Small semiovals in PG(2, q)

Abstract

A semioval in a projective plane is a nonempty subset S of points with the property that for every point P ∈ S there exists a unique line such that S ∩ = {P }. It is known that q + 1 ≤ |S| ≤ q √ q + 1 and both bounds are sharp. We say that S is a small semioval in if |S| ≤ 3(q + 1).Dover [5] proved that if S has a (q − 1)-secant, then 2q − 2 ≤ |S| ≤ 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. (2000): 51E21. Mathematics Subject Classification

On the Structure of Semiovals of Small Size

“…As corollary, with we obtain the following (see Corollary 3.1 of ): Corollary Let be a semioval in the Desarguesian plane PG(2, q ). If and has a ‐secant, then is a subset of a vertexless triangle.…”

“…In the following theorem is proven by algebraic methods (Theorem 1.1 of ). Theorem Let be a semioval in the Desarguesian plane PG(2, q ).…”