Double k-sets in symplectic generalized quadrangles

Abstract

In classical projective geometry, a double six of lines consists of 12 lines ℓ ₁, ℓ ₂, . . . , ℓ ₆, m ₁, m ₂, . . . , m ₆ such that the ℓ _i are pairwise skew, the m _i are pairwise skew, and ℓ _i meets m _j if and only if i ≠ j. In the 1960s Hirschfeld studied this configuration in finite projective spaces PG(3, q) showing they exist for almost all values of q, with a couple of exceptions when q is too small. We will be considering double-k sets in the symplectic geometry W(q), which is constructed from PG(3, q) using an alternating bilinear form. This geometry is an example of a generalized quadrangle, which means it has the nice property that if we take any line ℓ and any point P not on ℓ, then there is exactly one line through P meeting ℓ. We will discuss all of this in detail, including all of the basic definitions needed to understand the problem, and give a result classifying which values of k and q allow us to construct a double k-set of lines in W(q).