Hermitian variety

Definition

Let K be a field with an involutive automorphism θ . Let n be an integer ≥ 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 , … , e n be a basis of V. If a point p in the projective space has homogeneous coordinates ( X 0 , … , X n ) with respect to this basis, it is on the Hermitian variety if and only if :

∑ i , j = 0 n a i j X i X j θ = 0

where a i j = a j i θ and not all a i j = 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 θ = 0

where X = [ X 0 X 1 ⋮ X n ] .

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.