14th edition: Charmey January 13 -17, 2025
Tentative Schedule
Monday
January 13 |
Tuesday
January 14 |
Wednesday
January 15 |
Thursday
January 16 |
Friday
January 17 |
|
breakfast |
8:45-9:45
mini-course |
8:45-9:45
mini-course |
8:45-9:45
mini-course |
8:45-9:45
mini-course |
10:15-11:15
mini-course |
10:15-11:15
mini-course |
10:15-11:15
mini-course |
10:00-11:00
mini-course |
12:30
welcome |
11:45-12:45
mini-course |
11:45-12:45
mini-course |
11:45-12:45
mini-course |
11:15-12:15
mini-course |
| lunch |
bus at 12:40 |
14:30-15:30
Shinder |
time for discussion /
enjoying the mountain side |
|
16:00-17:00
Loeffler |
17:20-18:10
Stadlmayr |
17:20-18:20
Kurz |
17:20-18:10
Truter |
17:30-18:30
Xie |
18:30-19:20
Fu |
18:30-19:20
Bernard |
18:30-19:20
Kim |
19h
dinner |
19:30
dinner |
Mini-courses
-
Formalising arithmetic and geometry in Lean: an introduction
David Loeffler (UniDistance Suisse)
Abstract
This mini-course is an introduction to the Lean Theorem Prover, a computer program for formally checking the validity of mathematical proofs. The course will mostly focus on how to use the software in practice (rather than going too deep into the logical foundations). In particular, we'll see a tour of some of the theorems in number theory and algebraic geometry that have already been formalised in the huge "Mathlib" library, and try out formalising a few more lemmas for ourselves.
There will be lots of examples and exercises to try out, so you should bring a computer of your own that you can install Lean on. The software runs fine on Windows, Mac, and most Linux distributions (but is unlikely to work well on phones or tablets). Installing Lean on your computer yourself before arriving in Charmey is not obligatory, but might be a good idea if you want to get the most out of the course; see docs.lean-lang.org/lean4/doc/quickstart.html for instructions.
-
Derived categories of coherent sheaves
Evgeny Shinder (University of Sheffield)
Abstract
We will start with a review of standard homological algebra, such as injective and projective resolutions and Ext-groups. Then we discuss derived functors, mostly focussing on coherent sheaves, and in particular review the coherent sheaf cohomology. Finally we go into the construction of the derived category of coherent sheaves and study its properties.
-
Different cones of divisors: from Mori Dream Spaces to K-trivial varieties
Zhixin Xie (Université de Lorraine)
Abstract
Given a variety, we consider the real vector space spanned by Cartier divisors modulo numerical equivalence. From the point of view of birational geometry, we are interested in the shape of the nef cone and the movable cone in this vector space.
For a Fano variety, i.e. a variety with ample anticanonical class, it follows from the Mori’s cone theorem that the nef cone is rational polyhedral. More generally for a Mori Dream Space, both the nef cone and the movable cone are rational polyhedral. However, these cones can be very wild in general: when we turn to K-trivial varieties, i.e. varieties with numerically trivial canonical class, they behave less well than Fano varieties. For instance, we can construct examples of K-trivial varieties for which the nef or the movable cone is round. Nevertheless, the Kawamata-Morrison cone conjecture predicts that the nef and the movable cones on a K-trivial variety are rational polyhedral up to the action of natural groups acting on them.
In this course, we will introduce basic notions and techniques related to the description of different cones of divisors and we will illustrate them on many examples. In particular, we present tools from the Minimal Model Program and their application to connect different cones. We will introduce the Kawamata-Morrison conjecture and explain some known results. We will also discuss the cones of divisors of varieties with nef anticanonical class, which interpolate between Fano varieties and K-trivial varieties.
Talks
-
Seminormal Toric Varieties
François Bernard (Paris)
Abstract
Toric varieties are traditionally assumed to be normal due to the categorical equivalence between normal toric varieties and fans, which are combinatorial objects. In this talk, I will present a joint work with Antoine Boivin, where we study seminormal toric varieties. The seminormalization of an algebraic variety is a variant of normalization that remains bijective with the original variety. One of its interesting properties is, for example, that it completely characterizes binational maps which extend to the complex points as an homeomorphism for the strong topology. Our goal is to answer the following question: is there a combinatorial object that corresponds to seminormal toric varieties? Building on a theorem by Reid and Roberts, which shows in the affine case that seminormalization is achieved by saturating each face of the associated monoid, and on a construction by Teissier and González Pérez to deal with non-affine and non-normal toric varieties, we establish a categorical equivalence between seminormal toric varieties and certain combinatorial objects that we call "fans with attached groups."
-
On the elliptic curves with many common torsion x-coordinates
Hang Fu (Basel)
Abstract
Let $E_1$ and $E_2$ be elliptic curves over C, together with
double covers $\pi_1: E_1 \to P^1$ and $\pi_2: E_2 \to P^1$ such that $E_j[2]$
is the set of ramification points of $\pi_j$. It is known that when
$\pi_1(E_1[2]) \neq \pi_2(E_2[2])$, then the intersection
$\pi_1(E_1[\infty]) \cap \pi_2(E_2[\infty])$ is finite, where
$\pi_j(E_j[\infty])$ denotes the set of torsion points on $E_j$. In this
talk, we will discuss how large this intersection can be. We will
introduce the background and motivation and also explain how this
question is naturally connected with other questions. This is a joint
work with Michael Stoll.
-
Certain families of K3 surfaces and their modularity
Seoyoung Kim (Basel)
Abstract
We start with a double sextic family of K3 surfaces with four parameters with Picard number $16$. Then by geometric reduction (specializing at fiber) processes, we obtain three, two and one parameter families of K3 surfaces of Picard number $17, 18$ and $19$ respectively. All these families turn out to be of hypergeometric type in the sense that their Picard-Fuchs differential equations are given by hypergeometric or Heun functions. We will study the geometry of 2-parameter families and related finite hypergeometric series in detail. We will then discuss some relevant modularity questions. This is a joint work with A. Clingher, A. Malmendier, and N. Yui.
-
A determinant on birational maps of Severi-Brauer surfaces
Elias Kurz (Neuchâtel)
Abstract
We define a determinant on the automorphisms of non-trivial Severi-Brauer surfaces. Using the generators and relations, we extend this determinant to birational maps between Severi-Brauer surfaces. Using this determinant and a group homomorphism found in [BSY23] we can determine the abelianization of the Cremona group of a non-trivial Severi-Brauer surface. This is the first example of an abelianization of the Cremona group of a geometrically rational surface where the automorphism group is not trivial.
-
Which rational double points occur on del Pezzo surfaces?
Claudia Stadlmayr (Neuchâtel)
Abstract
Canonical surface singularities, also called rational double points (RDPs), can be classified according to their dual resolution graphs, which are Dynkin diagrams of types A, D, and E. Whereas in characteristic different from 2, 3, and 5, rational double points are "taut", that is, they are uniquely determined by their dual resolution graph, this is not necessarily the case in small characteristics. To such non-taut RDPs Artin assigned a coindex distinguishing the ones with the same resolution graph in terms of their deformation theory. In 1934, Du Val determined all configurations of rational double points that can appear on complex RDP del Pezzo surfaces. In order to extend Du Val’s work to positive characteristic, one has to determine the Artin coindices to distinguish the non-taut rational double points that occur. In this talk, I will explain how to answer the question "Which rational double points (and configurations of them) occur on del Pezzo surfaces?" for all RDP del Pezzo surfaces in all characteristics. This will be done by first reducing the problem to degree 1 and then exploiting the connection to (Weierstraß models of) rational (quasi-)elliptic surfaces.
-
Fano 4-fold hypersurfaces
Marc Truter (Warwick)
Abstract
In dimensions 1 and 2 the classification of Fano’s is complete. Fano 3-folds have been studied for nearly a century, yet their classification is still to be fully solved. However, some important cases have been completed; for example, we have 105 smooth Fano 3-folds and 95 Fano 3-fold hypersurface families. Even less is known about 4-folds. However, in 2016 Fano 4-fold hypersurfaces that are quasismooth were classified into 11,617 families. In this talk, I hope to paint a picture of the nonquasismooth case, where we don’t just encounter cyclic quotient singularities but now also the arrival of nastier hyperquotient singularities.
Participants
- Marc Abboud (Neuchâtel)
- Jefferson Baudin (Lausanne)
- François Bernard (Paris)
- Fabio Bernasconi (Rome)
- Tobias Bisang (Basel)
- Jérémy Blanc (Neuchâtel)
- Alice Bouillet (Lyon)
- Valerio Buttinelli (Rome/Dijon)
- Alessio Cangini (Basel)
- Mattia Cavicchi (Dijon)
- Victor Chachay (Dijon)
- Gabriel Dill (Neuchâtel)
- Adrien Dubouloz (Poitiers)
- Mani Esna Ashari (Basel)
- Andrea Fanelli (Bordeaux)
- Mattias Ferreira-Filoramo (Paris)
- Pascal Fong (Hannover)
- Hang Fu (Basel)
- Lyalya Guseva (Dijon)
- Philipp Habegger (Basel)
- Yijue Hu (Nottingham)
- Seoyoung Kim (Basel)
- Elias Kurz (Neuchâtel)
- Robin Lahni (Hannover)
- David Loeffler (Brig)
- Gebhard Martin (Bonn)
- Emre Alp Özavcı (Lausanne)
- Erik Paemurru (Saarbrücken)
- Gianni Petrella (Esch-sur-Alzette)
- Antoine Pinardin (Edinburgh)
- Victor Pinot (Lille)
- Quentin Posva (Düsseldorf)
- Anne Schnattinger (Neuchâtel)
- Julia Schneider (Sheffield)
- Evgeny Shinder (Sheffield)
- Claudia Stadlmayr (Neuchâtel)
- Marc Truter (Warwick)
- Nikolaos Tsakanikas (Lausanne)
- Christian Urech (Zürich)
- Immanuel van Santen (Basel)
- Francesco Veneziano (Genova)
- Henrik Wehrheim (Neuchâtel)
- Zhixin Xie (Nancy)
- Egor Yasinsky (Bordeaux)
- Sarah Zerbes (Zürich)
- Susanna Zimmermann (Orsay)
Organizers
Andrea Fanelli (Bordeaux)
Philipp Habegger (Basel)
Julia Schneider (Sheffield)
Logistic support: Adrien Dubouloz (Poitiers)
Support
ANR Fracasso
Swiss Mathematical Society
Swiss Academy of Sciences
University of Basel