  In a projective plane $\Pi_q$ of order $q$, a non-empty point set ${\cal
S}_t$ is a $t$-semiarc if the number of tangent lines to ${\cal S}_t$ at each
of its points is $t$. If ${\cal S}_t$ is a $t$-semiarc in $\Pi_q$, $t<q$, then
each line intersects ${\cal S}_t$ 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 $t\geq \sqrt{q-1}$. In $\mathrm{PG}(2,q)$ we
prove the lower bound $t\geq(q-1)/2$, with equality only if ${\cal S}_t$ is a
blocking set of R\'edei 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 $\mathrm{PG}(2,q)$.
