Exterior splashes and linear sets of rank 3

Abstract

In $\PG(2,q^3)$, let $\pi$ be a subplane of order $q$ that is exterior to $\li$. The exterior splash of $\pi$ is defined to be the set of $q^2+q+1$ points on $\li$ that lie on a line of $\pi$. This article investigates properties of an exterior \orsp\ and its exterior splash. We show that the following objects are projectively equivalent: exterior splashes, covers of the circle geometry $CG(3,q)$, Sherk surfaces of size $q^2+q+1$, and $\GF(q)$-linear sets of rank 3 and size $q^2+q+1$. 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.

References in corpus (1)

• An investigation of the tangent splash of a subplane of PG(2,q^3)

Cited by in corpus (4)

• An investigation of the tangent splash of a subplane of PG(2,q^3)
• On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$
• The exterior splash in $PG(6,q)$: Special conics
• The exterior splash in PG(6,q): Transversals