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I believe to have found a typo in Griffiths & Harris.

In the chapter on surfaces, section Rational Surfaces 1, I am trying to read the result that a holomorphic vector bundle over $\mathbb{P}^1_{\mathbb{C}}$ is a sum of invertible bundles.

What is the exact sequence that shows up at the start of his argument? Mine only has 2 terms and involves the fibres of E and H, which doesn't really make sense to me.

Any help would be appreciated.

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Consider the "term" in that exact sequence that makes no sense because it's two terms written next to each other. Now put an arrow between them. Presto---you're back on track! –  Kevin Buzzard Nov 1 '10 at 13:17
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Hi Robert, There are many many typos in Griffiths and Harris. I've only gone through Chapters 0 and 1, but marked up my book in many places. I can show you next term when you're in my Kahler manifolds class. –  Spiro Karigiannis Nov 1 '10 at 13:25
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For the proof in the algebraic case, and a clever remark, try section 10.5, page 129, of Algebraic varieties, by George Kempf. –  roy smith Nov 1 '10 at 15:41
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(I have removed some gendered language. I hope I have not changed any intended meanings.) –  Theo Johnson-Freyd Nov 1 '10 at 18:20
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2 Answers

up vote 5 down vote accepted

Let $H$ be the divisor corresponding to the point $x\in \mathbb{P}^1$. Tensoring the exact sequence $$ 0\to O_{\mathbb{P}^1}(-H)\to O_{\mathbb{P}^1}\to O_x\to 0. $$ with $E\otimes H^k$, gives $$ 0\to O_{\mathbb{P}^1}(E\otimes H^{k-1})\to O_{\mathbb{P}^1}(E\otimes H^{k})\to E_x\otimes H_x^{k}\to 0. $$Here GH writes $E_x\otimes H_x^{k}$ for $E\otimes H^k\otimes O_x$, or what is the same, the fiber of $E\otimes O(H^{k})$ over $x$.

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What exactly is the final sheaf that appears there? I don't understand. –  Robert Garbary Nov 2 '10 at 2:11
    
I edited my answer. –  J.C. Ottem Nov 2 '10 at 12:00
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I don't have my copy of Griffiths and Harris in front of me, but regarding Grothendieck's theorem, there's a nice elementary argument by Michiel Hazewinkel & Clyde F. Martin. Here is a link to ScienceDirect. It's only five pages, and mostly consists of linear algebra (and really, is closer to 3 pages, ignoring abstract, intro, and white space at the end).

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The classical theorem is called the Kronecker Pencil Lemma. There is a proof in one of Gantmacher's books on matrices. I also recall a modern proof in the book "Vector Bundles on Complex Projective Spaces", by Okonek et al. –  Deane Yang Nov 1 '10 at 15:44
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I think "only five papers" should be "only five pages." –  Charles Staats Nov 2 '10 at 0:52
    
@Charles: Ack! Thanks for the correction –  Charles Siegel Nov 2 '10 at 2:16
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