The hyperplanes of which arise from embedding

We show that there are six isomorphism classes of hyperplanes of the dual polar space Δ = D W ( 5 , 2 h ) which arise from the Grassmann-embedding. If h ≥ 2 , then these are all the hyperplanes of Δ arising from an embedding. If h = 1 , then there are 6 extra classes of hyperplanes as has been shown by Pralle [H. Pralle, The hyperplanes of D W ( 5 , 2 ) , Experiment. Math. 14 (2005) 373–384] with the aid of a computer. We will give a computer-free proof for this fact. The hyperplanes of D W ( 5 , q ) , q odd, arising from an embedding will be classified in the forthcoming paper [B.N. Cooperstein, B. De Bruyn, Points and hyperplanes of the universal embedding space of the dual polar space D W ( 5 , q ) , q odd, Michigan Math. J. (in press)].