Small semiovals in PG(2, q)

Abstract.

A semioval in a projective plane $\prod$ is a nonempty subset S of points with the property that for every point P ∈ S there exists a unique line ℓ such that $S \cap \le = \{P\}$. It is known that $q +1 \leq \|S\| \leq q\sqrt{q} + 1$ and both bounds are sharp. We say that S is a small semioval in $\prod$ if $\|S\| \leq 3(q + 1)$.

Dover [5] proved that if S has a (q − 1)-secant, then $2q-2 \leq \|S\| \leq 3q-3$, thus S is small, and if S has more than one (q − 1)-secant, then S can be obtained from a vertexless triangle by removing some subset of points from one side. We generalize this result and prove that if there exist integers 1  ≤ t and − 1  ≤ k such that $\|S\| = 2q - t + k, 2(t +k) < q, t +4(k+1) < q$ and S has a (q − t)-secant, then the tangent lines at the points of the (q − t)-secant are concurrent. Specially when t  = 1 then S can be obtained from a vertexless triangle by removing some subset of points from one side.