Semi — Quadratic sets in projective spaces

The purpose of this paper is to characterize semi-quadrics in projective spacesP of finite dimension 2 at least. A concept of semi-quadratic set inP is introduced: a semi-quadratic setQ inP is essentially a set of points ofP such that the union of all tangent lines at each pointp ofQ is either a hyperplane ofP orP itself. (A tangent line ofQ atp is a line contained inQ or meetingQ exactly inp). The main result is that a semi-quadratic set which is invariant under “many” perspectivities is a semi-quadric.