# Degree of the variety of independent matrices of rank $\leq r$?

Consider an $$m$$-by-$$n$$ matrix $$A$$ with entries in a field $$k$$; we can see $$A$$ as a point in the affine space $$\mathbb{A}^{m n}$$. The rank of $$A$$ will be $$\leq r$$ (where $$r<\min(m,n)$$) if and only if every $$(r+1)$$-by-$$(r+1)$$ minor of A is $$0$$. That tells us that the set of all such matrices $$A$$ forms a variety $$V$$. Moreover, by higher-dimensional Bézout, we obtain a bound on the sum of the degrees of the components $$V_i$$ of $$V$$: it is at most $$(r+1)^{k_{r,m,n}},$$ where $$k_{r,m,n} = \binom{n}{r+1} \binom{m}{r+1}$$ is the number of $$(r+1)$$-by-$$(r+1)$$ minors.

Unfortunately, that's quite a large upper bound. Is there another way to characterize $$V$$, resulting in a better upper bound for $$\sum_i \deg(V_i)$$?

The affine variety you described are called determinantal rings, and just about everything is known about them: dimension, singular loci, etc. The degree is also called the Hilbert-Samuel multiplicity, and it was computed as determinant of the matrix $$a_{ij}= \binom{m+n-i-j}{m-i}$$ for $$i,j=1,...,r$$. See this paper.

One can compute the degree explicitly, by using a natural resolution of singularities of the associated projective variety. Indeed, to give a matrix of a rank $$r$$ one needs to fix an $$r$$-dimensional subspace of $$k^n$$, i.e., a point of $$Gr(r,n)$$, and a surjective map from $$k^m$$ to this subspace, i.e., a full-rank vector in the fiber of the vector bundle $$k^m \otimes U \cong U^{\oplus m}$$, where $$U$$ is the tautological bundle of the Grassmannian. If we relax the surjectivity (full-rank) condition and projectivize, we obtain a map $$P_{Gr(r,n)}(U^{\oplus m}) \to P^{mn-1},$$ which is birational onto the projectivization of the variety of matrices of rank at most $$r$$. Now, if $$H$$ is the relative hyperplane class for the projective bundle, the required degree is equal to $$H^N$$, where $$N = r(n-r) + mr - 1 = r(n + m - r) - 1$$ is the dimension. The integer $$H^N$$ can be explicitly computed using the Grothendieck description of the cohomology of a projective bundle and Schubert calculus on $$Gr(r,n)$$.