Hermitian variety

Hermitian varieties are in a sense a generalisation of quadrics, and occur naturally in the theory of polarities.
Definition.
Let "K" be a field with an involutive automorphism formula_1. Let "n" be an integer formula_2 and "V" be an "(n+1)"-dimensional vector space over "K".
A Hermitian variety "H" in "PG(V)" is a set of points of which the representing vector lines consisting of isotropic points of a non-trivial Hermitian sesquilinear form on "V".
Representation.
Let formula_3 be a basis of "V". If a point "p" in the projective space has homogeneous coordinates formula_4 with respect to this basis, it is on the Hermitian variety if and only if :
formula_5
where formula_6 and not all formula_7
If one constructs the Hermitian matrix "A" with formula_8, the equation can be written in a compact way :
formula_9
where formula_10
Tangent spaces and singularity.
Let "p" be a point on the Hermitian variety "H". A line "L" through "p" is by definition tangent when it is contains only one point ("p" itself) of the variety or lies completely on the variety. One can prove that these lines form a subspace, either a hyperplane of the full space. In the latter case, the point is singular.