Timeline for Extension of ample vector bundles is ample
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11 events
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| Feb 28, 2016 at 11:52 | comment | added | Damian Rössler | @Count Dracula. Sorry I commented too quickly. The fact that the extension of an ample bundle by another ample bundle is ample is not very difficult to prove (see Jason's reference) but it is a subtle fact that in char. 0, a non-trivial extension of an ample bundle by a trivial bundle is ample. This is also proven in Lazarsfeld's book (and it false in char. p>0). | |
| Feb 27, 2016 at 15:16 | comment | added | Jason Starr | @chhan92: "Ample Vector Bundles" by Robin Hartshorne, Pub. Math. de l'IHÉS (1966) Vol. 29, pp 63-94. eudml.org/doc/103864 | |
| Feb 27, 2016 at 6:03 | comment | added | chhan92 | @JasonStarr For Hartshorne's Corollary 3.4, I am unable to locate such statement in Hartshorne's algebraic geometry book. Do you either mean Residues and Duality or some other reference by Hartshorne. Can you be more specific on this please? | |
| Feb 26, 2016 at 22:08 | comment | added | Count Dracula | @JasonStarr Yes you are right, that is better. | |
| Feb 26, 2016 at 20:22 | comment | added | Jason Starr | @CountDracula: You are correct; I commented too quickly. I looked at Proposition 6.1.13 without looking at Remark 6.1.17. Anyway, I think it is better to give a reference to a specific result in Hartshorne, namely, Corollary 3.4. Lazarsfeld does not specifically cite Corollary 3.4 for this step of the proof. | |
| Feb 26, 2016 at 20:00 | comment | added | Count Dracula | @JasonStarr This is why I referenced his Remark 6.1.17 where Lazarsfeld explicitly discusses char p. | |
| Feb 26, 2016 at 19:37 | comment | added | Jason Starr | @CountDracula: Lazarsfeld's standing hypothesis is that all schemes are over $\mathbb{C}$. Since many of his earlier preparatory arguments use Hodge theory, etc., this is a fairly important hypothesis in the book. Of course, Hartshorne's Corollary 3.4 has no such hypothesis. This is the correct reference. | |
| Feb 26, 2016 at 15:02 | comment | added | Count Dracula | @DamianRössler So you claim Lazarsfeld Remark 6.1.17 is wrong? I think you may be mistaken, because your reference is talking about extending an ample vector bundle by the trivial one. | |
| Feb 26, 2016 at 13:03 | comment | added | Damian Rössler | @abx This is no true in positive characteristic. For the general theory, see the first part of Martin-Dechamps, Propriétés de descente des variétés à fibré cotangent ample." Annales de l'institut Fourier 34.3 (1984). | |
| Feb 26, 2016 at 5:32 | comment | added | abx | This is true on any projective variety, see Lazarsfeld's Positivity in Algebraic Geometry II, Proposition 6.1.13 (ii). | |
| Feb 26, 2016 at 5:04 | history | asked | chhan92 | CC BY-SA 3.0 |