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

Abstract

A $t$-semiarc is a pointset ${\cal S}_t$ with the property that the number of tangent lines to ${\cal S}_t$ at each of its points is $t$. We show that if a small $t$-semiarc ${\cal S}_t$ in $\mathrm{PG}(2,q)$ has a large collinear subset ${\cal K}$, then the tangents to ${\cal S}_t$ at the points of ${\cal K}$ can be blocked by $t$ points not in ${\cal K}$. 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.