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determinantal variety
In algebraic geometry, determinantal varieties are spaces of matrices with a given upper bound on their ranks. Their significance comes from the fact that many examples in algebraic geometry are of this form, such as the Segre embedding of a product of two projective spaces.
==Definition==
Given ''m'' and ''n'' and ''r'' < min(''m'', ''n''), the determinantal variety ''Y'' ''r'' is the set of all ''m'' × ''n'' matrices (over a field ''k'') with rank ≤ ''r''. This is naturally an algebraic variety as the condition that a matrix have rank ≤ ''r'' is given by the vanishing of all of its (''r'' + 1) × (''r'' + 1) minors. Considering the generic ''m'' × ''n'' matrix whose entries are algebraically independent variables ''x'' ''i'',''j'', these minors are polynomials of degree ''r'' + 1. The ideal of ''k''() generated by these polynomials is a determinantal ideal. Since the equations defining minors are homogeneous, one can consider ''Y'' ''r'' either as an affine variety in ''mn''-dimensional affine space, or as a projective variety in (''mn'' − 1)-dimensional projective space.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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