# A characterisation of Baer subplanes

## TL;DR

This paper characterizes certain point sets in projective spaces as tangent Baer subplanes, linking geometric intersection properties with algebraic structures in finite projective geometries.

## Contribution

It provides a new characterization of tangent Baer subplanes in PG(2,q^2) through intersection properties with Baer-pencils.

## Key findings

- Sets of points with specific 3-space intersection properties are ruled cubic surfaces.
- Such sets correspond to tangent Baer subplanes via Bruck-Bose representation.
- The characterization aids in identifying tangent Baer subplanes in finite projective planes.

## Abstract

Let $K$ be a set of $q^2+2q+1$ points in $PG(4,q)$. We show that if every 3-space meets $K$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic, then $K$ is a ruled cubic surface. Moreover, $K$ corresponds via the Bruck-Bose representation to a tangent Baer subplane of $PG(2,q^2)$. We use this to prove a characterisation in $PG(2,q^2)$ of a set of points $B$ as a tangent Baer subplane by looking at the intersections of $B$ with Baer-pencils.