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As I read Huybrechts-Lehn's book on Moduli of Sheaves, it is making a claim that extensions of several (at least 2) ample vector bundles (on curves) is again ample. Somehow, I am unable to see this right away from Hartshorne's definition of ample vector bundle. Can someone either show me a proof or direct me to reference for this?

P.S. Is it necessary to assume that the base is a curve for above statement to hold?

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    $\begingroup$ This is true on any projective variety, see Lazarsfeld's Positivity in Algebraic Geometry II, Proposition 6.1.13 (ii). $\endgroup$
    – abx
    Commented Feb 26, 2016 at 5:32
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    $\begingroup$ @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). $\endgroup$
    – Damian Rössler
    Commented Feb 26, 2016 at 13:03
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    $\begingroup$ @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. $\endgroup$
    – Count Dracula
    Commented Feb 26, 2016 at 15:02
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    $\begingroup$ @chhan92: "Ample Vector Bundles" by Robin Hartshorne, Pub. Math. de l'IHÉS (1966) Vol. 29, pp 63-94. eudml.org/doc/103864 $\endgroup$
    – Jason Starr
    Commented Feb 27, 2016 at 15:16
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    $\begingroup$ @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). $\endgroup$
    – Damian Rössler
    Commented Feb 28, 2016 at 11:52

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