Timeline for When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 17, 2022 at 14:42 | comment | added | PrimeRibeyeDeal | @WillSawin This clears things up. Thank you. | |
| Mar 17, 2022 at 11:33 | comment | added | Will Sawin | For the other way, we are lookiong for submodules of $\mathcal L_2/\mathcal L_1$, which is a sheaf supported on a curve. As long as this curve is nonempty, it is easy to see there are infinitely many submodules - e.g. the submodule of sections vanishing on some finite set of points. Only finitely many can have inverse image in $\mathcal L_2$ a line bundle. | |
| Mar 17, 2022 at 11:32 | comment | added | Will Sawin | There's no reference, but it's easy: Fix a meromorphic section $s$ of $\mathcal L_1$ and look at its divisor as a section of $\mathcal L_1, \mathcal F, \mathcal L_2$. If $\mathcal F$ is a line bundle, the divisor of $\mathcal F$ is sandwiched between the divisor of $\mathcal L_1$ and $\mathcal L_2$, but these differ on only finitely many irreducible components, so there are only finitely many possibilities. | |
| Mar 17, 2022 at 2:20 | comment | added | PrimeRibeyeDeal | @WillSawin Ok, this is interesting, and verily dashes my hopes. Can you provide a rationale or reference for that fact? | |
| Mar 17, 2022 at 0:13 | comment | added | Will Sawin | "Almost never" would be a reasonable answer. For any fixed $\mathcal L_1 \subset \mathcal L_2$, there are only finitely many line bundles between them, but usually infinitely many non-line bundles. | |
| Mar 16, 2022 at 20:55 | comment | added | PrimeRibeyeDeal | @sdr Thank you. This is along the lines of the conditions I am looking for, but unfortunately in my case the two divisors will be the same divisor $D$. I was hoping the fact that it is integral (reduced + irreducible) would be enough to draw a conclusion, but I suppose it's not. | |
| Mar 16, 2022 at 19:52 | comment | added | sdr | Here's a geometric condition. Like others, I'm not sure what you want, so I don't know if it's good for you. Note that $\mathcal{L_2}/\mathcal{L_1}$ is supported on some divisor $D$. Also, $\overline{\mathcal{L}}$ is supported on another divisor $\widetilde{D}$. It seems to me you're okay if $D$ and $\widetilde{D}$ intersect transversally in their smooth loci. | |
| Mar 16, 2022 at 19:44 | comment | added | Sasha | The question is too vague. And as @user1092847 explained, typically such $F$ is not locally free (essentially, the condition that $F$ contains a locally free subsheaf is non-restrictive). | |
| Mar 16, 2022 at 19:11 | answer | added | user1092847 | timeline score: 1 | |
| Mar 16, 2022 at 18:29 | history | asked | PrimeRibeyeDeal | CC BY-SA 4.0 |