Timeline for Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces
Current License: CC BY-SA 2.5
14 events
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| Apr 21, 2010 at 15:09 | comment | added | damiano | From the definition of ACM surface, it follows that any smooth surface in characteristic zero with $h^1(O)=0$ is an ACM surface for some very ample bundle. Since this ample bundle plays no role in the subsequent part, it seems that you could simply assume that your surface has vanishing irregularity, right? | |
| Jan 19, 2010 at 2:47 | vote | accept | Hailong Dao | ||
| Dec 9, 2009 at 18:14 | comment | added | Hailong Dao | $L$ can't be trivial because of the condition on $X$. By torsion I just mean a torsion element in $Pic(X)$. | |
| Dec 9, 2009 at 18:13 | history | edited | Hailong Dao | CC BY-SA 2.5 |
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| Dec 9, 2009 at 16:13 | comment | added | Dmitri Panov | Do I understand correctly that by $\mathcal O(k))$ for all $k$ you mean powers of $O(1)$, where by $O(1)$ is some ample bundle over your surface $X$ (that you have right to chose)? This is not 100% clear from the question. And also you want your $L$ be torsion line, but not a trivial line bundle? | |
| Dec 9, 2009 at 15:58 | history | edited | Hailong Dao | CC BY-SA 2.5 |
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| Dec 9, 2009 at 14:51 | answer | added | Dmitri Panov | timeline score: 4 | |
| Dec 9, 2009 at 3:09 | comment | added | Hailong Dao | Hi Dmitri! Would you explain or give a reference for that fact? | |
| Dec 9, 2009 at 0:33 | comment | added | Dmitri Panov | What is the ground field that you consider in this question? If this is over $C$ and the surface is smooth then a globally generated torsion line bundle is trivial. | |
| Dec 6, 2009 at 1:10 | comment | added | Hailong Dao | In general, ACM varieties have all the intermediate cohomolgy vanish. They correspond to Cohen-Macaulay rings, which have maximal depth, hence the name. From both algebraic (maximal depth) and geometric (no cohomology) I think they are nice. There are no wiki entry for them though (: | |
| Dec 6, 2009 at 0:56 | comment | added | Ilya Nikokoshev | I wonder what is so interesting about ACM surfaces (I'm sure there is something if you're asking)? | |
| Dec 6, 2009 at 0:42 | history | edited | Hailong Dao | CC BY-SA 2.5 |
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| Dec 6, 2009 at 0:27 | comment | added | Andrew Critch | Some (hopefully) positive criticism: You should add (edit) some discussion to motivate your question and show some thoughts/progress you have about it, even if it is very little. This serves many purposes: 1) it gets people interested, 2) it shows you care about the problem (which I believe you do), and 3) it makes it easier for others to start working on it. | |
| Dec 5, 2009 at 22:39 | history | asked | Hailong Dao | CC BY-SA 2.5 |