Hermitian variety

From Wikipedia, the free encyclopedia

Hermitian varieties are in a sense a generalisation of quadrics, and occur naturally in the theory of polarities.

[edit] Definition

Let K be a field with an involutive automorphism θ. 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.

[edit] 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}^{\theta} =0

where a_{i j}=a_{j i}^{\theta} and not all aij = 0

If one construct the (Hermitian) matrix A with Aij = aij, the equation can be written in a compact way :

XtAXθ = 0

where X= \begin{bmatrix} X_0 \\ X_1 \\ \vdots \\ X_n \end{bmatrix}.

[edit] 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.

 This mathematics-related article is a stub. You can help Wikipedia by expanding it.