Timeline for Families of ideal sheaves: What's the correct definition?
Current License: CC BY-SA 3.0
13 events
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| Oct 5, 2012 at 6:21 | comment | added | temp | I see a potential problem for the morphism $Hilb(X)\rightarrow M_I(X)$ to fail to be an isomorphism: We have $I\hookrightarrow I^{∗∗}$, but the latter may fail to be a locally free sheaf. It is a reflexive sheaf of rank one, so it is locally free in the smooth case. In general assume $X$ is normal, then $I^{**}$ is of the form $\mathcal{O}_X(D)$ for a Weil divisor $D$, which may not be Cartier. If one tries to define the determinant on the smooth locus and then pushforward, then there is no natural map from $I$ to $\det I$. | |
| Oct 2, 2012 at 1:02 | comment | added | temp | It appear to me the determinant of a coherent sheaf can be defined over a normal scheme. Just push-forward the determinant of the restriction of the coherent sheaf to the smooth locus (complement of a codimension 2 thing). And then $M_I(X)$ makes sense. Did I miss something here? | |
| Sep 19, 2012 at 17:07 | comment | added | 36min | I missed something. I think I do need the singular case. Sorry. | |
| Sep 19, 2012 at 10:08 | comment | added | MartinG | @Sasha: You are right. I did point out that I didn't know about the singular case (the OP seemed interested in the smooth case also), but it is worthwhile to note that the very definition of $M_I(X)$ via determinants applies only in the smooth case. Thanks. | |
| Sep 19, 2012 at 9:14 | comment | added | Sasha | There is a problem with the determinant map. You know, the scheme $X$ in Bridgeland's paper is NOT smooth and ideal sheaves he considers are not perfect complexes, so they don't have a locally free resolution and so their determinant is just NOT DEFINED. | |
| Sep 19, 2012 at 7:49 | history | edited | MartinG | CC BY-SA 3.0 |
New answer to edited question.
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| Sep 19, 2012 at 6:42 | comment | added | MartinG | @36min: And to answer your new question 2: With this definition, you can realize M_I as the fibre over $\mathcal{O}_X$ for the determinant map $M\to \mathrm{Pic}(X)$, where $M$ is the Simpson moduli space for stable/torsion free rank one sheaves. | |
| Sep 19, 2012 at 6:40 | comment | added | 36min | The relavent space is smooth here. Also I updated my question. | |
| Sep 19, 2012 at 6:38 | comment | added | MartinG | @36min: Yes, I meant to say that moduli of rank one sheaves with trivial determinant is more or less the standard definition. But, on autopilot, I assumed $X$ to be smooth, I guess Bridgeland does not. If singularitites are allowed, I do not know whether the natural map from the Hilbert scheme is an isomorphism. (And even for $X$ smooth it is not entirely obvious.) | |
| Sep 19, 2012 at 6:29 | comment | added | 36min | I think he may be taking the thing you mentioned in the second paragraph as definition. And the only case of interest is the moduli space of ideal sheaves that are numerically equivalent to a point in a 3-fold. (He quoted Simpson's paper a lot. That's where the definition of moduli of ideal sheaves is from.) However if the moduli space is isomorphic to the Hilbert scheme then I don't see why it is necessary to consider it. | |
| Sep 18, 2012 at 22:34 | comment | added | 36min | Now I'm more confused... In the Inventiones paper by Bridgeland, he really said "the scheme $M_I(X)$ is the moduli space of ideal sheaves on $X$". (See 7.3 of that paper.) He didn't say what that is and how it is constructed... I thought he meant some well-known thing. | |
| Sep 18, 2012 at 18:40 | history | edited | MartinG | CC BY-SA 3.0 |
Forgot: $I_Z$ is $S$-flat
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| Sep 18, 2012 at 17:04 | history | answered | MartinG | CC BY-SA 3.0 |