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 $\sum_i \deg(V_i)$?