Totaro’s question on zero-cycles on torsors

Abstract

Let $G$ be a smooth connected linear algebraic group, and let $X$ be a $G$-torsor. Totaro asked: If $X$ admits a zero-cycle of degree $d\ge 1$, then does $X$ have a closed étale point of degree dividing $d$? While the literature contains affirmative answers in some special cases, we give examples to show that the answer is negative in general.