# Conics in Baer subplanes

## TL;DR

This paper explores the geometric relationship between conics in tangent Baer subplanes of PG(2,q^2) and normal rational curves in PG(4,q), revealing a correspondence mediated by transversal lines of the regular spread.

## Contribution

It establishes a bijective correspondence between conics in tangent Baer subplanes and certain normal rational curves in PG(4,q), linking two geometric frameworks.

## Key findings

- Conics in tangent Baer subplanes correspond to normal rational curves meeting transversal lines.
- Every such normal rational curve in PG(4,q) corresponds to a conic in a tangent Baer subplane.
- The study clarifies the geometric structure of conics via the André/Bruck-Bose representation.

## Abstract

This article studies conics and subconics of $PG(2,q^2)$ and their representation in the Andr\'e/Bruck-Bose setting in $PG(4,q)$. In particular, we investigate their relationship with the transversal lines of the regular spread. The main result is to show that a conic in a tangent Baer subplane of $PG(2,q^2)$ corresponds in $PG(4,q)$ to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3 and 4-dimensional normal rational curve in $PG(4,q)$ that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of $PG(2,q^2)$.