Quadratic sets of a $3$-dimensional locally projective regular planar space

Abstract

In this paper quadratic sets of a $3$-dimensional locally projective regular planar space $(\cal S,\cal L,\cal P)$ of order $n$ are studied and classified. It is proved that if in $(\cal S,\cal L,\cal P)$ there is a non-degenerate quadratic set $\bf H$, then the planar space is either $\mathop{\rm{PG}}(3,n)$ or $\mathop{\rm{AG}}(3,n)$. Moreover in the first case $\bf H$ is either an ovoid or an hyperbolic quadric, in the latter case $\bf H$ is either a cylinder with base an oval or a pair of parallel planes.