On the intersection of a Hermitian surface with an elliptic quadric

Abstract

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 constructed.