Timeline for Projectivity of blowups
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 26, 2011 at 13:11 | comment | added | Karl Schwede | anon, no small resolutions will still be projective in most cases you write down. You are right, that the blow-up of $Z$ in $\mathbb{P}^n$ will have the strict transform you want. The problem is that if blowing up $Z$ gives a small resolution, then $Z$ had divisorial components, and wasn't just the singular locus. | |
| Jul 26, 2011 at 12:07 | comment | added | anon | In fact, maybe this embedding inside projective space trick works in general. Won't the blowup of X at Z be the proper transform of X inside the blowup of projective space at Z? In which case the ample bundle Sandor's argument provides (since projective space is Q-factorial) would work for X as well? Not sure why this wouldn't apply in the small resolution setting - are those always non-projective? | |
| Jul 26, 2011 at 11:25 | vote | accept | anon | ||
| Jul 26, 2011 at 11:25 | comment | added | anon | Nice argument - thanks! What are some other sufficient conditions for this to hold? For instance, it seems to me that no matter how singular $X$ is, blowups at smooth centers will have this property (by embedding inside the blowup of projective space at that center). | |
| Jul 26, 2011 at 2:21 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
added 2 characters in body
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| Jul 25, 2011 at 23:56 | history | answered | Sándor Kovács | CC BY-SA 3.0 |