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PreprintarXiv licencearXiv

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 PSL2(𝔽11)-unirational, establishing that the essential dimension of PSL2(𝔽11) 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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Cite

BibTeXRIS
BibTeX
@misc{ballard2026equivariant,
  title = {{Equivariant Unirationality of Cubic Threefolds}},
  author = {Matthew Ballard and Alexander Duncan and Zhijia Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09962},
  eprint = {2610.09962},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09962},
}
RIS
TY  - UNPB
TI  - Equivariant Unirationality of Cubic Threefolds
AU  - Matthew Ballard
AU  - Alexander Duncan
AU  - Zhijia Zhang
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09962
UR  - https://arxiv.org/abs/2610.09962
AB  - 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 PSL_2(𝔽_11)-unirational, establishing that the essential dimension of PSL_2(𝔽_11) is $3$. This disproves a conjecture of Dolgachev that the essential dimension of a group is at least its Cremona dimension.
ER  -