Hermitian varieties are in a sense a generalisation of quadric

Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

s, and occur naturally in the theory of polarities.

Definition

Let K be a field with an involutive automorphism

Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

 . Let n be an integer and V be an (n+1)-dimensional vectorspace 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 be a basis of V. If a point p in the projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 has homogenous coordinates with respect to this basis, it is on the Hermitian variety if and only if :



where and not all

If one construct 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 (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.