#### Vol. 17, No. 2, 2019

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Conics in Baer subplanes

### Susan G. Barwick, Wen-Ai Jackson and Peter Wild

Vol. 17 (2019), No. 2, 85–107
##### Abstract

This article studies conics and subconics of $PG\left(2,{q}^{2}\right)$ and their representation in the André/Bruck–Bose setting in $PG\left(4,q\right)$. 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 $PG\left(2,{q}^{2}\right)$ corresponds in $PG\left(4,q\right)$ to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3- and 4-dimensional normal rational curve in $PG\left(4,q\right)$ that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of $PG\left(2,{q}^{2}\right)$.

##### Keywords
Bruck–Bose representation, Baer subplanes, conics, subconics
Primary: 51E20
##### Milestones
Revised: 4 December 2018
Accepted: 29 December 2018
Published: 14 March 2019
##### Authors
 Susan G. Barwick School of Mathematical Sciences University of Adelaide Australia Wen-Ai Jackson School of Mathematical Sciences University of Adelaide Australia Peter Wild University of London United Kingdom