Hyperplanes of Hermitian dual polar spaces of rank 3 containing a quad

Let F and F' be two fields such that F' is a quadratic Galois extension of F. If vertical bar F vertical bar >= 3, then we provide sufficient conditions for a hyperplane of the Hermitian dual polar space DH(5, F') to arise from the Grassmann embedding. We use this to give an alternative proof for the fact that all hyperplanes of DH(5, q(2)), q not equal 2, arise from the Grassmann embedding, and to show that every hyperplane of DH(5, F') that contains a quad Q is either classical or the extension of a non-classical ovoid of Q. We will also give a classification of the hyperplanes of DH(5, F') that contain a quad and arise from the Grassmann embedding.

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