# Zero-cycles of degree one on Skorobogatov's bielliptic surface

## TL;DR

This paper proves that Skorobogatov's bielliptic surface, which counters the Hasse principle without Brauer-Manin obstruction, indeed has a zero-cycle of degree one, confirming a conjecture by Colliot-Thélène.

## Contribution

It demonstrates the existence of a zero-cycle of degree one on Skorobogatov's bielliptic surface, supporting the conjecture that such cycles exist even when the Hasse principle fails.

## Key findings

- Confirmed the existence of a zero-cycle of degree one on the surface.
- Supported Colliot-Thélène's conjecture regarding zero-cycles.
- Provided insights into the arithmetic of bielliptic surfaces.

## Abstract

Skorobogatov constructed a bielliptic surface which is a counterexample to the Hasse principle not explained by the Brauer-Manin obstruction. We show that this surface has a $0$-cycle of degree 1, as predicted by a conjecture of Colliot-Th\'el\`ene.