#### Vol. 14, No. 1, 2015

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

### Bence Csajbók, Tamás Héger and György Kiss

Vol. 14 (2015), No. 1, 1–26
##### Abstract

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

##### Keywords
finite plane, semiarc, semioval, blocking set, Sz”onyi–Weiner Lemma
##### Mathematical Subject Classification 2010
Primary: 51E20, 51E21