Characterising the secant lines of Q(4,q), q even

Abstract

We show that a set A of lines in PG(4,q), q even, is the set of secant lines of a parabolic (non-singular) quadric if and only if A satisfies the following three conditions:(I)every point of PG(4,q) lies on 0,12q3 or q3 lines of A;(II)every plane of PG(4,q) contains 0, 12q(q+1) or q2 lines of A; and(III)every hyperplane of PG(4,q) contains 12q2(q2+1), 12q3(q+1) or 12q2(q+1)2 lines of A.