Totaro’s question on zero-cycles on torsors

Abstract

Let G be a smooth connected linear algebraic group and X be a G-torsor. Totaro asked: if X admits a zero-cycle of degree d ≥ 1, then does X have a closedétale point of degree dividing d? While the literature contains affirmative answers in some special cases, we give an example to show that the answer is negative in general.

Closed points on cubic hypersurfaces

“…• Serre asked the question for X being homogeneous space of algebraic groups, see [Ser95], [GSS18], [Bla11] and [Par05] for various results.…”