Timeline for Open subset of the moduli space of stable sheaves on a noetherian scheme
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 20, 2014 at 8:12 | vote | accept | User3773 | ||
| Aug 19, 2014 at 16:32 | answer | added | Sasha | timeline score: 1 | |
| Aug 19, 2014 at 11:38 | comment | added | Jason Starr | If your schemes are proper, then local freeness should be open, and I am sure it is written down somewhere in EGA. First of all, using "openness of flatness", the locus in the total space of your family where the universal coherent sheaf is flat is open. Thus its complement is closed. Since the family is proper over the base, the image in the base of this closed subset is also a closed subset. The open complement is the open subset that you want. | |
| Aug 19, 2014 at 10:45 | history | edited | User3773 | CC BY-SA 3.0 |
added 136 characters in body
|
| Aug 19, 2014 at 10:41 | comment | added | User3773 | @JasonStarr: you are right. I am sorry, maybe I'd be more precise. When I talk about stable or Hilbert polynomial, I have in mind projective schemes. But I was interested also in general about the locally free condition, so your first answer was very useful and I would be very happy if you can give me a reference for that. Thank you! P.S.:I'll edit the question to avoid misunderstanding. | |
| Aug 19, 2014 at 2:34 | comment | added | bananastack | @JasonStarr: if I may say so, I think the OP wouldn't mind assuming X to be proper (while in the meantime he/she tries to make sense of the broader question). | |
| Aug 18, 2014 at 18:47 | comment | added | Jason Starr | How do you define the Hilbert polynomial for a sheaf with non-proper support? If you scheme is non-proper, how can any (nonzero) locally free sheaf have proper support? | |
| Aug 18, 2014 at 18:14 | history | edited | User3773 | CC BY-SA 3.0 |
edited body
|
| Aug 18, 2014 at 18:14 | comment | added | User3773 | No, my scheme is not proper in general, but a positive answer in the proper case is very useful. For the definition of stable sheaves, I am using the one given in Huybrechts-Lehn "The geometry of moduli spaces", that a stable sheaf is a pure sheaf such that the reduced Hilbert polynomial of every proper subsheaf is strictly minor than its. Do you have a reference for that? Thanks! | |
| Aug 18, 2014 at 18:04 | comment | added | Jason Starr | Is your scheme proper? How are you defining stable? For coherent sheaves on a non-proper scheme, "locally free" is not always an open condition. | |
| Aug 18, 2014 at 18:00 | history | asked | User3773 | CC BY-SA 3.0 |