Speaker: Jean-Louis Colliot-Thélène
Affiliation: Université Paris-Saclay
Title of Talk: On the stable rationality of certain real threefolds
Date: February 02, 2026
Time: 14:30:00 Hours
Venue: Guest House Seminar Room

Abstract: Over the reals, we investigate retract rationality and stable rationality of threefolds of the following types : intersections of two quadrics and quadric surface bundles over the projective line, assuming that the set of real points is connected. Over the the reals, there are singular examples which are not stably rational. Over the field $\mathbb{R}$ of real Puiseux series (a real closed field), we construct smooth varieties of each of these types which are not stably rational but for which the space $X(\mathbb{R})$ of $\mathbb{R}$-points is semi-algebraically connected. The question of constructing such examples over the field of real numbers remains open. We also consider specific quadric bundles over the reals from the point of view of decomposition of the diagonal (a property weaker than stable rationality). A specific case is given by the innocent looking affine equations $x^2 + y^2 + z^2 = u.p(u)$ over the reals, with $p(u)$ a positive polynomial of degree~2. Joint works with Alena Pirutka and with Federico Scavia.