Timeline for Is there a stable reduction for a family of vector bundles?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 2, 2022 at 14:26 | vote | accept | AG learner | ||
| Jan 2, 2022 at 7:57 | comment | added | Sasha | I am not sure how to answer this. Maybe the reason is that the authomorphism group of a stable bundle is $\mathbb{G}_{\mathrm{m}}$, hence connected, while in the curve case it is not. | |
| Jan 1, 2022 at 19:48 | comment | added | AG learner | Got it! So it turns out that we don’t need any base change to modify the family and get the semi-stable sheaf on the special fiber. Is there a reason why there is no local obstruction to get a “local universal family”? For stable reduction of a family of curves (i.e., family of plane curves degenerate to a cuspidal curve), base change is usually needed since the local monodromy is non-trivial. | |
| Jan 1, 2022 at 19:32 | comment | added | Sasha | This is the composition of the canonical morphism $E \to i_*i^*E$ with the pushforward of the morphism $i^*E = E_0 \to E_0/F_0$. | |
| Jan 1, 2022 at 19:20 | comment | added | AG learner | Thanks, Sasha. Maybe this is trivial, but I'm confused on why there is a surjective morphism $E\to i_*(E_0/F_0)$ (even for the simplest case $E$ is constant structure sheaf $\mathcal{O}_{X\times B}/\pi^{*}\mathcal{O}_B$ over $B$, if it surjects onto the torsion sheaf $i_*\mathcal{O}_X$, then it seems to send sections to something discontinuous.) | |
| Jan 1, 2022 at 9:16 | history | answered | Sasha | CC BY-SA 4.0 |