Ruled sets of plane–type $$\varvec{(m, h)_2}$$ ( m , h ) 2 in $$\mathbf{PG}\varvec{(3, q)}$$ PG ( 3 , q ) with $$\varvec{m\le q}$$ m ≤ q

Abstract

Thas (Geom Dedicata 1(2):236–240, 1973) proved that a set $${\mathcal {K}}$$ of points of $${\mathrm {PG}}(d, q)$$ intersected by any hyperplane in 1 or h points (and hyperplanes with exactly one point in $${\mathcal {K}}$$ exist) is either a line or $$d = 3$$ and $${\mathcal {K}}$$ is an ovoid. In this paper, we present two generalizations of this result in $${\mathrm {PG}}(3, q)$$ .