Family of curves

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 $\theta$. Let n be an integer $\ge 1$ 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 $e_0,e_1,\dots ,e_n$ be a basis of V. If a point p in the projective space has homogenous coordinates $(X_0,\dots ,X_n)$ with respect to this basis, it is on the Hermitian variety if and only if :

$\sum _{i,j=0}^n{a}_{ij}{X}_i{X}_j^{\theta }=0$

where $a_{ij}=a_{ji}^{\theta }$ and not all $a_{ij}=0$

If one construct the Hermitian matrix A with $A_{ij}=a_{ij}$, the equation can be written in a compact way :

$X^{t}A{X}^{\theta }=0$

where $X=\left[\begin{array}{c}{X}_0\\ X_1\\ ⋮\\ X_n\end{array}\right].$

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.

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