Hermitian variety

Hermitian varieties are in a sense a generalisation of quadrics, and occur naturally in the theory of polarities. 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 θ {displaystyle heta } . Let n be an integer ≥ 1 {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 e 0 , e 1 , … , e n {displaystyle e_{0},e_{1},ldots ,e_{n}} be a basis of V. If a point p in the projective space has homogeneous coordinates ( X 0 , … , X n ) {displaystyle (X_{0},ldots ,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 {displaystyle sum _{i,j=0}^{n}a_{ij}X_{i}X_{j}^{ heta }=0} where a i j = a j i θ {displaystyle a_{ij}=a_{ji}^{ heta }} and not all a i j = 0 {displaystyle a_{ij}=0} If one construct the Hermitian matrix A with A i j = a i j {displaystyle A_{ij}=a_{ij}} , the equation can be written in a compact way : X t A X θ = 0 {displaystyle X^{t}AX^{ heta }=0} where X = [ X 0 X 1 ⋮ X n ] . {displaystyle X={egin{bmatrix}X_{0}\X_{1}\vdots \X_{n}end{bmatrix}}.}