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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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    $\begingroup$ To me it looks like the OP is describing a map (not an embedding) from $\mathbb CP^{m+n-1}\to \mathbb CP^{m-1}*\mathbb CP^{n-1}$ where $X*Y$ denotes the join of $X$ and $Y$. The Segre embedding is an embedding $P^{m-1}\times P^{n-1}\hookrightarrow P^{mn-1}$. What is the connection? $\endgroup$ – Gregory Arone Jun 15 at 16:52
  • $\begingroup$ It is not the Segre embedding, that is (approximately) multiplicative in the dimension and this needs to be additive. The join is more like it. The problem is how to define the function... $\endgroup$ – Edwin Beggs Jun 15 at 18:13
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    $\begingroup$ There is a homeomorphism $S^{2(m+n)-1}\cong S^{2m-1}*S^{2n-1}$. You can define your map as the following composition: $$\mathbb CP^{m+n-1}\cong S^{2(m+n)-1}/_{S^1}\xrightarrow{\cong} (S^{2m-1}*S^{2n-1})/_{S^1}\to S^{2m-1}*S^{2n-1}/_{S^1\times S^1}\cong S^{2m-1}/_{S^1}*S^{2n-1}/_{S^1}\cong \mathbb CP^{m-1}*\mathbb CP^{n-1}$$ $\endgroup$ – Gregory Arone Jun 15 at 18:20
  • $\begingroup$ This is the map in the post I think. A problem is to extend this to an algebraic result (in terms of algebraic geometry), which might have a $\mathrm{CP}^1$ instead of $[0,1]$. The problem is to try to get the map to be injective (ish) and maybe smooth. $\endgroup$ – Edwin Beggs Jun 15 at 19:42
  • $\begingroup$ Notice that the dimension of the source is higher than the dimension of the target. So it is hard to see how the map can be injective. Also note that the target is definitely not a manifold. So it is hard to see how the map can be smooth. I have a hunch that the map is null-homotopic, but have not checked. $\endgroup$ – Gregory Arone Jun 15 at 20:17

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