Equivariant Unirationality of Cubic Threefolds

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 ${\mathsf{P}\mathsf{S}\mathsf{L}}_2({\mathbb{F}}_{11})$-unirational, establishing that the essential dimension of ${\mathsf{P}\mathsf{S}\mathsf{L}}_2({\mathbb{F}}_{11})$ is $3$. This disproves a conjecture of Dolgachev that the essential dimension of a group is at least its Cremona dimension.