On the intersection of a Hermitian surface with an elliptic quadric

Abstract

We investigate the intersection between the generalized quadrangle arising from a Hermitian surface H(3, q2) and an elliptic quadric Q-(3, q2) of PG(3, q2). In odd characteristic we determine the possible intersection sizes between H(3, q2) and Q-(3, q2) under the hypothesis that they share the same tangent plane at a common point. When the characteristic is even, we determine the configuration arising from the intersection of H(3, q2) and Q-(3, q2), provided that the generators of H(3, q2) that are tangents with respect to Q-(3, q2) are the extended lines of a symplectic generalized quadrangle W(3, q) embedded in H(3, q2). As a by-product, new infinite families of hyperovals on H(3, q2) are constructe

Intersections of the Hermitian Surface with Irreducible Quadrics in Even Characteristic

“…In [1], we determined the possible intersection numbers between Q and H in PG(3, q 2 ) under the assumption that q is an odd prime power and Q and H share at least one tangent plane. The same problem has been studied independently also in [6] for Q an elliptic quadric; this latter work contains also some results for q even.…”

“…In the next three lemmas we denote by Ξ the quadric of PG(4, q) of equation (6), whereas by Ξ ∞ its section at infinity that is, the quadric of PG(3, q) of equation (7). Lemma 3.4.…”

“…Then, N = q 2 + q + 1 − (q + 1) = q 2 . Now we are going to use the same group theoretical arguments as in [1, Lemma 2.3] in order to be able to fix the values of some of the parameters in (6) without losing in generality. If Q is a cone, we can assume without loss of generality:…”