A Characterisation of Tangent Subplanes of PG(2,q^3)

TL;DR

This paper characterizes tangent subplanes of PG(2,q^3) by establishing a correspondence with certain ruled surfaces in PG(6,q), extending previous work on their representation in the Bruck-Bose model.

Contribution

It proves that any ruled surface with specific properties corresponds to a tangent order-q-subplane in PG(2,q^3), completing the characterization.

Findings

Ruled surfaces correspond to tangent order-q-subplanes.

The converse of previous representation results is established.

Provides a geometric criterion for identifying tangent subplanes.

Abstract

In: S.G. Barwick and W.A. Jackson. Sublines and subplanes of PG(2,q^3) in the Bruck--Bose representation in PG(6,q). Finite Fields Th. App. 18 (2012) 93--107., the authors determine the representation of order-q-subplanes and order-q-sublines of PG(2,q^3) in the Bruck-Bose representation in PG(6,q). In particular, they showed that an order-q-subplane of PG(2,q^3) corresponds to a certain ruled surface in PG(6,q). In this article we show that the converse holds, namely that any ruled surface satisfying the required properties corresponds to a tangent order-q-subplane of PG(2,q^3).