On sets of points in general position that lie on a cubic curve in the plane and determine lines that can be pierced by few points

TL;DR

This paper investigates the structure of point sets in the plane lying on cubic curves, showing they have algebraic properties related to subgroups, and relates these findings to problems about directions and configurations of points.

Contribution

It establishes a connection between point sets on cubic curves and subgroup cosets, providing bounds and counterexamples relevant to classical conjectures in combinatorial geometry.

Findings

Point sets on cubic curves with few piercing points have subgroup coset structure.

The bound |R| < 3/2 n is tight for some results.

Counterexamples to Jamison's conjecture are constructed.

Abstract

Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$. We show that if $|R| < \frac{3}{2}n$ and $P \cup R$ is contained in a cubic curve $c$ in the plane, then $P$ has a special property with respect to the natural group action on $c$. That is, $P$ is contained in a coset of a subgroup $H$ of $c$ of cardinality at most $|R|$. We use the same approach to show a similar result in the case where each of $B$ and $G$ is a set of $n$ points in general position in the plane and every line through a point in $B$ and a point in $G$ passes through a point in $R$. This provides a partial answer to a problem of Karasev. The bound $|R| < \frac{3}{2}n$ is best possible at least for part of our results. Our extremal constructions provide a counterexample to an old conjecture attributed to Jamison about point sets that determine few directions. Jamison conjectured that if $P$ is a set of $n$ points in general position in the plane that determines at most $2n-c$ distinct directions, then $P$ is contained in an affine image of the set of vertices of a regular $m$-gon. This conjecture of Jamison is strongly related to our results in the case the cubic curve $c$ is reducible and our results can be used to prove Jamison's conjecture at least when $m-n$ is in the order of magnitude of $O(\sqrt{n})$.