Semiarcs with long secants

Abstract

In a projective plane ${\Pi }_q$ of order $q$, a non-empty point set ${\mathcal{S}}_t$ is a $t$-semiarc if the number of tangent lines to ${\mathcal{S}}_t$ at each of its points is $t$. If ${\mathcal{S}}_t$ is a $t$-semiarc in ${\Pi }_q$, $t<q$, then each line intersects ${\mathcal{S}}_t$ in at most $q+1-t$ points. Dover proved that semiovals (semiarcs with $t=1$) containing $q$ collinear points exist in ${\Pi }_q$ only if $q<3$. We show that if $t>1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\ge \sqrt{q-1}$. In $\mathrm{PG}(2,q)$ we prove the lower bound $t\ge (q-1)/2$, with equality only if ${\mathcal{S}}_t$ is a blocking set of Rédei type of size $3(q+1)/2$. We call the symmetric difference of two lines, with $t$ further points removed from each line, a $V_t$-configuration. We give conditions ensuring a $t$-semiarc to contain a $V_t$-configuration and give the complete characterization of such $t$-semiarcs in $\mathrm{PG}(2,q)$.

12 pages