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

Abstract

In $PG(2,q^3)$, let $\pi$ be a subplane of order $q$ that is tangent to ${\ell }_{i}nfty$. The tangent splash of $\pi$ is defined to be the set of $q^2+1$ points on ${\ell }_{i}nfty$ that lie on a line of $\pi$. 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 $PG(2,q^3)$ in $PG(6,q)$. We show that a tangent splash of $PG(1,q^3)$ is a $GF(q)$-linear set of rank 3 and size $q^2+1$; this allows us to use results about linear sets from \cite{lavr10} to obtain properties of tangent splashes.

References in corpus (2)

• Exterior splashes and linear sets of rank 3
• The tangent splash in $\PG(6,q)$

Cited by in corpus (2)

• Exterior splashes and linear sets of rank 3
• Subgeometries and linear sets on a projective line