An investigation of the tangent splash of a subplane of PG(2, q3)

In PG(2, q3), let π be a subplane of order q that is tangent to `∞. The tangent splash of π is defined to be the set of q2 + 1 points on `∞ that lie on a line of π. This article investigates properties of the tangent splash. We show that all tangent splashes are projectively equivalent, investigate sublines contained in a tangent splash, and consider the structure of a tangent splash in the Bruck-Bose representation of PG(2, q3) in PG(6, q). We show that a tangent splash of PG(1, q3) is a GF(q)-linear set of rank 3 and size q2 + 1; this allows us to use results about linear sets from [17] to obtain properties of tangent splashes.