Title preseminar: Rationality of varieties over non-closed fields

Abstract preseminar: An algebraic variety over a field k is said to be k-rational if it admits a birational map to projective space over k. Determining the rationality of a variety is a classical problem, especially when k is the complex numbers. When the field k is not algebraically closed, there are additional considerations in the rationality problem, coming from the arithmetic of k. In this talk, we'll present examples of rational and non-rational varieties, and we'll see some obstructions to rationality.

Title seminar: Rationality problems for linear spaces on pencils of quadrics

Abstract seminar: The linear spaces contained in the base locus of a pencil of quadrics encode a lot of interesting geometry. For pencils of even-dimensional quadrics, there is a deep relationship between these linear spaces and hyperelliptic curves, dating back to Weil. This has found numerous applications, e.g. to rational points and to moduli spaces of sheaves on curves. In this talk, we study rationality questions for the Fano schemes of these linear spaces, and we generalize some results of Reid, Colliot-Thélène–Sansuc–