The Veldkamp Space Of W(Q)

The geometric hyperplanes of generalized quadrangles and their pairwise intersections are studied using combinatorial techniques. Particular attention is paid to the family of regular generalized quadrangles W(q), when q is a power of an odd prime number. In this case, the geometric hyperplanes of W(q) are exactly the perps of points of W(q), and the Veldkamp space V(W(q)) is shown to be a linear space whose points are the perps of points of W(q) and whose lines have cardinality q + 1.