Assume a smooth cubic surface is defined over a field $k$ characteristic $0$, that it has line defined over $k$ and that its arithmetic Picard rank over $k$ is maximal i.e. $7$. Does this imply that the surface has to contain a second line defined over $k$ which does not meet the first ?
2026-04-02 14:49:41.1775141381
Arithmetic picard rank of smooth cubic surfaces
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