Unirationality and $R$-equivalence for conic bundles over quasi-finite fields

TL;DR

This paper investigates the unirationality and $R$-equivalence of conic bundle surfaces over quasi-finite fields, providing new conditions under which these properties hold, especially over finite fields.

Contribution

It establishes that regular conic bundles over $ extbf{P}^1_k$ are unirational when all non-split fibers are over rational points, extending previous results for large finite fields.

Findings

Unirationality holds under specified fiber conditions over quasi-finite fields.

All rational points on the conic bundle are $R$-equivalent under the same conditions.

Results improve upon previous work by Mestre for large finite fields.

Abstract

Yanchevski\u{i} had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular conic bundle $X$ over $\mathbf{P}^1_k$ is unirational if all non-split fibres lie over rational points. For large finite fields $k$, this beats a previous result of Mestre. Under the same assumption, we also prove that all rational points of $X$ are $R$-equivalent.