This is a question about projective spaces which is either well known or totally misconceived, and it would be nice to know which. It arose from looking at the pure state spaces on finite dimensional $C^*$-algebras, which is why it is initially written using norm one notation (with corresponding quotient by $U_1$). The idea is to relate $\mathbb{CP}^{n+m-1}$ to a product of $\mathbb{CP}^{n-1}$ and $\mathbb{CP}^{m-1}$. This is of course impossible by using a cartesian product, the problem is how to modify it.

Write a nonzero vector $v\in \mathbb{C}^{n+m}$ with $|v|^2=1$ as $v=(u,w)$ for $u\in \mathbb{C}^{n}$ and $w\in \mathbb{C}^{m}$. There is a map $$\mathbb{CP}^{n+m-1}\to \frac{\mathbb{CP}^{n-1}\times \mathbb{CP}^{m-1}\times[0,1]}{\sim}$$ which sends $$ [(u,w)] \mapsto \Big[ [u/\sqrt{|u|}],[w/\sqrt{|w|}],|u|^2\Big] $$ where the equivalence relation $\sim$ collapses one projective space at 0 and the other at 1 to avoid problems with zero vectors.

I am hoping that this is well known (with apologies to experts in the field), and that there is an algebraic version of the construction which is better behaved (this map is certainly neither injective, as it loses the relative phase of $u$ and $w$, nor is it smooth). I actually need the result on finitely many factors, but this should work if two factors work.

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