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 $heta$. Let "n" be an integer $geq\; 1$ 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 vectorlines consist of isotropic points of a nontrivial 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 homogenous coordinates $(X\_0,ldots,X\_n)$ with respect to this basis, it is on the Hermitian variety if and only if :

$sum\_\{i,j\; =\; 1\}^\{n\}\; a\_\{ij\}\; X\_\{i\}\; X\_\{j\}^\{\; heta\}\; =0$

where $a\_\{i\; j\}=a\_\{j\; i\}^\{\; heta\}$ and not all $a\_\{ij\}=0$

If one construct the (Hermitian) matrix "A" with $A\_\{i\; j\}=a\_\{i\; j\}$, the equation can be written in a compact way :

$X^t\; A\; X^\{\; heta\}=0$

where

Tangent spaces and singularity

Let "p" be a point on the Hermitian variety "H". A line "L" through "p" is by definion 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.