Hermitian variety

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 X= egin{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 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.


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