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In PG ( 2 , q 3 ) , let π be a subplane of order q that is exterior to ? ∞ . The exterior splash of π is defined to be the set of q 2 + q + 1 points on ? ∞ that lie on a line of π . This article investigates properties of an exterior order- q -subplane?and its exterior splash. We show that the following objects are projectively equivalent: exterior splashes, covers of the circle geometry C G ( 3 , q ) , Sherk surfaces of size q 2 + 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.

The article gives a combinatorial characterisation of a ruled cubic surface in PG ( 4 , q ) . In PG ( 4 , q ) , q odd, q ≥ 9 , a set K of q 2 + 2 q + 1 points is a ruled cubic surface if and only if K has class [ q + 1 , 2 q + 1 , 3 q + 1 ] 3 such that every plane that contains four points of K contains at least q + 1 points of K .

Let E be a set of solids (hyperplanes) in PG ( 4 , q ) , q even, q > 2 , such that every point of PG ( 4 , q ) lies in either 0, 1 2 ( q 3 − q 2 ) or 1 2 q 3 solids of E , and every plane of PG ( 4 , q ) lies in either 0, 1 2 q or q solids of E . This article shows that E is either the set of solids that are disjoint from a hyperoval, or the set of solids that meet a non-singular quadric Q ( 4 , q ) in an elliptic quadric.

Let B be a subplane of PG ( 2 , q 3 ) of order q that is tangent to ? ∞ . Then the tangent splash of B is defined to be the set of q 2 + 1 points of ? ∞ that lie on a line of B . Tangent splashes of PG ( 2 , q 3 ) are all projectively equivalent, and are equivalent to GF ( q ) -linear sets of rank 3 and size q 2 + 1 . In the Bruck-Bose representation of PG ( 2 , q 3 ) in PG ( 6 , q ) , we investigate the interaction between the ruled surface corresponding to B and the planes corresponding to the tangent splash of B . We then give a geometric construction of the unique order- q -subplane determined by a given tangent splash and a fixed order- q -subline.

Let C¯ be a conic in PG(2,q2) and suppose we derive with respect to a derivation set or multiple derivation set. This paper looks at whether the conic C¯ gives rise to an inherited arc in the derived plane. Further, we construct two families of conics which give rise to inherited arcs after certain double derivations. Finally we construct a family of complete (q2+1)-arcs in certain André planes.