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arXiv · 2610.09962

Equivariant Unirationality of Cubic Threefolds

Matthew Ballard·Alexander Duncan·Zhijia Zhang
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Abstract

Using the recent proof of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces, we finish the classification of $G$-unirational complex cubic threefolds. In particular, we prove that the Klein cubic threefold is $\mathsf{PSL}_2(\mathbb{F}_{11})$-unirational, establishing that the essential dimension of $\mathsf{PSL}_2(\mathbb{F}_{11})$ is $3$. This disproves a conjecture of Dolgachev that the essential dimension of a group is at least its Cremona dimension.

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Matthew Ballard, Alexander Duncan, Zhijia Zhang. 2026-10-07. Equivariant Unirationality of Cubic Threefolds. https://arxiv.org/abs/2610.09962

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