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 ${\displaystyle \theta }$. Let n be an integer ${\displaystyle \geq 1}$ 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 ${\displaystyle e_{0},e_{1},\ldots ,e_{n}}$ be a basis of V. If a point p in the projective space has homogeneous coordinates ${\displaystyle (X_{0},\ldots ,X_{n})}$ with respect to this basis, it is on the Hermitian variety if and only if :

${\displaystyle \sum _{i,j=0}^{n}a_{ij}X_{i}X_{j}^{\theta }=0}$

where ${\displaystyle a_{ij}=a_{ji}^{\theta }}$ and not all ${\displaystyle a_{ij}=0}$

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

${\displaystyle X^{t}AX^{\theta }=0}$

where ${\displaystyle X={\begin{bmatrix}X_{0}\\X_{1}\\\vdots \\X_{n}\end{bmatrix}}.}$

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.