A new graphic characterization of non singular quadrics of PG(r,q)

Abstract

LetK be ak-set of class [0, 1,m,n]1 of anr-dimensional projective Galois space PG(r, q) of orderq. We prove that: Ifr = 2s (s ≥2),k = θ2s−1 and if through each point ofK there are exactlyq2(s−1) tangent lines and at most θ2s−3n-secant lines, thenK is a non singular quadric of PG(2s,q). Ifr = 2s−1 (s≥2),k=θ2(s−1) +qs−1 and if at each point ofK there are exactlyq2s−3 −qs−2 tangents and at most θ2(s−2)+qs−2n-secant lines, thenK is a hyperbolic quadric of PG(2s−1,q).