13th edition: Charmey January 29 - February 2, 2024
Mini-courses
-
Mock rational points in Chabauty--Kim loci
Jennifer Balakrishnan (Boston)
Abstract
We will explore the computation of rational points on curves
via the p-adic Chabauty--Kim sets, starting with the classical
Chabauty--Coleman set. We will first describe how to compute these
sets and study the mock rational points (those finitely many points
that are not rational) that may arise in these sets in view of a
conjecture of Kim. We will also discuss conditions under which one
can find algebraic mock rational points. Throughout, we will present a
number of examples in the case of curves of punctured elliptic curves,
as well as curves of genus 2 and 3. (This minicourse is based on work
of Bianchi, as well as ongoing joint work with Betts, joint work with
Bianchi and Dogra, and joint work with Bianchi, Ciperiani,
Cantoral-Farfan, and Etropolski.)
-
Uniformization of complex algebraic varieties
Yohan Brunebarbe (Bordeaux)
Abstract
In an attempt to understand which complex analytic spaces can be realised as the universal covering of a complex algebraic variety, Shafarevich asked whether the universal covering $\tilde{X}$ of any smooth projective variety $X$ is holomorphically convex. In other words, does there exist a proper holomorphic map from $\tilde{X}$ to a Stein analytic space? The goal of this series of lectures will be to revisit (some part of) the work of Eyssidieux-Katzarkov-Pantev-Ramachandran, that answers positively
Shafarevich question for smooth projective varieties whose fundamental group admits a faithful complex linear representation. The main tools are complex non-abelian Hodge theory and its non-archimedean counterpart, that I will introduce. Time permitting, I will present some recent developments dealing with universal coverings of smooth quasi-projective varieties.
-
Tropical geometry in moduli theory
Francesca Carocci (Geneva)
Abstract
Many introductions to Logarithmic and Tropical geometry begin with
the following statement "Logarithmic geometry and its combinatorial counterpart,
tropical geometry, were developed to deal with two fundamental and related problems
in algebraic geometry: compactifications and degenerations”.
In particular, tropical geometry, comes equipped with a convenient
“toric toolkit” which has been proved extremely useful in studying
several questions regarding the geometry of moduli spaces and their
invariants.
We will introduce some of the basic notions of logarithmic and tropical
geometry with a bias towards applications to moduli theory, such as:
construction of modular compactification of moduli of curves and maps,
computations of Gromov-Witten invariants, tropical Brill-Noether theory
and Kodaira dimension of $\bar{M}_{g}$.
Talks
-
Field extensions and the Manin-Mumford conjecture
Tobias Bisang (Basel)
Abstract
Algebraic number theory is a mathematical subfield that uses algebraic geometry as an important tool, for instance in solving Diophantine equations. This talk presents the Manin-Mumford conjecture (proven by Raynaud in 1983).
The simplest case, Manin-Mumford states as follows :
Let F be an irreducible polynomial in two variables with coefficients in the rationals or any other number field. Then the zero locus V(F) contains only finitely many points with all coordinates being roots of unity, unless V(F) has the special shape of a torsion coset.
In this talk, some tools of algebraic number theory are presented and applied to prove the Manin-Mumford conjecture in easy cases.
-
Automorphism groups of del Pezzo surfaces of degree $5$
Aurore Boitrel (Orsay)
Abstract
Del Pezzo surfaces and their automorphism groups play a key role in the study of algebraic subgroups of the Cremona group of the plane. Over an algebraically closed field, it is classically known that a del Pezzo surface is either isomorphic to $\mathbb{P}^{1} \times \mathbb{P}^{1}$ or to the blow-up of $\mathbb{P}^{2}$ in at most $8$ points in general position, and in this case, automorphism groups of del Pezzo surfaces of any degree are known. In particular, there is only one isomorphism class of del Pezzo surfaces of degree $5$ over an algebraically closed field. In this talk, we will focus on del Pezzo surfaces of degree $5$ defined over a perfect field. In this case, there are many more extra surfaces (as one can see already if the ground field is the field of real numbers), and the classification as well as the description of the automorphism groups of those surfaces over a perfect field $\mathbf{k}$ is reduced to understanding the possible actions of the Galois group $\operatorname{Gal}(\overline{\mathbf{k}}/\mathbf{k})$ on the graph of $(-1)$-curves.
-
Infinitesimal rational actions
Bianca Gouthier (Bordeaux)
Abstract
For any $k$-group scheme of finite type $G$, if there exists a
generically free rational $G$-action on a $k$-variety $X$, then the
dimension of $Lie (G)$ is upper bounded by the dimension of the variety.
This inequality turns out to be also a necessary condition to the
existence of such actions, when $k$ is a perfect field of positive
characteristic and $G$ is infinitesimal commutative trigonalizable. In
this talk, we will overview the motivation for this problem and explain
the result in the case of the $p$-torsion of a supersingular elliptic curve.
-
Classifying parabolic subgroups in positive characteristic
Matilde Maccan (Grenoble/Rennes)
Abstract
We present the classification of all rational projective homogeneous varieties over algebraically closed fields; such a variety can always be written as the quotient of a semisimple algebraic group by a parabolic subgroup (scheme). We introduce the problem and its history, finally getting to a statement describing all parabolics in a uniform way, independent of the Dynkin type of the group and of the characteristic of the base field. For this, we focus on the most exotic cases, namely on characteristics two and three. We then move on to illustrate a few geometric consequences.
-
$G$-solid rational surfaces
Antoine Pinardin (Edinburgh)
Abstract
A rational surface is a surface S such that there exists a birational map between S and the projective plane. Given a rational surface S and a finite subgroup G of Aut(S), we are interested in determining whether or not there exists a G-equivariant birational map between S and a G-conic bundle. If not, we say that S is G-solid. The Minimal Model Program for surfaces implies that it is enough to consider the case where S is a smooth Del Pezzo surface. After introducing this formalism, we will present the full classification of pairs (G,S) such that the surface S is G-solid. This classification is motivated by the long lasting problem of classifying the conjugacy classes of finite subgroups of the group of birational self maps of the projective space in dimension 2 and 3.
-
Rational angles in plane lattices
Francesco Veneziano (Genova)
Abstract
Generalising classical questions about regular polygons with vertices on a plane lattice,
we are interested in pairs of points $A,B$ on a lattice such that the angle $\widehat{AOB}$
is a rational multiple of $\pi$. This problem leads to diophantine-trigonometric equations
that in turn involve the study of rational points on curves of genus $0,1,2,3,5$.
I will present the full classification of plane lattices according to how many independent rational
angles they contain and in which configurations they appear.
This is a joint work with R. Dvornicich, D. Lombardo and U. Zannier
Participants
- Ahmed Abouelsaad (Basel)
- Eduardo Alves da Silva (Orsay)
- Jennifer Balakrishnan (Boston)
- Jefferson Baudin (Lausanne)
- Marta Benozzo (London)
- François Bernard (Paris)
- Fabio Bernasconi (Basel)
- Tobias Bisang (Basel)
- Jérémy Blanc (Basel)
- Aurore Boitrel (Orsay)
- Antoine Boivin (Angers)
- Anna Bot (Basel)
- Naoufal Bouchareb (Aix-Marseille)
- Alice Bouillet (Lyon)
- Yohan Brunebarbe (Bordeaux)
- Jean-Baptiste Campesato (Angers)
- Alessio Cangini (Basel)
- Francesca Carocci (Genève)
- Victor Chachay (Dijon)
- Tiago Duarte Guerreiro (Orsay)
- Adrien Dubouloz (Poitiers)
- Mani Esna Ashari (Basel)
- Andrea Fanelli (Bordeaux)
- Mattias Ferreira-Filoramo (Paris)
- Enrica Floris (Poitiers)
- Pascal Fong (Orsay)
- Bianca Gouthier (Bordeaux)
- Philipp Habegger (Basel)
- Liana Heuberger (Bath)
- Andrés Jaramillo Puentes (Essen)
- Guillaume Kineider (Aix-Marseille)
- Crislaine Kuster (Dijon/IMPA)
- Matilde Maccan (Grenoble/Rennes)
- Frédéric Mangolte (Marseille)
- Felipe Monteiro (Dijon)
- Lucy Moser-Jauslin (Dijon)
- Erik Paemurru (Saarbrücken)
- Gianni Petrella (University of Luxembourg)
- Anaëlle Pfister (Leipzig)
- Antoine Pinardin (Edinburgh)
- Isabel Rendell (London)
- Linus Rösler (Lausanne)
- Anne Schnattinger (Basel)
- Julia Schneider (Zürich)
- Marc Truter (Warwick)
- Nikolaos Tsakanikas (Lausanne)
- Christian Urech (Zürich)
- Immanuel van Santen (Basel)
- Stefania Vassiliadis (London)
- Francesco Veneziano (Genova)
- Henrik Wehrheim (Basel)
- Egor Yasinsky (Bordeaux)
- Sokratis Zikas (Poitiers)
- Susanna Zimmermann (Orsay)
Organizers
Andrea Fanelli (Bordeaux)
Julia Schneider (Zurich)
Philipp Habegger (Basel)
Susanna Zimmermann (Orsay)
Logistic support: Adrien Dubouloz (Poitiers)
