# Totaro's Question on Zero-Cycles on Torsors

## TL;DR

This paper investigates Totaro's question on the existence of closed étale points of certain degrees on torsors with zero-cycles, providing a counterexample that shows the answer is negative in general.

## Contribution

The paper constructs a specific example demonstrating that a torsor with a zero-cycle of degree d need not have a closed étale point of degree dividing d, answering Totaro's question negatively.

## Key findings

- Counterexample to Totaro's question
- Zero-cycle existence does not imply étale point of dividing degree
- Clarification of conditions under which the question holds

## 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 $d \geq 1$, then does $X$ have a closed \'etale 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.