Hermitian variety

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

Let K be a field with an involutive automorphism ${\displaystyle \theta }$. Let n be an integer ${\displaystyle \geq 1}$ 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 vector lines consisting of isotropic points of a non-trivial Hermitian sesquilinear form on V.

Let ${\displaystyle e_{0},e_{1},\ldots ,e_{n}}$ be a basis of V. If a point p in the projective space has homogeneous coordinates ${\displaystyle (X_{0},\ldots ,X_{n})}$ with respect to this basis, it is on the Hermitian variety if and only if :

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