Timeline for Determinant of the conormal bundle
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2023 at 11:35 | comment | added | Invariance | Dear @abx, nice example! Thanks a lot! | |
| Mar 2, 2023 at 10:43 | comment | added | abx | There are many! A typical example: blow up a point in a surface, you get a rational curve $E$ with $E^2=-1$, hence $N^*_E=\mathscr{O}_E(1)$, which of course has no nontrivial map to $\mathscr{O}_E$. | |
| Mar 2, 2023 at 9:55 | comment | added | Invariance | Ok thanks @abx, so in general the inclusion should fail, right? Can you provide some examples illustrating it? Thanks.. | |
| Mar 2, 2023 at 9:16 | comment | added | abx | Yes, that's exactly what I meant. | |
| Mar 2, 2023 at 9:02 | comment | added | Invariance | Hi @abx, thanks for this criterion. Here is my attempts, please point out any possible mistakes. Since $T_X$ is globally generated, we have $$\bigoplus_i \mathcal O_X \twoheadrightarrow T_X,$$ the pull back operator gives then $$\bigoplus_i \mathcal O_Y \twoheadrightarrow {T_X}|_Y\twoheadrightarrow N_Y.$$ Thus $\operatorname{det} N_Y$ is also globally generated. Selecting a section of $\operatorname{det} N_Y$ gives the desired non-canonical injection. | |
| Mar 2, 2023 at 1:41 | comment | added | LSpice |
TeX notes: $\det N_Y$ \det N_Y spaces automatically, avoiding the need for manual spacing in \text{det } N_Y. For ad hoc operators, you can use \operatorname, like \operatorname{det} if \det weren't already defined. In $\mathcal O(Y)\,|\,_Y$ \mathcal O(Y)\,|\,_Y, the subscript attaches to the thin space, not to the vertical bar. The vertical alignment in \mathcal O(Y)|_Y is slightly different (side by side: $\vphantom{\,}_Y\vphantom|_Y$). To keep the thin space, you can use $\mathcal O(Y){\,|\,}_Y$ \mathcal O(Y){\,|\,}_Y. I have edited accordingly.
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| Mar 2, 2023 at 1:34 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise title; TeX; links to comments
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| Mar 1, 2023 at 19:27 | comment | added | Jason Starr | I feel like I heard Winkelmann and Campana talk about some other cases related to Gromov's h-principle. Maybe these are cases of compact homogeneous manifolds . . . | |
| Mar 1, 2023 at 17:47 | comment | added | abx | One case where you have such an inclusion is when $X$ is a compact homogeneous manifold: then $T_X$ is globally generated, hence so are $N_Y$ and $\det N_Y$, so picking a section of $\det N_Y$ gives an injection $\det N_Y^*\hookrightarrow \mathscr{O}_Y$ (non canonical, as Jason points out). | |
| Mar 1, 2023 at 17:30 | comment | added | Invariance | Dear @JasonStarr, thanks for your correction! I have corrected the question. | |
| Mar 1, 2023 at 17:29 | comment | added | Invariance | Dear @abx, you are right! I have modified the question. The case where $r\geq 2$ is surely I want to ask. | |
| Mar 1, 2023 at 17:26 | history | edited | Invariance | CC BY-SA 4.0 |
added 72 characters in body
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| Mar 1, 2023 at 17:07 | comment | added | Jason Starr | The pullback to $Y$ of the natural inclusion $\mathcal{O}(-Y)\hookrightarrow \mathcal{O}_X$ is a zero homomorphism. So you do not have any (natural) inclusion of $N_Y^*$ into $\iota^*\mathcal{O}_X$, not even in codimension $1$. | |
| Mar 1, 2023 at 17:04 | comment | added | abx | Don't you want $\ \det N^*_Y\subset \mathscr{O}_Y$? This is what we have for $r=1$. | |
| Mar 1, 2023 at 16:52 | history | asked | Invariance | CC BY-SA 4.0 |