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 
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 
be a basis of V. If a point p in the projective space has homogenous coordinates 
with respect to this basis, it is on the Hermitian variety if and only if :
where 
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 
[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. |
