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'''Hermitian varieties''' are in a sense a generalisation of [[quadric]]s, and occur naturally in the [[theory of polarities]]. | |||
==Definition== | |||
Let ''K'' be a field with an involutive [[automorphism]] <math>\theta</math>. Let ''n'' be an integer <math>\geq 1</math> 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 <math>e_0,e_1,\ldots,e_n</math> be a basis of ''V''. If a point ''p'' in the [[projective space]] has homogenous coordinates <math>(X_0,\ldots,X_n)</math> with respect to this basis, it is on the Hermitian variety if and only if : | |||
<math>\sum_{i,j = 0}^{n} a_{ij} X_{i} X_{j}^{\theta} =0</math> | |||
where <math>a_{i j}=a_{j i}^{\theta}</math> and not all <math>a_{ij}=0</math> | |||
If one construct the [[Hermitian matrix]] ''A'' with <math>A_{i j}=a_{i j}</math>, the equation can be written in a compact way : | |||
<math>X^t A X^{\theta}=0</math> | |||
where <math>X= \begin{bmatrix} X_0 \\ X_1 \\ \vdots \\ X_n \end{bmatrix}. </math> | |||
==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. | |||
[[Category:Algebraic varieties]] | |||
{{Algebra-stub}} | |||
Latest revision as of 20:48, 15 March 2013
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 homogenous 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.