Program
Schedule
| Time | Monday | Tuesday | Wednesday | Thursday | Friday |
|---|---|---|---|---|---|
| 9:15-10:45 | Kebekus 1 (starts at 9:45) |
Kresch 1 | Bresciani | Kresch 2 | Kebekus 3 |
| 10:45-11:20 | Coffee | Coffee | Coffee | Coffee | Coffee |
| 11:20-12:20 | Pieropan | Vistoli 1 | Kebekus 2 | Porta | Corvaja |
| 12:20-15:00 | Lunch | Lunch | Lunch | Lunch | Lunch |
| 15:00-16:30 | Viviani (termine à 16h) | Vistoli 2 | Free | Open problems and short talks |
Free |
| 16:30- | Coffee | Coffee | Coffee |
Mini-courses
Orbifoldes géométriques over the complex numbers and over global function fields
Stefan Kebekus
Abstract: The lectures introduce Campana’s theory of Orbifoldes Géométriques and discusses them in the complex setting and over global function fields. The first two lecture work over the complex numbers and survey recent attempts (joint with Rousseau) to develop an analogue of the Albanese and the beginnings of a Nevanlinna theory. The last lecture surveys work (joint with Pereira and Smeets) where we use Campana’s theory to construct examples of algebraic varieties over global fields for which Skorobogatov’s etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.
The analogy between birational geometry over a non algebraically closed field and equivariant birational geometry
Andrew Kresch
Résumé: In these lectures I will discuss rational maps of orbifolds over a field of characteristic zero and explain some of the issues that arise when formulating for orbifolds a notion of birational equivalence. It will be explained how recent advances around orbifolds, such as functorial destackification, contribute to a better understanding of the birational geometry of orbifolds.
Fields of moduli, the arithmetic of quotient singularities, and the Lang-Nishimura theorem for algebraic stacks
Angelo Vistoli and Giulio Bresciani
Abstract: Given a perfect field k with algebraic closure k’ and a variety X over k’, possibly with some additional structure, the field of moduli of X is the subfield of k’ of elements fixed by elements s of the Galois group of k’ over k such that the twist Xs is isomorphic to X. The field of moduli is contained in every algebraic extension of k over which X is defined. The notion of field of moduli was first introduced for polarized varieties by T. Matsusaka in 1958; the definition was clarified and extended by G. Shimura and S. Koizumi. It has been intensively studied since then, particularly for curves and abelian varieties.
The question that I will treat is: when is X defined over its field of moduli?
My course will be divided as follows. All the results in parts 4, 5 and 6 are due to Giulio Bresciani and myself.
1) Generalities on fields of moduli, and review of some of the known results.
2) The stack theoretic formalism, residual gerbes, and applications of non-abelian cohomology.
3) The result of Dèbes and Emsalem on smooth curves, with applications.
4) The extension of this to higher dimensional varieties.
5) The arithmetic of quotient singularities: this is necessary technology to obtain meaningful applications of the result in part 4. Here the basic question is: given a variety X over k, with a quotient singularity p, and a resolution of singularities Z —> X, when is there a rational point of Z over p? I will explain some criteria that ensure the existence of such a rational point, and obtain criteria for a variety, possibly with additional structure, to be defined over its field of moduli.
6) The Lang-Nishimura theorem for algebraic stacks, and sketches of the proofs of the results in parts 3, 4 and 5.
Research Talks
GCD estimates in algebraic geometry
Pietro Corvaja
Abstract: Around twenty years ago, Bugeaud, Zannier and myself proved that for multiplicativley independent positive integers a,b, the greatest common divisor of \(a^n-1,b^n-1\) can be bounded as \(\gcd(a^n-1,b^n-1) \ll \exp(\epsilon n)\) for every \(\epsilon>0\). We shall discuss geometric analogues of the above result, providing (sometimes conjectural) upper bounds for singularities of algebraic curves in semi-abelian varieties in terms of their degree and Euler characteristic.
Campana points on Fano varieties: theory and results
Marta Pieropan
Abstract: This talk introduces Campana points, an arithmetic notion, first studied by Campana and Abramovich, that interpolates between the notions of rational and integral points. Campana points are expected to satisfy suitable analogs of Lang's conjecture, Vojta's conjecture and Manin's conjecture, and their study introduces new number theoretic challenges of a computational nature. I will illustrate the definition on examples, recall the latest results and discuss a range of open questions.
Categorified Beauville-Laszlo theorem (and related problems)
Mauro Porta
Abstract: Sheaves of Azumaya algebras were introduced by Grothendieck to represent classes in the cohomological Brauer group of schemes, i.e. \(Br(X) := H^2_{ét}(X;G_m)\), along the same lines every class in \(H^1_{ét}(X;G_m)\) is representable by a line bundle on \(X\). However, it turns out that not every class in \(Br(X)\) can be represented by a sheaf of Azumaya algebras, as shown in the case of Mumford's normal surface. In much more recent times, Toën introduced the notion of sheaf of derived Azumaya algebra, and proved that these objects represent even nontorsion classes in \(Br(X)\).
In collaboration with Federico Binda we studied two problems related to derived Azumaya algebras: the Grothendieck existence and the Beauville-Laszlo theorems. In this talk, I will survey both questions and explain how our categorified approach allows to go beyond a classical injectivity result of Grothendieck. I will finish with a brief discussion of the consequences of categorified Beauville-Laszlo that will be the object of a future work.
On the Picard group of the stack of G-bundles on families of curves
Filippo Viviani
Abstract: For a given connected linear algebraic group G and a family of curves, we study the Picard group of the stack of relative G-principal bundles on the given family. This is a joint work with R. Fringuelli.