# Codimension of a subset of $Hom(E,F)$

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?

## 1 Answer

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.