Subquadrangle m-regular systems on generalized quadrangles

摘要

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 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 a?(3, q~2), q odd, and two infinite families of symplectic subquadrangle hemisystems of _3(q~2), q even. Some sporadic examples of symplectic subquadrangle regular systems of a? H(3, q~2) are also presented.