论文标题
Elekes-szabó用于立方体表面的共线性
Elekes-Szabó for collinearity on cubic surfaces
论文作者
论文摘要
我们在立方表面上研究果园问题。我们将可能还原的立方体表面分类为$ x \ subseteq \ mathbb {p}^3（\ c）$与平滑的组件，在上面存在有限套件（无界尺寸的）家族，具有四倍的3次富含3条线的线，这些线在任何投影平面上都不集中（自然意义上）。也就是说，我们证明当$ x $是共享共同线的三个飞机的结合时，就确切地存在这样的家庭。 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但是有限的许多固定点。
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.
