Hermitian variety

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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 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 $e_{0},e_{1},\ldots ,e_{n}$ 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 :

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

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

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

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

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.

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