Timeline for Linear projection from a point preserves flatness
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 12, 2019 at 13:40 | comment | added | Chen | @JasonStarr Thanks. I was interested in the case $S$ is non-reduced. | |
| Feb 12, 2019 at 13:15 | comment | added | Jason Starr | "... will there exist a non-empty open subset $U$ of $S$ such that $\mathcal{Y}_U$ ... is flat over $U$?" I am not certain what you are asking. If $S$ is a reduced Noetherian scheme, that is true by Grothendieck's Generic Flatness Theorem. However, if $S$ is nonreduced, that can fail. | |
| Feb 12, 2019 at 13:08 | comment | added | Chen | @JasonStarr Can we comment about generic flatness i.e., will there exist a non-empty open subset $U$ of $S$ such that $\mathcal{Y}_U$ (the preimage of $U$ under the natural morphism from $\mathcal{Y}$ to $S$) is flat over $U$? | |
| Feb 12, 2019 at 11:33 | comment | added | Jason Starr | Linear projection does not always preserve flatness. Images of morphisms do not always preserve flatness. The reduced image of a linear projection can have strictly lower degree than the domain. This forces the linear projection to have non-reduced structure if you want to preserve flatness. The non-reduced structure depends on more than the fiber, and this quickly leads to examples where the linear projection is not flat. | |
| Feb 12, 2019 at 9:16 | comment | added | Laurent Moret-Bailly | As I understand it, the linear projection from a point lands into $\mathbb{P}^{n-1}_\mathbb{C}$, not $\mathbb{C}^{n-1}$. | |
| Feb 11, 2019 at 21:44 | history | asked | Chen | CC BY-SA 4.0 |