“Hermitian variety”的意思、由来-开放百科全书

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Hermitian variety

释义

Definition
Representation
Tangent spaces and singularity

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 . 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 vector lines consisting of isotropic points of a non-trivial Hermitian sesquilinear form on V.

Representation

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

where and not all

If one construct the Hermitian matrix A with , the equation can be written in a compact way :

where

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.

1 : Algebraic varieties

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