Universal torsors

Abstract: In this talk, we focus on torsors under algebraic tori. For an algebraic variety $X$ with discrete $\operatorname{Pic}(X_{\bar k})$ (for instance, when $X_{\bar k}$ is rationally connected), the dual group $T = \operatorname{Hom}(\operatorname{Pic}(X_{\bar k}), \mathbb{G}_m)$ is a natural algebraic torus associated to $X$. Universal torsors are a class of torsors over $X$ under $T$ which, on the one hand, can capture the set of all rational points on $X$ and, on the other, often admit an explicit description. After developing the theory, we will see how universal torsors can be used to prove the existence of rational points.

References:
- Wittenberg, Rational points and zero-cycles on rationally connected varieties over number fields, Section 3.3.
- Skorobogatov, Torsors and Rational Points, Section 2.3, p.25 and the references there.

algebraic geometrynumber theory

Audience: advanced learners

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2026 topic: The descent obstruction.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.