Semiarcs with a long secant in PG(2,q)

Abstract

A $t$-semiarc is a point set ${\mathcal{S}}_t$ with the property that the number of tangent lines to ${\mathcal{S}}_t$ at each of its points is $t$. We show that if a small $t$-semiarc ${\mathcal{S}}_t$ in $PG\left(2,q\right)$ has a large collinear subset $\mathcal{K}$, then the tangents to ${\mathcal{S}}_t$ at the points of $\mathcal{K}$ can be blocked by $t$ points not in $\mathcal{K}$. In fact, we give a more general result for small point sets with less uniform tangent distribution. We show that in $PG\left(2,q\right)$ small $t$-semiarcs are related to certain small blocking sets and give some characterization theorems for small semiarcs with large collinear subsets.