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Preprint Number 2317
2317. Martin Bays, Jan Dobrowolski, Tingxiang Zou Elekes-Szabó for collinearity on cubic surfaces E-mail: 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. Mathematics Subject Classification: Keywords and phrases: |

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