User:Jeremydover/sandbox/Desarguesian plane of order 3

Construction
This plane can be constructed using the standard vector space construction with a three-dimensional vector space over the finite field GF(3).

Ovals and Hyperovals
This plane has odd order, so by Qvist's theorem it does not have a hyperoval. Being a Desarguesian, this plane does have ovals, and by Segré's theorem, all ovals in this plane are conics, being the set of absolute points of an orthogonal polarity.

Subplanes
Since this plane is a Desarguesian plane of prime order, it has no proper subplanes.

Unitals
Since the order of this plane is not square, it does not have a unital.