Timeline for Singularities of reflexive sheaves
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 26, 2018 at 14:19 | comment | added | User43029 | Dear @Mohan, the tangent bundle is not trivial, but locally it is. The question is not wheter or not the sheaf is trivial on the whole projective space, but how one can characterize reflexive sheaves on open sets that contain the singular locus. | |
| Mar 26, 2018 at 12:35 | comment | added | Mohan | Take the tangent bundle of the projective space. Then the singular locus as you describe it is empty, but the tangent bundle is not the sum of copies of the structure sheaf. | |
| Mar 26, 2018 at 11:45 | comment | added | User43029 | Dear @Sasha, thank you for your comment! So I just need to compute such linear conditions to have the local description of my sheaf? | |
| Mar 26, 2018 at 11:43 | comment | added | User43029 | Dear @abx by explicit I mean explicit in that open, not in the whole $\mathbb{P}^3$, e.g., a not trivial locally free sheaf is not sum of the trivial on whole $\mathbb{P}^3$, but it is in some open subset. | |
| Mar 26, 2018 at 11:41 | comment | added | User43029 | Dear @Mohan, I mean the restricted sheaf in the open is the sum of the structural sheaf of the open. This should happens by the definition of locally free sheaf, don't it? Is there a counter example? | |
| Mar 26, 2018 at 6:26 | comment | added | Sasha | A reflexive sheaf is the kernel of a morphism of locally free sheaves. Locally, you can assume the locally free sheaves to be trivial, so you can think of a reflexive sheaf as a subsheaf of $\mathcal{O}^{\oplus n}$ given by a finite number of fiberwise linear conditions. | |
| Mar 26, 2018 at 5:34 | review | Close votes | |||
| Apr 1, 2018 at 3:02 | |||||
| Mar 26, 2018 at 5:15 | comment | added | abx | What do you mean by explicit? It won't be more explicit that the description of the sheaf on the whole $\Bbb{P}^3$. | |
| Mar 26, 2018 at 1:46 | comment | added | Mohan | The line ` the restricted sheaf should (be) isomorphic to copies of the trivial (line bundle) ' is false. | |
| Mar 25, 2018 at 21:22 | history | asked | User43029 | CC BY-SA 3.0 |