Exterior splashes and linear sets of rank 3

In PG(2,q3), let ππ be a subplane of order qq that is exterior to ℓ∞ℓ∞. The exterior splash of ππ is defined to be the set of q2+q+1q2+q+1 points on ℓ∞ℓ∞ that lie on a line of ππ. This article investigates properties of an exterior order-qq-subplane and its exterior splash. We show that the following objects are projectively equivalent: exterior splashes, covers of the circle geometry CG(3,q)CG(3,q), Sherk surfaces of size q2+q+1q2+q+1, and scattered linear sets of rank 3. We compare our construction of exterior splashes with the projection construction of a linear set. We give a geometric construction of the two different families of sublines in an exterior splash, and compare them to the known families of sublines in a scattered linear set of rank 3.