On sets of odd type of and the universal embedding of the dual polar space

A set of points of the projective space P G ( n , 4 ) , n ≥ 0 , is said to be of odd type, if it intersects each line at an odd number of points. The number of sets of odd type of P G ( n , 4 ) , n ≥ 0 , is known to be equal to 2 a n , where a n = 1 3 ( n + 1 ) ( n 2 + 2 n + 3 ) . In the present paper, we give an alternative more geometric proof of this property. The additional information revealed by this proof will allow us to prove some facts regarding the hyperplanes and the universal embedding of the Hermitian dual polar space D H ( 2 n − 1 , 4 ) .