Timeline for blow-up proper varieties to projective ones
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 30, 2011 at 18:42 | comment | added | Karl Schwede | shenghao, I tried to write up some details on this, let me know if things make sense. I still don't have any good ideas on how to make the blow-ups at smooth centers. I'll let you know if I have any thoughts. | |
| Jan 30, 2011 at 18:40 | answer | added | Karl Schwede | timeline score: 3 | |
| Jan 30, 2011 at 3:22 | comment | added | Karl Schwede | shenghao, I agree, the restriction of $I$ to $U_i$ is not $J_i$ anymore, but the blow-up will still dominate the blow-up of $J_i$. (The blow-up of a product of ideals behaves in quite controllable ways, in particular, it always dominates the blow-up of each ideal in the product). | |
| Jan 30, 2011 at 3:18 | comment | added | Karl Schwede | shenghao, I agree, it is not $V(J_i)$ anymore, but the blow-up will still dominate the blow-up of $J_i$. (The blow-up of a product of ideals behaves in quite controllable ways, in particular, it always dominates the blow-up of each ideal in the product). | |
| Jan 29, 2011 at 23:55 | comment | added | shenghao | Dear Karl: when extending the ideal sheaf J_i to I_i I guess we are taking the closure of V(J_i) in X, and this will introduce new points in other opens U_j. So, I don't think the restriction of V(I) to U_i is V(J_i) anymore. | |
| Jan 29, 2011 at 22:20 | comment | added | Karl Schwede | Sorry, the $U_i$ should have been an affine cover. | |
| Jan 29, 2011 at 19:11 | comment | added | Karl Schwede | Actually, why doesn't the following work: $f : Z \to X$ is projective and birational and $U_i$ is a cover of $X$ with $V_i = f^{-1}(U_i)$. Each $f_i : V_i \to U_i$ is projective and birational and so the blow-up of some ideal $J_i \subset O_{U_i}$. We can find ideal an sheaf $I_i \subset O_{X}$ which restricts to $J_i$. Then consider the blow-up of $I = \prod I_i$. This should be the same as the blow-up of $f^{-1}(I) \cdot \O_Z$ on $Z$. | |
| Jan 29, 2011 at 19:05 | comment | added | Karl Schwede | shenghao, thanks, I didn't realize the problem was the quasi-projectivity. I was using II, 7.17 as you said, but I guess I hadn't realized that one needed the quasi-projectiveness (you can't glue the ideals you construct together for instance). | |
| Jan 29, 2011 at 18:49 | comment | added | shenghao | Dear Karl: sorry, I sort of misunderstood your comment. Maybe you were using Hartshorne II 7.17, which, I'm afraid, does not apply since the variety X I am considering is not quasi-projective. It'll be great if we could impose additional conditions on the centers. For instance in char. 0, one may take smooth centers (Moishezon). | |
| Jan 29, 2011 at 17:34 | comment | added | shenghao | The map Z --> X is indeed projective (I also mentioned it in the question). But I don't think every projective map is a sequence of blow-ups: e.g. a projective space bundle; not true even for projective birational maps: at least we need the inverse image of the locus where the map is not an isomorphism to be a divisor. The center being reduced or not is not an issue: as long as two ideals are "cofinal". | |
| Jan 29, 2011 at 17:07 | comment | added | Karl Schwede | shenghao, I'm confused. We have $Z \to X \to k$ which we know is projective. $X \to k$ is also proper and so separated. Does this not imply that $Z \to X$ is projective - and so a blow-up of some ideal? See Exercise 4.8(e) in Hartshorne. Do you intend that this blow-up should be a sequence of at some reduced centers or something? | |
| Jan 29, 2011 at 12:17 | history | asked | shenghao | CC BY-SA 2.5 |