Jérémy Blanc - Universität Basel - Mathematik
Basel-Dijon seminar
Basel, May 9-10, 2017
All talks in Kollegienhaus, HR119.
May 9
May 10
 Rémi Bignalet-Cazalet

 Christian Urech
 lunch  lunch
 Olivier Benoist

 Francesco Veneziano

 Philippe Gille

 social dinner

 Ulrike Rieß

Olivier BENOIST (Strasbourg) - Noether-Lefschetz loci and sums of three squares
It is a theorem of Hilbert that a real polynomial in two variables that is non-negative is a sum of four squares of rational functions. Moreover, Cassels, Ellison and Pfister have shown that there exist such polynomials that are not sums of three squares of rational functions. In this talk, we will explain that those polynomials that may be written as sums of three squares are however dense in the set of non-negative polynomials. The proof is Hodge-theoretic.

Rémi BIGNALET-CAZALET (Dijon) - Rational transformations with codimension one base locus
To an homogeneous polynomial f in K[x_0,..., x_n] where K is an algebraically closed field, one can associate a rational transformation of KP^n given by the partial derivatives of f and called the polar of f. If f is not reduced, the associated polar is different from the polar associated to the reduction of f, because there is a codimension 1 component to remove for the polar of f to be defined. It is a problem posed by I.V.Dolgachev in the birational complex case, to know if the topological degree of the polar of the reduction of f is always equal to the topological degree of the polar of f. I will extend this problem to a generalisation of the polar associated to a polynomial and I will give some examples in positive characteristic where it is not the case.

Philippe GILLE (Lyon) - Isotopes of octonion algebras and Triality
It is a joint work with Seidon Alsaody, see hal-01507255v1. Coming back to a theme of Knus/Parimala/Sridharan, we are interested in octonion algebras defined over a ring R. Given such an algebra C, McCrimmon defined certain deformations of C called isotopes sharing the same octonionic norm than C. We appeal to triality (i.e. an outer automorphism of order 3 of Spin(8)) to show that those deformations parametrize actually all octonion algebras having the same norm than C. We give also an isomorphism criterion for those algebras.

Ulrike RIEẞ (ETHZ) - Base loci of big and nef line bundles on irreducible symplectic varieties
In this talk, we present results on base loci of big and nef line bundles on irreducible symplectic varieties, which were motivated by Mayer's remarkable statements for K3 surfaces. We begin with a brief introduction to irreducible symplectic varieties. Then we focus on conditions for the existence of base divisors for big and nef line bundles on certain irreducible symplectic varieties.

Christian URECH (Basel) - Simple subgroups of the Cremona group
The Cremona group is the group of birational transformations of the complex projective plane. In 2012 Cantat and Lamy proved that the Cremona group is not simple, which answered a long standing open question. In this talk we will refine their results and show that a simple group can be embedded into the plane Cremona group if and only if it can be embedded into PGL(3,C). Our techniques also yield new insights into the structure of subgroups consisting of elliptic elements. In particular, we will see that torsion subgroups are abelian up to finite index and that the Cremona group satisfies Tits' alternative for arbitrary subgroups; this extends a result by Cantat.

Francesco VENEZIANO (Basel) - Anomalous intersections and the effective Mordell-Lang conjecture
I will present the topic of anomalous intersections and show the links between this subject and the general problem of finding rational points on curves, with special attention to the matter of effectivity. Results in collaboration with Sara Checcoli and Evelina Viada.

Olivier Benoist (Strasbourg)
Rémi Bignalet-Cazalet (Dijon)
Gabriel Dill (Basel)
Jérémy Blanc (Basel)
Frédéric Déglise (Dijon)
Adrien Dubouloz (Dijon)
Daniele Faenzi (Dijon)
Enrica Floris (Basel)
Philippe Gille (Lyon)
Philipp Habegger (Basel)
Mattias Hemmig (Basel)
Hanspeter Kraft (Basel)
Andrew Kresch (Zürich)
Rosa Maria Miro-Roig (Barcelona)
Pierre-Marie Poloni (Bern)
Ulrike Rieß (ETHZ)
Julia Schneider (Basel)
Aline Steiner (Basel)
Ronan Terpereau (Dijon)
Christian Urech (Basel)
Francesco Veneziano (Basel)
Send an email to Jeremy Blanc unibas ch if you would like to participate.

Adrien Dubouloz (Dijon)
Jérémy Blanc (Basel)

Financial support
We gratefully acknowledge support from:
University of Basel