Timeline for Products of projective spaces
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2021 at 20:26 | comment | added | Edwin Beggs | The target would have to be altered - something like putting in $\mathbb{CP}^1$ instead of $[0,1]$. | |
| Jun 15, 2021 at 20:17 | comment | added | Gregory Arone | 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. | |
| Jun 15, 2021 at 19:42 | comment | added | Edwin Beggs | 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. | |
| Jun 15, 2021 at 18:20 | comment | added | Gregory Arone | 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: $$\mathbb CP^{m+n-1}\cong S^{2(m+n)-1}/_{S^1}\xrightarrow{\cong} (S^{2m-1}*S^{2n-1})/_{S^1}\to S^{2m-1}*S^{2n-1}/_{S^1\times S^1}\cong S^{2m-1}/_{S^1}*S^{2n-1}/_{S^1}\cong \mathbb CP^{m-1}*\mathbb CP^{n-1}$$ | |
| Jun 15, 2021 at 18:13 | comment | added | Edwin Beggs | 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... | |
| Jun 15, 2021 at 16:52 | comment | added | Gregory Arone | 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? | |
| Jun 15, 2021 at 12:41 | history | asked | Edwin Beggs | CC BY-SA 4.0 |