Exterior splashes and linear sets of rank 3

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.

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