A Theorem of Sylvester--Gallai Type for Abelian Groups

F. K. Nilov ; A. A. Polyanskii

Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 99-109

Résumé

A finite subset $X$ of an Abelian group $A$ with respect to addition is called a Sylvester–Gallai set of type $m$ if $|X|\ge m$ and, for every distinct $x_1,\dots,x_{m-1} \in X$, there is an element $x_m \in X \setminus \{x_1,\dots,x_{m-1}\}$ such that $$ x_1+\dots+x_m=o_A, $$ where $o_A$ stands for the zero of the group $A$. We describe all Sylvester–Gallai sets of type $m$. As a consequence, we obtain the following result: if $Y$is a finite set of points on an elliptic curve in $\mathbb P^2(\mathbb C)$ and (A) if, for every two distinct points $x_1,x_2 \in Y$, there is a point $x_3 \in Y \setminus \{x_1,x_2\}$ collinear to $x_1$ and $x_2$, then either $Y$ is a Hesse configuration of an elliptic curve or $Y$ consists of three points lying on the same line; (B) if, for every five distinct points $x_1,\dots,x_5 \in Y$, there is a point $x_6 \in Y \setminus \{x_1,\dots,x_{5}\}$ such that $x_1,\dots,x_6$ lie on the same conic, then $Y$ consists of six points lying on the same conic.

Détail

Keywords: Sylvester–Gallai theorem, configurations of points and conics, elliptic curves.
Mots-clés : configurations of points and lines