Semiarcs with long secants

In a projective plane πq of order q, a non-empty point set St is a t-semiarc if the number of tangent lines to St at each of its points is t. If St is a t-semiarc in πq, t < q, then each line intersects St in at most q + 1-t points. Dover proved that semiovals (semiarcs with t = 1) containing q collinear points exist in πq only if q ≤ 3. We show that if t > 1, then t-semiarcs with q + 1-t collinear points exist only if t ≥ √q-1. In PG(2; q) we prove the lower bound t ≥ (q-1)/2, with equality only if St 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 Vt-configuration. We give conditions ensuring a t-semiarc to contain a Vt-configuration and give the complete characterization of such t-semiarcs in PG(2; q).