I'm trying to prove that variety $$X=\{(A,C,B)\vert AC=CA=BC=CB=0, \text{rk}(C)\leq1, \det(A)=\det(B)=0\}$$ is irreducible. Here $C$ considered up to multiplication by constant i.e. projectivisation of variety of matrices with rank $\leq 1$.
I want to apply irreducibility criteria from Vakil's note. So I need to show that projection map $$\text{Mat}_{n\times n}\times \mathbb{P}(\text{Mat}_{n\times n})\times \text{Mat}_{n\times n}\longrightarrow \mathbb{P}(\text{Mat}_{n\times n})$$ projecting $(A,C,B)\longrightarrow C$ is of proper type. Alternatively I need to show that this map is finite, hence proper. Another book states that this map should be closed. But i haven't come up with any idea yet.
Probably useful: $f$ is of proper type iff $f$ can be decomposed to closed embedding and projection $X\longrightarrow P\times Y \longrightarrow Y$. I have projection, so probably should do something with closed embedding of matrices.
Probably the answer I was looking for. Consider $\mathbb{P}(\text{Mat}_{n\times n}\times \text{Mat}_{n\times n})\times \mathbb{P}(\text{Mat}_{n\times n})$. Given equations defines projective variety in this space. And now projection is a map from projective variety hence proper. Which means that projective variety is irreducible. Our initial variety $X$ (X is quasi projective as variety in product of affine and projective spaces) is an open subset of irreducible variety hence irreducible by Irreducibility of an Affine Variety and its Projective Closure.