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. Let n be an integer and V be an -dimensional vector space over K.
A Hermitian variety H in PG 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 be a basis of V. If a point p in the projective space has homogeneous coordinates with respect to this basis, it is on the Hermitian variety if and only if :
where and not all
If one constructs the Hermitian matrix A with, the equation can be written in a compact way :
where

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 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.