces. The surfaces from the corrected list turn out to be very interesting from the point of view of complex geometry. For example, although their algebraic automorphisms group is usually trivial, they admit a lot of nice holomrphic automorphisms. More precisely, some of them were shown to satisfy certain kinds of algebraic density property. In this setting, they are very similar to the torus (C*)^2, whose holomorphic automorphisms are still far from being understood.

 
Harry Schmidt - tba
tba

 
tba - tba
tba

 

Participants

Angelo Bianchi (Sao Paulo)
Arthur Bik (Bern)
Rémi Bignalet (Dijon)
Cinzia Bisi (Ferrara)
Jérémy Blanc (Basel)
Anna Bot (ETH)
Jung Kyu Canci (Lucerne)
Cinzia Casagrande (Torino)
Mattia Cavicchi (Dijon)
Benoit Dejoncheere (Alberta)
Gabriel Dill (Basel)
Adrien Dubouloz (Dijon)
Marta Dujella (Basel)
Daniele Faenzi (Dijon)
Andrea Fanelli (Bordeaux)
Enrica Floris (Poitiers)
Pascal Fong (Basel)
Jean-Philippe Furter (Bordeaux)
Pierre-Alexandre Gillard (Dijon)
Richard Griffon (Basel)
Douglas Guimaraes (Dijon)
Isac Hedén (Warwick)
Philipp Habegger (Basel)
Lucy Jauslin-Moser (Dijon)
Pierre Le Boudec (Basel)
Anne Lonjou (Basel)
Orlando Marigliano (Leipzig)
Myrto Mavraki (Basel)
Jan Nagel (Dijon)
Türkü Özlüm (Leipzig)
Erik Paemurru (Loughborough)
Maxime Pelletier (Nice)
Joachim Petit (Basel)
Tomasz Pelka (Bern)
Pierre-Marie Poloni (Basel)
Joan Pons (Torino)
Quentin Posva (EPFL)
Harry Schmidt (Basel)
Julia Schneider (Basel)
Ursina Schweizer (EPFL)
Bernd Sturmfels (Berkeley)
Ronan Terpereau (Dijon)
Immanuel van Santen (Basel)
Francesco Veneziano (Genova)
Christian Urech (Lausanne)
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 Burgundy)

The swiss-french workshops in Algebraic Geometry

Here are the previous ones:

1st, 2nd , 3rd, 4th, 5th, 6th, 7th, 8th swiss-french workshop in Algebraic Geometry

Financial support

We gratefully acknowledge support from:
ANR FIBALGA
Institut de Mathématiques de Bourgogne
Swiss Academy of Sciences
Swiss doctoral program (cuso)
Swiss mathematical society
University of Basel
Université d'Angers