Locally singular hyperplanes in thick dual polar spaces of rank 4

We study (i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ ( 2 n , K ) , then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ ( 8 , K ) and show that every locally singular hyperplane of DQ ( 8 , K ) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.