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 $\mathbf{A}^{mn} \times \mathbf{Gr}(r,m)$ over $\mathbf{A}^{mn}$ where $\mathbf{Gr}(r,m)$ is the Grassmannian of r-planes in an m-dimensional vector space, and consider the subspace $Z_r = \{ (A, W) \mid A(k^n) \subseteq W \}$, which is a desingularization of Y_r (over the open set of matrices with rank exactly r, this map is an isomorphism), and Z_r is a vector bundle over $\mathbf{Gr}(r,m)$ which is isomorphic to $\mathrm{Hom}(k^n, \mathcal{R})$ where $\mathcal{R}$ is the tautological bundle over the Grassmannian. So dim Y_r = dim Z_r since they are birationally equivalent, and $\dim Z_r = \dim \mathbf{Gr}(r,m) + nr = r(m-r) + nr$ since the fiber of $\mathrm{Hom}(k^n, \mathcal{R})$ has dimension nr.

The above shows that the matrices of rank <r contains the singular locus of Y_r, 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 $G = \mathbf{GL}(m) \times \mathbf{GL}(n)$, a product of general linear groups. The problem of determining the syzygies of Y_r, 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).

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, 202–237.
• 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.