Characterization of quadric cones in a Galois projective space

AbstractIn an r-dimensional projective Galois space, PG(r,q), of order q, let K be a k-set of class [0,1,m,n]1, with respect to the lines. We prove that: if r=2s−1(s⩾2 and q=2,q=4 or q odd if s=2), k=θ2s−1 and there exists a point V of K through which exactly q2(s−1) 1-secant lines pass and through any other point of K pass q2s−3 1-secants, then K is a quadric cone projecting from V a non-singular quadric of a PG(2(s−1),q) skew with V; if r=2(s−1) (s⩾3),k=θ2s−3+qs−1 and there exists a point V of K through which exactly q2s−3−qs−2 1-secant lines pass and through any other point of K q2(s−2)−qs−2 1-secants pass, then K is a quadric cone projecting from V a hyperbolic quadric of a PG(2s−3,q) skew with V