Semiarcs with a long secant in $\mathrm{PG}(2,q)$

Abstract

A $t$-semiarc is a pointset ${\mathcal{S}}_t$ 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 $\mathrm{PG}(2,q)$ 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 $\mathcal{K}$. We also show that small $t$-semiarcs are related to certain small blocking sets. Some characterization theorems for small semiarcs with large collinear subsets in $\mathrm{PG}(2,q)$ are given.

15 pages, revised version. Most importantly, we corrected the definition of b(q) in Theorem 4.7