Timeline for Is the pull-back of canonical sheaf invertible (modulo torsion)?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 15, 2021 at 9:33 | vote | accept | user45397 | ||
| Oct 14, 2021 at 21:16 | comment | added | user45397 | My problem is, I cannot imagine a torsion-free sheaf of rank 1 over a smooth variety which is not invertible but if we tensor it twice it becomes invertible. I am thinking using the natural short exact sequence that you get from a torsion-free sheaf to its reflexive hull with non-trivial cokernel. I cannot imagine this cokernel vanishing if we tensor it twice. I find the language used in the literatures a bit ambiguous. In particular, a lot of the times by $f^*K_X$ they mean actually the reflexive hull of this sheaf, which is of course invertible. | |
| Oct 13, 2021 at 12:05 | comment | added | Francesco Polizzi | I mean, a torsion sheaf is supported on a proper subvariety. But $2f^*K_X=2K_Y+E$ is a line bundle, hence its torsion part is zero. So also the torsion part of $f^*K_X$ is zero. | |
| Oct 13, 2021 at 11:58 | comment | added | Francesco Polizzi | In my example, it seems to me that $f^*K_X$ is torsion-free, so the answer should be again no. Am I missing something? | |
| Oct 13, 2021 at 9:34 | comment | added | user45397 | Thanks for the answer. Can you also say something about the second question (i.e., pullback modulo torsion is invertible?) | |
| Oct 12, 2021 at 12:17 | history | answered | Francesco Polizzi | CC BY-SA 4.0 |