Determinantal variety

Properties

The ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of a generic square matrix, that is a matrix whose entries are distinct indeterminates. This ideal is a prime ideal in the polynomial ring in these indeterminates (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 ${\displaystyle {\mathbf{A}}^{mn}\times \mathbf{G}\mathbf{r}(r,m)}$ over ${\displaystyle {\mathbf{A}}^{mn}}$ where ${\displaystyle \mathbf{G}\mathbf{r}(r,m)}$ is the Grassmannian of r-planes in an m-dimensional vector space, and consider the subspace ${\displaystyle Z_r=\{(A,W)\mid A(k^n)\subseteq W\}}$, which is a desingularization of ${\displaystyle Y_r}$ (over the open set of matrices with rank exactly r, this map is an isomorphism), and ${\displaystyle Z_r}$ is a vector bundle over ${\displaystyle \mathbf{G}\mathbf{r}(r,m)}$ which is isomorphic to ${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(k^n,\mathcal{R})}$ where ${\displaystyle \mathcal{R}}$ is the tautological bundle over the Grassmannian. So ${\displaystyle \dim Y_r=\dim Z_r}$ since they are birationally equivalent, and ${\displaystyle \dim Z_r=\dim \mathbf{G}\mathbf{r}(r,m)+nr=r(m-r)+nr}$ since the fiber of ${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(k^n,\mathcal{R})}$ has dimension nr.

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