Families of quadratic sets on the Klein quadric

Abstract

Consider the Klein quadric $Q^+(5,q)$ in $\text{ PG }(5,q)$. A set of points of $Q^+(5,q)$ is called a quadratic set if it intersects each plane $\pi$ of $Q^+(5,q)$ in a possibly reducible conic of $\pi$, i.e. in a singleton, a line, an irreducible conic, a pencil of two lines or the whole of $\pi$. A quadratic set is called good if at most two of these possibilities occur as $\pi$ ranges over all planes of $Q^+(5,q)$. Good quadratic sets can come into 15 possible types and in earlier work we already discussed 11 of these types. The present paper is devoted to the four remaining types. We will describe several infinite families of good quadratic sets of $Q^+(5,q)$. This will show that there are examples of quadratic sets for each of these four types and for each value of the prime power q.