Hermitian variety

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

[math]\displaystyle{ \sum_{i,j = 0}^{n} a_{ij} X_{i} X_{j}^{\theta} =0 }[/math]

where [math]\displaystyle{ a_{i j}=a_{j i}^{\theta} }[/math] and not all [math]\displaystyle{ a_{ij}=0 }[/math]

If one constructs the Hermitian matrix A with [math]\displaystyle{ A_{i j}=a_{i j} }[/math], the equation can be written in a compact way :

[math]\displaystyle{ X^t A X^{\theta}=0 }[/math]

where [math]\displaystyle{ X= \begin{bmatrix} X_0 \\ X_1 \\ \vdots \\ X_n \end{bmatrix}. }[/math]

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.