A survey on semiovals

A semioval in a finite projective plane is a non-empty pointset S with the property that for every point in S there exists a unique line tP such that S ∩ tP = {P}. This line is called the tangent to S at P . Semiovals arise in several parts of finite geometries: as absolute points of a polarity (ovals, unitals), as special minimal blocking sets (vertexless triangle), in connection with cryptography (determining sets). We survey the results on semiovals and give some new proofs. 1. The beginning, semi quadratic sets and semi-ovoids Semiovals first appeared as special examples of semi-quadratic sets. Let Π be a projective space and Q = (P,L) be a pair consisting of a set P of points of Π, and a set L of lines of Π. A tangent to Q at P ∈ P is a line ` ∈ L such that P is on `, and either ` ∩ P = {P}, or ` ∈ L. Q is called semi quadratic set (SQS), if every point on a line of L belongs to P, and for all P ∈ P the union TP of all tangents to Q at P is either a hyperplane or the whole space Π. A lot of attempts were made to classify all SQS, but the problem is still open in general. For the known results about SQS we refer to [8] and [20]. An SQS Q = (P,L) is called a semi-ovoid (or semioval if dim Π = 2), if L = ∅ and P contains at least 2 points. The complete characterization of semi-ovoids was given by J. Thas [32]. Using elementary double counting arguments, he proved the following results.