Jérémy Blanc - Universität Basel - Mathematik
  Algebraic and Arithmetic Geometry - "Birpol6"
Basel, December 11-13, 2013
The conference was held in Basel from December 11 to 13, 2013.

Alexander DUNCAN
Damiano TESTA
Paris 7

December 11
December 12
December 13
 On cubic birational
maps of 3()

 coffee break

 Field generators
 Pliability of Mori fibre
spaces in dimension 3

 coffee break
Topology of real algebraic
varieties of dimension 3

 Examples of
in dimension 3

 coffee break

 Andriy REGETA
 Lie subalgebras of
vector fields and
the Jacobian conjecture

 Damiano TESTA
 Unirationality of certain
del Pezzo surfaces
of degree two


 Alexander DUNCAN
 Equivariant Unirationality

 Unverzweigte Kohomologie
 Social dinner

Hamid AHMADINEZHAD - Pliability of Mori fibre spaces in dimension 3
I begin with recalling the definition of pliability for Mori fibre spaces, and Fano varieties in particular. Then I explore the strong connection between this invariant and the finite generation of certain Cox rings. This, for instance, highlights the geometry behind the method of maximal singularities, used to prove birational rigidity. Some applications in proving birational rigidity (and hence non-rationality) and birational non-rigidity for some Fano threefolds will be mentioned.
Serge CANTAT - Examples of (pseudo-)automorphisms in dimension 3
I shall describe examples of complex projective threefolds with a nice group of automorphisms or pseudo-automorphisms, the dynamics of which is non-trivial. These examples are due to Keiji Oguiso and his co-authors.
Pierrette CASSOU-NOGUES - Field generators
Let k be a field. A field generator is a polynomial F∈k[X,Y] satisfying k(F,G)=k(X,Y), for some G∈k(X,Y). If G can be chosen in k[X,Y], we call F a "good field generator", otherwise F is a "bad field generator". These notions were first studied by Abhyankar, Jan and Russell in the 70s. Until 2005, only two examples of bad field generators were known. We study the behavior of bad field generators under birational morphisms. This lead to the construction of new bad field generators with arbitrary number of dicriticals.
   Joint work with Daniel Daigle, U. Ottawa.
Jean-Louis COLLIOT-THÉLÈNE - Unverzweigte Kohomologie
This talk will be part of the Perlen-Kolloquium (classical colloquium of the institute).
Die Brauergruppe von Varietäten kann man zu verschieden Zwecken benutzen :
-- Zeigen, das eine Varietät nicht rational ist, d. h., ihr Funktionenkörper nicht rein transzendant ist.
-- Wenn der Grundkörper ein Zahlkörper ist, zeigen, daβ das Lokal-Global Prinzip für rationale Punkte nicht gilt (Brauer-Maninsche Hindernis).
-- Die Picardgruppe, d.h. die Klassengruppe von Divisoren, untersuchen (Tatesche Vermutung).
Unverzweigte Kohomologiegruppen sind Verallgemeinerungen der Brauergruppe. Ich werde erklären, wie man sie zu ähnlichen Zwecken anwenden kann.
Julie DÉSERTI - On cubic birational maps of 3()
I will deal with the irreducible components of the set of birational maps of bidegree (3,3), (3,4) and (3,5).
Alexander DUNCAN - Equivariant Unirationality
A variety X is unirational if there exists a dominant rational map from an affine space V to X. I discuss a generalization of this notion which I call ``G-unirationality.'' Consider a variety X with an action of a linear algebraic group G. We say that X is G-unirational if there exists an equivariant dominant rational map from a linear representation V of G. I will discuss how existing unirationality constructions in the arithmetic case can be used to provide constructions in the equivariant case. I also discuss how progress in the equivariant setting may yield results in the arithmetic setting. The main objects of study will be del Pezzo surfaces.
Frédéric MANGOLTE - Topology of real algebraic varieties of dimension 3
We know since Nash (1952) and Tognoli (1973) that any compact smooth manifold M admits a real algebraic model. Namely, given a manifold M, there exists polynomials with real coefficients whose locus of common real zeroes is diffeomorphic to M.
Bochnak and Kucharz proved later that there exists in fact an infinite number of distincts models for a given M. We try therefore to find "simpler" models than the others in a meaning to be specified. In this talk, I will describe the state-of-the-art concerning this research program about "simple" real algebraic models for low dimensional varieties.
The situation for curves and surfaces is quite well understood now, and the surface case is already interesting. For real algebraic threefolds, János Kollár opened in 1999 a direction of research thanks to his solution of Minimal Model Program over the reals. We will discuss several Kollár?s conjectures solved since then.
Andriy REGETA - Lie subalgebras of vector fields and the Jacobian conjecture
We study Lie subalgebras L of the vector fields Vecc(2) of the affine 2-space 2 of constant divergence, and we classify those L which are isomorphic to the Lie algebra aff2 of the group Aff2(K) of affine transformations of 2. We then show that the following statements are equivalent:
(a) The Jacobian Conjecture holds in dimension 2;
(b) All Lie subalgebras L ⊂ Vecc(2) isomorphic to aff2 are conjugate under Aut(2);
(c) All Lie subalgebras L ⊂ Vecc(2) isomorphic to aff2 are algebraic.
Damiano TESTA - Unirationality of certain del Pezzo surfaces of degree two
A consequence of the Segre-Manin Theorem is that a del Pezzo surface of degree two is unirational over the ground field as soon as it contains a general point. I will report on joint work with C. Salgado and T. Várilly-Alvarado where we find precise conditions implying unirationality. In particular, we prove that all del Pezzo surfaces of degree two over finite fields are unirational with at most three possible exceptions.

Hamid Ahmadinhezhad (Linz)
Cinzia Bisi (Ferrara)
Jérémy Blanc (Basel)
Serge Cantat (Rennes)
Jung-Kyu Canci (Basel)
Pierrette Cassou-Noguès (Bordeaux)
Jean-Louis Colliot-Thélène (Paris-Sud)
Julie Déserti (Basel)
Adrien Dubouloz (Dijon)
Alexander Duncan (Michigan)
Eric Edo (Nouméa)
Jean-Philippe Furter (La Rochelle)
Julien Grivaux (Marseille)
Isac Héden (Basel)
Hanspeter Kraft (Basel)
Kevin Langlois (Grenoble)
Frédéric Mangolte (Angers)
Stefan Maubach (Bremen)
Jean-Philippe Monnier (Angers)
Lucy Moser-Jauslin (Dijon)
Karol Palka (Warsaw)
Charlie Petitjean (Dijon)
Pierre-Marie Poloni (Basel)
Abdul Rauf (Bremen)
Andriy Regeta (Basel)
Maria Fernanda Robayo (Basel)
Immanuel Stampfli (Bremen)
Damiano Testa (Warwick)
Christian Urech (Rennes)
Susanna Zimmerman (Basel)

Financial support
We gratefully acknowledge support from:
Swiss national Science Foundation and
French ANR Birpol