# On sets of $n$ points in general position that determine lines that can be pierced by $n$ points

## TL;DR

This paper proves that for a set of n points in general position with a corresponding set of n points on lines through each pair, the combined set lies on a cubic curve, revealing geometric structure.

## Contribution

It establishes a novel geometric characterization linking point sets and cubic curves based on line piercing properties.

## Key findings

- P ∪ R is contained in a cubic curve.
- The structure of point sets with line piercing points is constrained.
- Provides a new connection between combinatorial geometry and algebraic curves.

## Abstract

Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$ and $y$. We show that $P \cup R$ must be contained in a cubic curve.