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Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces

  • Valery Alexeev
  • Stefan Schreieder
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Abstract

The Cassels--Swinnerton-Dyer conjecture asserts that a cubic hypersurface contains a rational point if and only if it contains a point of degree coprime to $3$, or, equivalently, a zero-cycle of degree $1$. The case of smooth cubic surfaces in characteristic zero has been reduced by Coray (1976) and Voisin (2026) to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument to extend the result to smooth cubic surfaces over arbitrary fields. We further give a separate argument for the case of singular cubic surfaces, extending previous work of Coray over perfect fields. Altogether, this proves the Cassels--Swinnerton-Dyer conjecture for cubic surfaces.

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Cite

BibTeXRIS
BibTeX
@misc{alexeev2026proofs,
  title = {{Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces}},
  author = {Valery Alexeev and Stefan Schreieder},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15930},
  eprint = {2609.15930},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15930},
}
RIS
TY  - UNPB
TI  - Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces
AU  - Valery Alexeev
AU  - Stefan Schreieder
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.15930
UR  - https://arxiv.org/abs/2609.15930
AB  - The Cassels--Swinnerton-Dyer conjecture asserts that a cubic hypersurface contains a rational point if and only if it contains a point of degree coprime to $3$, or, equivalently, a zero-cycle of degree $1$. The case of smooth cubic surfaces in characteristic zero has been reduced by Coray (1976) and Voisin (2026) to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument to extend the result to smooth cubic surfaces over arbitrary fields. We further give a separate argument for the case of singular cubic surfaces, extending previous work of Coray over perfect fields. Altogether, this proves the Cassels--Swinnerton-Dyer conjecture for cubic surfaces.
ER  -