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 $\theta$ . Let n be an integer $\geq 1$ 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 $e_0,e_1,\ldots,e_n$ be a basis of V. If a point p in the projective space has homogeneous coordinates $(X_0,\ldots,X_n)$ with respect to this basis, it is on the Hermitian variety if and only if :

$\sum_{i,j = 0}^{n} a_{ij} X_{i} X_{j}^{\theta} =0$

where $a_{i j}=a_{j i}^{\theta}$ and not all $a_{ij}=0$

If one constructs the Hermitian matrix A with $A_{i j}=a_{i j}$ , the equation can be written in a compact way :

$X^t A X^{\theta}=0$

where $X= \begin{bmatrix} X_0 \\ X_1 \\ \vdots \\ X_n \end{bmatrix}.$

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.