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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''&nbsp;<&nbsp;min(''m'',&nbsp;''n''), the '''determinantal variety''' ''Y''<sub>&nbsp;''r''</sub> is the set of all ''m''&nbsp;&times;&nbsp;''n'' matrices (over a field&nbsp;''k'') with rank ≤&nbsp;''r''. This is naturally an [[algebraic variety]] as the condition that a matrix have rank ≤&nbsp;''r'' is given by the vanishing of all of its (''r''&nbsp;+&nbsp;1)&nbsp;&times;&nbsp;(''r''&nbsp;+&nbsp;1) [[minor (linear algebra)|minors]]. Considering the generic ''m''&nbsp;&times;&nbsp;''n'' matrix whose entries are [[algebraic independence|algebraically independent]] variables ''x''<sub>&nbsp;''i'',''j''</sub>, these minors are polynomials of degree ''r''&nbsp;+&nbsp;1. The ideal of ''k''[''x''<sub>&nbsp;''i'',''j''</sub>] generated by these polynomials is a '''determinantal ideal'''. Since the equations defining minors are homogeneous, one can consider ''Y''<sub>&nbsp;''r''</sub> either as an [[affine variety]] in ''mn''-dimensional [[affine space]], or as a [[projective variety]] in (''mn''&nbsp;&minus;&nbsp;1)-dimensional [[projective space]].
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==Properties==
 
The radical ideal defining the determinantal variety is generated by the (''r''&nbsp;+&nbsp;1)&nbsp;&times;&nbsp;(''r''&nbsp;+&nbsp;1) minors of the matrix (Bruns-Vetter, Theorem 2.10).
 
Assuming that we consider ''Y''<sub>&nbsp;''r''</sub> as an affine variety, its dimension is ''r''(''m''&nbsp;+&nbsp;''n''&nbsp;&minus;&nbsp;''r''). One way to see this is as follows: form the product space <math>\mathbf{A}^{mn} \times \mathbf{Gr}(r,m)</math> over <math>\mathbf{A}^{mn}</math> where <math>\mathbf{Gr}(r,m)</math> is the [[Grassmannian]] of ''r''-planes in an ''m''-dimensional vector space, and consider the subspace <math>Z_r = \{ (A, W) \mid A(k^n) \subseteq W \}</math>, which is a [[resolution of singularities|desingularization]] of <math>Y_r</math> (over the open set of matrices with rank exactly ''r'', this map is an isomorphism), and <math>Z_r</math> is a vector bundle over <math>\mathbf{Gr}(r,m)</math> which is isomorphic to <math>\mathrm{Hom}(k^n, \mathcal{R})</math> where <math>\mathcal{R}</math> is the tautological bundle over the Grassmannian. So <math>\dim Y_r = \dim Z_r</math> since they are birationally equivalent, and <math>\dim Z_r = \dim \mathbf{Gr}(r,m) + nr = r(m-r) + nr</math> since the fiber of <math>\mathrm{Hom}(k^n, \mathcal{R})</math> has dimension ''nr''.
 
The above shows that the matrices of rank <''r'' contains the [[Singular point of an algebraic variety|singular locus]] of <math>Y_r</math>, and in fact one has equality. This fact can be verified using that the radical ideal is given by the minors along with the [[Jacobian criterion]] for nonsingularity.
 
The variety ''Y''<sub>&nbsp;''r''</sub> naturally has an action of <math>G = \mathbf{GL}(m) \times \mathbf{GL}(n)</math>, a product of [[general linear group]]s. The problem of determining the [[Syzygy (mathematics)|syzygies]] of <math>Y_r</math>, when the [[characteristic (algebra)|characteristic]] of the field is zero, was solved by [[Alain Lascoux]], using the natural action of&nbsp;''G''.
 
==Related topics==
 
One can "globalize" the notion of determinantal varieties by considering the space of linear maps between two vector bundles on an algebraic variety. Then the determinantal varieties fall into the general study of [[degeneracy loci]]. An expression for the cohomology class of these degeneracy loci is given by the [[Thom-Porteous formula]], see (Fulton-Pragacz).
 
==References==
* Winfried Bruns, Udo Vetter, ''Determinantal rings'', Lecture Notes in Mathematics, '''1327'''. Springer-Verlag, Berlin, 1988.
* William Fulton, Piotr Pragacz, ''Schubert varieties and degeneracy loci'' Lecture Notes in Mathematics, '''1689''' Springer-Verlag, Berlin, 1998.  
* Alain Lascoux, Syzygies des variétés déterminantales, ''[[Advances in Mathematics]]'' '''30''' (1978), no. 3, 202&ndash;237.
* Ezra Miller, Bernd Sturmfels, ''Combinatorial Commutative Algebra'', Graduate Texts in Mathematics '''227''', Springer-Verlag, New York, 2005.
* Jerzy Weyman, ''Cohomology of vector bundles and syzygies'', Cambridge Tracts in Mathematics '''149''', [[Cambridge University Press]], Cambridge, 2003.
 
[[Category:Algebraic geometry]]
[[Category:Algebraic varieties]]

Latest revision as of 16:50, 20 April 2014


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