On hyperovals of polar Grassmannians

Abstract

A hyperoval of a point–line geometry is a nonempty set of points meeting each line in either 0 or 2 points. In this paper, we study hyperovals in line Grassmannians of finite polar spaces of rank 3, hereby often imposing some extra regularity conditions. We determine an upper bound and two lower bounds for the size of such a hyperoval. If equality occurs in one of these bounds, then there is an associated interesting point set of the polar space, like a tight set, an m-ovoid or a set of points having two possible intersection sizes with generators. With the aid of a computer, we have determined all hyperovals of the line Grassmannians of Q+(5,2), Q(6,2) and Q−(7,2). Several of the bounds we have found are actually tight for these geometries.