A Characterisation of Tangent Subplanes of PG(2,q³)

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).