Subquadrangle m-regular systems on generalized quadrangles

Abstract

Abstract:We introduce the notion of subquadrangle regular system of a generalized quadrangle. A subquadrangle regular system of order m on a generalized quadrangle of order (s, t) is a set R of embedded subquadrangles with the property that every point lies on exactly m subquadrangles of R. If m is one half of the total number of subquadrangles on a point, we call R a subquadrangle hemisystem. We construct two infinite families of symplectic subquadrangle hemisystems of the Hermitian surface H(3, q 2 ), q odd, and two infinite families of symplectic subquadrangle hemisystems of W 3 (q 2 ), q even. Some sporadic examples of symplectic subquadrangle regular systems of H(3, q 2 ) are also presented. q

On the intersection of a Hermitian surface with an elliptic quadric

“…(4, 2 ) acts transitively on the symplectic subquadrangles contained in H(3, 2 ), see [11], we can assume without loss of generality that W(3, ) is the symplectic subquadrangle consisting of all points of (3, ) together with the totally isotropic lines of (3, ) with respect to the alternating form:…”

“…By[11] there are 2 ( 3 + 1) symplectic subquadrangles embedded in H(3, 2 ), therefore there are 2 ( 3 + 1) 4 ( 4 − 1)/2 = 6 ( 3 + 1)( 4 − 1)/2 elliptic quadrics Q − (3, 2 ) such that the generators of H(3, 2 ) that are tangent with respect to Q − (3, 2 ) are extended lines of a general linear complex.Remark 3.6. By Theorem 3.3, the intersection between a Hermitian surface H(3, 2 ) and an elliptic ovoid Q − (3, 2 ) such that the generators of H(3, 2 ) that are tangent with respect to Q − (3, 2 ) are extended lines of a symplectic generalized quadrangle W(3, ) embedded in H(3, 2 ) consists of 3 + 1 points, and + 1 of them lie on a Baer conic C. In W(3, ) the conic C is the base locus of a pencil of quadrics of which /2 are elliptic and /2 are hyperbolic.…”