Timeline for Line bundles on fibrations
Current License: CC BY-SA 3.0
13 events
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| Aug 20, 2011 at 10:02 | history | edited | rita | CC BY-SA 3.0 |
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| Aug 19, 2011 at 15:48 | comment | added | Donu Arapura | Matt, proper is enough. | |
| Aug 19, 2011 at 15:48 | comment | added | Donu Arapura | Rita, this is a very nice answer. I vaguely recall using something similar long ago. After I checked, I found a related result and proof on p 54 of Mumford's Abelian Varieties. | |
| Aug 19, 2011 at 15:46 | comment | added | Matt | Do you use that $f$ is projective? I think the only step is the semi-continuity one which works for proper. | |
| Aug 19, 2011 at 15:03 | comment | added | rita | @Ulrich: I've edited my answer and included that condition that the fibers be reduced. | |
| Aug 19, 2011 at 15:02 | history | edited | rita | CC BY-SA 3.0 |
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| Aug 19, 2011 at 8:11 | vote | accept | The Chopper | ||
| Aug 18, 2011 at 12:40 | comment | added | naf | Thanks for the reference. Given counterexamples with non-reduced fibres or base what you say seems to be the best possible. | |
| Aug 18, 2011 at 12:10 | comment | added | rita | It is a line bundle if $h^0(X_y, M)=1$ for all $y$ by Corollary 12.9 in Hartshorne, `Algebraic geometry'. | |
| Aug 18, 2011 at 11:56 | comment | added | naf | I don't have a counterexample in nonzero characteristics but I also don't see why $f*_M$ is a line bundle without some additional assumptions. | |
| Aug 18, 2011 at 11:10 | comment | added | rita | @Ulrich: I think you are right about the reduced fibers, thank you very much. Does the characteristic really matter here? | |
| Aug 18, 2011 at 10:44 | comment | added | naf | I guess you are assuming that all fibres are reduced (and maybe also characteristic zero)? | |
| Aug 18, 2011 at 9:25 | history | answered | rita | CC BY-SA 3.0 |