A new characterization of projections of quadrics in finite projective spaces of even characteristic

We will classify, up to linear representations, all geometries fully embedded in an affine space with the property that for every antiflag {p,L}{p,L} of the geometry there are either 0, αα, or qq lines through pp intersecting LL. An example of such a geometry with α=2α=2 is the following well known geometry HTn. Let Qn+1Qn+1 be a nonsingular quadric in a finite projective space PG(n+1,q), n≥3n≥3, qq even. We project Qn+1Qn+1 from a point r∉Qn+1r∉Qn+1, distinct from its nucleus if n+1n+1 is even, on a hyperplane PG(n,q) not through rr. This yields a partial linear space HTn whose points are the points pp of PG(n,q), such that the line 〈p,r〉〈p,r〉 is a secant to Qn+1Qn+1, and whose lines are the lines of PG(n,q) which contain qq such points. This geometry is fully embedded in an affine subspace of PG(n,q) and satisfies the antiflag property mentioned. As a result of our classification theorem we will give a new characterization theorem of this geometry.