The hyperplanes of DW(5,2h)DW(5,2h) which arise from embedding

We show that there are six isomorphism classes of hyperplanes of the dual polar space Δ=DW(5,2h)Δ=DW(5,2h) which arise from the Grassmann-embedding. If h≥2h≥2, then these are all the hyperplanes of ΔΔ arising from an embedding. If h=1h=1, then there are 6 extra classes of hyperplanes as has been shown by Pralle [H. Pralle, The hyperplanes of DW(5,2)DW(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 DW(5,q)DW(5,q), qq 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 DW(5,q),qDW(5,q),q odd, Michigan Math. J. (in press)].