Michigan Math. J. 58 (2009) Points and Hyperplanes of the Universal Embedding Space of the Dual Polar Space DW(5, q), q Odd B. N. Cooperstein & B. De Bruyn Dedicated to the memory of Donald G. Higman 1. Introduction A partial linear rank-2 incidence geometry, also called a point-line geometry, is a pair Ŵ = ( P, L) consisting of a set P whose elements are called points and a col- lection L of distinguished subsets of P whose elements are called lines, such that any two distinct points are contained in at most one line. The point-collinearity graph of Ŵ is the graph with vertex set P where two points are adjacent if they are collinear (i.e., lie on a common line). By a subspace of Ŵ we mean a subset S of P such that, if l ∈ L and l ∩ S contains at least two points, then l ⊂ S. A sub- space S is singular if each pair of points in S is collinear—that is, if S is a clique in the collinearity graph of Ŵ. We say that ( P, L) is a Gamma space (see [13]) if, for every x ∈ P, { x }∪ Ŵ(x) is a subspace. A subspace S  = P is a geometric hyperplane if it meets every line. Let e be a positive integer, p a prime, and V a 6-dimensional vector space over the finite field F q , q = p e , equipped with a nondegenerate alternating form f. Then every vector ¯ u ∈ V is isotropic, that is, satisfies f( ¯ u, ¯ u) = 0. A subspace U of V is called totally isotropic (with respect to f) if f( ¯ u 1 , ¯ u 2 ) = 0 for all ¯ u 1 , ¯ u 2 ∈ U. Associated with (V, f) is a polar space denoted by W(5, q). Here, by a polar space we mean a point-line geometry (P, L) that satisfies the following properties: 1. (P, L) is a Gamma space and, for every point p and line l , p is collinear with some point of l (this means that p is collinear with one point or all points of l); 2. no point p is collinear with every other point; and 3. there is an integer n called the rank of (P, L) such that, if S 0 ⊂ S 1 ⊂···⊂ S k is a properly ascending chain of singular subspaces, then k ≤ n. When n = 2, (P, L) is said to be a generalized quadrangle. The points (resp. lines) of W(5, q) are the 1-dimensional (resp. 2-dimensional) subspaces of V that are totally isotropic with respect to f and where incidence is containment. In W(5, q), two points 〈¯ u 1 〉 V and 〈¯ u 2 〉 V are collinear if and only if f( ¯ u 1 , ¯ u 2 ) = 0 (i.e., iff ¯ u 1 and ¯ u 2 are orthogonal). Received April 17, 2007. Revision received June 13, 2008. The research of the second author is supported by a grant of the Fund for Scientific Research – Flanders (Belgium). 195