Semiovals from unions of conics

Abstract

A semioval in a projective plane $\pi$ is a collection of points $\mathcal{O}$ with the property that for every point $P$ of $\mathcal{O}$, there exists exactly one line of $\pi$ meeting $\mathcal{O}$ precisely in the point $P$. There are many known constructions of and theoretical results about semiovals, especially those that contain large collinear subsets.

In a Desarguesian plane $\pi$ a conic, the set of zeroes of some nondegenerate quadratic form, is an example of a semioval of size $q+1$ that also forms an arc (i.e., no three points are collinear). As conics are minimal semiovals, it is natural to use them as building blocks for larger semiovals. Our goal in this work is to classify completely the sets of conics whose union forms a semioval.