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Let $E$ and $F$ be vector spaces over the field of complex numbers. Consider

$$W=\{f\in Hom(E,F)\,:\, \textrm{dim }ker(f)\geq c\}\,.$$

The claim is that $W$ is a closed subvariety of $Hom(E,F)$ of codimension $max\{0,c(m+c)\}$ where $m=\text{dim } F-\text{dim }E$. How do we prove this?

I have first been trying to prove the case when $E=F$. In this case I need to prove that $W$ is of codimension $c^2$ in $Hom(E,E)$. I need to cut down by $c^2$ equations. Are they equations of certain minors?

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Yes, essentially. If $\dim \ker f\ge c$, let $r=\dim E-c$, so $\dim\ker f \ge c$ is equivalent to $\operatorname{rank} f \le r$. Then any $r+1\times r+1$ minor of $f$ must be $0$, and conversely if every $r+1\times r+1$ minor of $f$ is $0$, then the rank of $f$ is at most $r$, so the kernel is at least $c$ dimensional.

How many minors are there? Well if $n=\dim E$, then there are $(n-r)(n+m-r)=c(m+c)$ minors, giving the desired codimension. I assume that you can handle the edge cases ofc.

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