$R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence

ArXi:2603.19215v1 Announce Type: cross Let $V$ be a smooth cubic surface over a $p$-adic field $k$ with good reduction. Swinnerton-Dyer proved that $R$-equivalence is trivial on $V(k)$ except perhaps if $V$ is one of three special types--those whose $R$-equivalence he could not bound by proving the universal (admissible) equivalence is trivial. We consider all surfaces $V$ currently known to have non-trivial universal equivalence.