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 = θ_2s−1 and if through each point ofK there are exactlyq ^2(s−1) tangent lines and at most θ_2s−3 n-secant lines, thenK is a non singular quadric of PG(2s,q). Ifr = 2s−1 (s≥2),k=θ_2(s−1) +q ^s−1 and if at each point ofK there are exactlyq ^2s−3 −q ^s−2 tangents and at most θ_2(s−2)+q ^s−2 n-secant lines, thenK is a hyperbolic quadric of PG(2s−1,q).