Timeline for dualizing sheaf of a nodal curve
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2011 at 9:10 | comment | added | HNuer | Dear Matt, as often seems to happen, I realized my error like ten minutes after I asked that question. I always get confused which sign to put inside $\mathcal O(D)$ to get what I want. Thanks for the hint how to prove the residue theorem, I think I have it. | |
| Mar 16, 2011 at 5:39 | comment | added | Emerton | Dear HNuer, Regarding $\Omega^1_{\tile{C}}(x+y)$: if $\mathcal F$ is any sheaf and $D$ is a divisor, then $\mathcal F(D)$ usually denotes sections with poles along $D$. This is the convention I am following. (In particular, $\mathcal O(D)$ --- which Hartshorne instead calls $\mathcal L(D)$ --- is not the ideal sheaf of $D$, but rather its dual.) As for the stronger statement, try proving it by Riemann--Roch plus Serre duality. (I don't know a reference off the top of my head, but the proof isn't too hard.) Regards, Matt | |
| Mar 16, 2011 at 5:24 | comment | added | HNuer | Dear Matt, I am indeed unfamiliar with that statement, and only know of the result I quoted from Hartshorne's brief discussion in III.7 of his book. Do know of a reference for the stronger statement? Also, in you answer above, isn't $\Omega^1_{\tilde{C}}(x+y)$ the differentials with zeroes at x and y? Wouldn't $\Omega^1_{\tilde{C}}(-x-y)$ be the sheaf of differentials with poles there? This may just be a stupid question, but I want to make sure I understand. Thanks for all the help. | |
| Mar 16, 2011 at 0:15 | comment | added | Emerton | Dear HNuer, I am referring to the stronger result which says that summing to zero is the only obstruction for finding a differential on a smooth projective curve with at worst simple poles with prescribed residues at some finite set of points (and holomorphic everywhere else). If you aren't already familiar with this statement, I'll leave it as an exercise. Best wishes, Matt | |
| Mar 16, 2011 at 0:12 | comment | added | Emerton | Dear Georges, Thank you for the correction and for the kind words. Best wishes, Matt | |
| Mar 16, 2011 at 0:12 | history | edited | Emerton | CC BY-SA 2.5 |
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| Mar 15, 2011 at 22:15 | vote | accept | HNuer | ||
| Mar 15, 2011 at 21:58 | comment | added | HNuer | How does the residue theorem imply the existence of differentials with non-zero residues? I thought it just implied that the sum of the residues of any meromorphic differential was zero. | |
| Mar 15, 2011 at 21:06 | comment | added | Georges Elencwajg | Dear Matt, in the middle of the displayed short exact sequence in the tenth line, I think you meant $\omega_C$ rather than $\omega_{\tilde C}$. Let me use the occasion to thank you for the recurring pleasure of reading your always lucid and beautifully written posts. | |
| Mar 15, 2011 at 19:14 | history | answered | Emerton | CC BY-SA 2.5 |