Polar spaces having some line of cardinality two

Abstract

The nonthick geometries of type C n and D n or equivalently all polar spaces having at least one line of cardinality 2 are classified. It turns out that there are two classes of such polar spaces. On the one hand, decomposable polar spaces or polar spaces which are direct sums of two or more polar spaces are obtained. On the other hand, polar spaces arising from the interval lattice of an irreducible projective geometry which can also be seen as being partitioned by a pair of disjoint maximal singular subspaces can be gotten.