Submission date: 28 December 2022

Abstract:

We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces with smooth components X ⊆ ℙ^3(ℂ) on which there exist families of finite sets (of unbounded size) with quadratically many 3-rich lines which do not concentrate (in a natural sense) on any projective plane. Namely, we prove that such a family exists precisely when X is a union of three planes sharing a common line.
Along the way, we prove a purely algebrogeometric result saying that if the composition of four Geiser involutions through sufficiently generic points a,b,c,d on a smooth irreducible cubic surface has infinitely many fixed points, then almost all these fixed points together with a,b,c,d lie on a single plane, and for the reducible case we obtain a general result about nilpotency of groups admitting an algebraic action satisfying an Elekes-Szabó condition.

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Full text arXiv 2212.14059: pdf, ps.