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Elekes-Szabó for collinearity on cubic surfaces

Martin Bays, Jan Dobrowolski, Tingxiang Zou

TL;DR

This work extends the Elekes-Szabó paradigm to the 3D orchard problem on complex cubic surfaces, achieving a full classification of when quadratic growth in collinear triples can occur without plane concentration. It combines model-theoretic coarse pseudofinite dimension methods with geometric and incidence-geometry techniques: a coherent-action/nilpotence argument handles reducible cubics, while a Szemerédi–Trotter–type energy analysis paired with fixed-point geometry of Geiser involutions handles irreducible cubics. The main finitary takeaway is that, for large finite point sets on a cubic surface not equal to three planes through a common line, substantial triple-line configurations must concentrate on a single plane, thereby mirroring the planar orchard phenomenon in a higher-dimensional setting. These results illuminate the higher-dimensional Elekes-Szabó landscape by identifying nilpotent-group actions as the natural mechanism behind quadratic incidences in the collinearity setting on cubic surfaces. The methods open avenues for broader applications to Ind-definable group actions and to other ternary incidence relations arising from algebraic surfaces.

Abstract

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.

Elekes-Szabó for collinearity on cubic surfaces

TL;DR

This work extends the Elekes-Szabó paradigm to the 3D orchard problem on complex cubic surfaces, achieving a full classification of when quadratic growth in collinear triples can occur without plane concentration. It combines model-theoretic coarse pseudofinite dimension methods with geometric and incidence-geometry techniques: a coherent-action/nilpotence argument handles reducible cubics, while a Szemerédi–Trotter–type energy analysis paired with fixed-point geometry of Geiser involutions handles irreducible cubics. The main finitary takeaway is that, for large finite point sets on a cubic surface not equal to three planes through a common line, substantial triple-line configurations must concentrate on a single plane, thereby mirroring the planar orchard phenomenon in a higher-dimensional setting. These results illuminate the higher-dimensional Elekes-Szabó landscape by identifying nilpotent-group actions as the natural mechanism behind quadratic incidences in the collinearity setting on cubic surfaces. The methods open avenues for broader applications to Ind-definable group actions and to other ternary incidence relations arising from algebraic surfaces.

Abstract

We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces 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 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 on a smooth irreducible cubic surface has infinitely many fixed points, then a single plane contains and all but finitely many of the fixed points.
Paper Structure (34 sections, 51 theorems, 57 equations)

This paper contains 34 sections, 51 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Background and motivation
  4. Orchard problems
  5. Higher-dimensional Elekes-Szabó
  6. Overview
  7. Outline of the paper
  8. Acknowledgements
  9. Preliminaries
  10. Non-standard setup
  11. Coarse pseudofinite independence
  12. Weak general position
  13. Elekes-Szabó triangles
  14. The reducible case
  15. Non-collinear planes
  16. ...and 19 more sections

Key Result

Theorem 1.1

For any $\epsilon > 0$ there exists $\eta > 0$ and $N_0 \in {\mathbb{N}}$ such that if $S \subseteq \mathbb{P}^3({\mathbb{C}})$ is a cubic surface with smooth irreducible components which is not the union of three planes sharing a common line, and $A \subseteq S$ is a finite subset with $|A| \geq

Theorems & Definitions (143)

  • Theorem 1.1: Corollary \ref{['c:mainOrchardFinite']}, symmetric case
  • Lemma 2.1
  • proof
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 133 more