A Severi-Brauer variety is a smooth projective variety X defined over a field k such that after base change to an algebraic closure k it becomes isomorphic to a projective space. The Brauer group of a field k can be defined as a monoid of central simple algebras over k (i.e. finite-dimensional algebras over k with the center being equal to k and no non-trivial two-sided ideals) modulo so-called Brauer equivalence. It turns out that the Brauer group also classifies Severi-Brauer varieties. Moreover, the Brauer group is also isomorphic to the second Galois cohomology of k with the coefficients in k*. This way the topic unites in itself the study of finite dimensional (non-commutative) algebras, group (Galois) cohomology and algebraic geometry.

The first part of the seminar will cover the correspondence described above with many examples and some of the proofs repeated by algebraic and geometric arguments. The second part will consist in discussing various topics: Albert's theorem on central simple algebras of period 2 and index 4, residue maps on Brauer groups, in particular, Faddeev's exact sequence, and, most notably, an application of the unramified Brauer group to the Lüroth's problem. Also the topic of the Brauer group of number fields and the Brauer-Manin obstruction could be included here according to the wishes of the participants on the prelimiary meeting.

Detailed program will appear on my website later.

Literature

• Gille P., Szamuely T. "Central Simple Algebras and Galois Cohomology", Vol. 165. Cambridge University Press, 2017;
• Artin M. "Brauer-Severi varieties." Brauer groups in ring theory and algebraic geometry. Springer, Berlin, Heidelberg, 1982. 194-210;
• Kollár J. "Severi-Brauer varieties; a geometric treatment." arXiv preprint arXiv:1606.04368 (2016);

• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk