Elekes-Szabó for collinearity on cubic surfaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866908621349060608 |
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| author | Bays, Martin Dobrowolski, Jan Zou, Tingxiang |
| author_facet | Bays, Martin Dobrowolski, Jan Zou, Tingxiang |
| contents | We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ 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-Szabó 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_14059 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Elekes-Szabó for collinearity on cubic surfaces Bays, Martin Dobrowolski, Jan Zou, Tingxiang Logic Combinatorics 52C10 (Primary) 03C98, 14C17 (Secondary) We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ 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-Szabó 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. |
| title | Elekes-Szabó for collinearity on cubic surfaces |
| topic | Logic Combinatorics 52C10 (Primary) 03C98, 14C17 (Secondary) |
| url | https://arxiv.org/abs/2212.14059 |