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

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 and Voisin to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument of Ma 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.