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"["x" "i","j"] 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.
Properties.
The radical ideal defining the determinantal variety is generated by the ("r" + 1) × ("r" + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).
Assuming that we consider "Y" "r" as an affine variety, its dimension is "r"("m" + "n" − "r"). One way to see this is as follows: form the product space formula_1 over formula_2 where formula_3 is the Grassmannian of "r"-planes in an "m"-dimensional vector space, and consider the subspace formula_4, which is a desingularization of formula_5 (over the open set of matrices with rank exactly "r", this map is an isomorphism), and formula_6 is a vector bundle over formula_3 which is isomorphic to formula_8 where formula_9 is the tautological bundle over the Grassmannian. So formula_10 since they are birationally equivalent, and formula_11 since the fiber of formula_8 has dimension "nr".
The above shows that the matrices of rank <"r" contains the singular locus of formula_5, 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" "r" naturally has an action of formula_14, a product of general linear groups. The problem of determining the syzygies of formula_5, when the characteristic of the field is zero, was solved by Alain Lascoux, using the natural action of "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).