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arXiv · 2410.19686

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

Elyes Boughattas
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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.

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Elyes Boughattas. 2024-10-25. Unirationality and $R$-equivalence for conic bundles over quasi-finite fields. https://arxiv.org/abs/2410.19686

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