Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Alexeev, Valery, Schreieder, Stefan
Institutions: University of Georgia, Leibniz University Hannover
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Geometry of KSBA spaces, G5089335471
OpenAlex ID: W7213417630