Timeline for glueing flat families of objects over a blow-up
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 23, 2013 at 21:58 | comment | added | Will Sawin | I'm not sure. It may depend on exactly what sort of moduli space it is. | |
| Jan 23, 2013 at 15:47 | comment | added | IMeasy | @will again: but if the classifying map on E is constant, then I can glue them, right? | |
| Jan 23, 2013 at 14:41 | comment | added | IMeasy | There's something related to the fine-ness of the moduli space going on here. In the case I am considering, the locus I blow up is the non semi-stable non-fine locus. | |
| Jan 20, 2013 at 19:31 | comment | added | Will Sawin | Probably unique is a better word. For instance, if i have two finite set bundles (etale covers) over a curve, that are isomorphic on that curve minus a point, they are isomorphic over that point. Moreover the isomorphism extends to an isomorphism over the whole curve, which is unique. On the other hand, if I have two vector bundles over a curve that are isomorphic everywhere but a point, though they are trivially isomorphic over that curve, that isomorphism does not necessarily extend to the whole vector bundle. | |
| Jan 20, 2013 at 15:10 | comment | added | IMeasy | By $\mathbb{F}_1$ I meant the first Hizerbruch surface, namely the blow up of $P^2$. In my case the fibers are isomorphic, yes. In what sense do you mean "natural"? | |
| Jan 19, 2013 at 22:54 | comment | added | Will Sawin | Just because there do exist nontrivial gluings does not imply that nice conditions that guarantee their existence can be defined. | |
| Jan 19, 2013 at 22:53 | comment | added | Will Sawin | I guess in your case you know the fibers are isomorphic, but the question is whether the isomorphism is natural? | |
| Jan 19, 2013 at 22:52 | comment | added | Will Sawin | Well I haven't studied this sort of thing so I don't fully understand your description of this moduli space, but I think the argument to prove whatever it is that's true goes like this: Take a line on $B$ that passes through some point on $E$. Look at its image in $M$. We can define two families on it - the restriction of the glued bundle from $E$, and the pullback from $M$. These families are the same except at one point. So the key question is, how different can they be at that point? For projective flat families, for instance, they must be the same. | |
| Jan 19, 2013 at 22:31 | comment | added | IMeasy | In fact, in my own case, the family over $E$ is given by strictly semistable bundles. The whole family is sent to one S-equivalence class in the semi-stable boundary of the moduli space, so it is constant (even if the moduli space is not fine). So this may well be one of the "constant" cases that are outisde of the hypotheses of your answer. | |
| Jan 19, 2013 at 22:17 | comment | added | Will Sawin | What's $\mathbb F_1$ in this context? | |
| Jan 19, 2013 at 21:55 | comment | added | IMeasy | Thank you, Will. But there must be some kind of conditions, because ecamples exist. For instance, I don't think that any family of curves parametrized by $\mathbb{F}_1$ is constant along the $(-1)$-curve. | |
| Jan 19, 2013 at 19:22 | history | answered | Will Sawin | CC BY-SA 3.0 |