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- W4313442780 abstract "We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $Xsubseteq mathbb{P}^3(C)$ with smooth components 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 obtain a general result about nilpotency of groups admitting an algebraic action satisfying an Elekes-Szab'o condition, and we prove the following purely algebrogeometric statement: 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 a single plane contains $a,b,c,d$ and all but finitely many of the fixed points." @default.
- W4313442780 created "2023-01-06" @default.
- W4313442780 creator A5041235660 @default.
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- W4313442780 date "2022-12-28" @default.
- W4313442780 modified "2023-09-26" @default.
- W4313442780 title "Elekes-Szab'o for collinearity on cubic surfaces" @default.
- W4313442780 doi "https://doi.org/10.48550/arxiv.2212.14059" @default.
- W4313442780 hasPublicationYear "2022" @default.
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