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I am going to enroll in a seminar with the topic "vector bundle on algebraic curve". Except Algebraic Geometry (which I think GTM 52 by Hartshorne is the main source), which topic I should prepare in order to learn that topic?

Could you please recommend me any textbooks/lecture notes on the topic? I have searched on Google but it gave me very few results.

Thanks.

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    $\begingroup$ Why not ask the instructor? $\endgroup$
    – Steven Landsburg
    Commented Jun 6, 2013 at 5:14
  • $\begingroup$ @Steven Landsburg : He is a foreign professor and I know that he is very busy. Other point ís I want to see bigger picture on that topic and(I think) asking question on MO may help. $\endgroup$
    – trequartista
    Commented Jun 7, 2013 at 3:51
  • $\begingroup$ If in addition to the big picture offered by the papers cited in the wonderful answers below, you may like a few precise facts of the small picture, Beauville proves the classification of rank two bundles on rational and elliptic curves, in less than one page, p.34, of his lovely Complex algebraic surfaces. Basically one analyzes vector bundles in terms of line bundles, by either decomposing, or filtering (via "extensions"), or considering the Narasimhan-Ramanan invariant associating to a bundle E, the divisor of line bundles that twist E to become effective. $\endgroup$
    – roy smith
    Commented Jun 10, 2013 at 15:37

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The modern theory of vector bundles over a curve starts with Grothendieck's article Sur la classification des fibrés holomorphes sur la sphère de Riemann. American Journal of Mathematics, 79, 121–138, 1957.
(Actually Geyer and others afterwards realized that the gist of the theorem, in another formulation of course, goes back to results of Dedekind-Weber back in 1882! See Scharlau's article below)
Atiyah then classified, also in 1957, vector bundles on an elliptic curve in this splendid article.
There was then in the next decades an intense activity in getting results on moduli spaces for curves of genus $\geq 2$.
Leaders in the field were among others Narasimhan and Seshadri and here too one can find older predecessors, notably André Weil with his 1938 article article Généralisation des fonctions abéliennes.

Bibliography
Some pleasant didactical references :

For the classification of vector bundles on complete curves in the spirit of Geometric Invariant Theory there is Chapter 5 of Newstead's book

Here is a great survey by Newstead, one of the creators of the theory.

Scharlau's history of Grothendieck's classification. And a proof of that classification can be found on page 23 of Montserrat Teixidor's survey (which by the way is one of the best texts I can recommend as an answer to your question).
An elementary survey by Cautis, emphasising the comparison of holomorphic, and topological vector bundles on Riemann surfaces.

And finally, another survey by one of the historical masters of the field.

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    $\begingroup$ In my thesis I needed a normal form for a pair of matrices, and my adviser told me to look in Gantmacher’s book for the Kronecker pencil lemma. He also said in passing that it was equivalent to the Grothendieck splitting theorem. I recall verifying this and in fact discussing it in my thesis. Kronecker’s paper is in 1890. $\endgroup$
    – Deane Yang
    Commented Nov 7, 2018 at 17:46
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    $\begingroup$ Dear @Deane: yes I had also heard about Kronecker's result. Scharlau attributes to Dedekind-Weber the algebraic result underlying Grothendieck's result. And Birkhoff is also supposed to be another predecessor of the splitting theorem ... $\endgroup$
    – Georges Elencwajg
    Commented Nov 7, 2018 at 19:14
  • $\begingroup$ Dear @GeorgesElencwajg, most of the survey links are dead. Could you please share the surveys again, may be by email? $\endgroup$
    – trequartista
    Commented Nov 22 at 14:59
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Geometry of albraic curves http://www.amazon.com/Geometry-Algebraic-Curves-mathematischen-Wissenschaften/dp/0387909974

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Le Potier, Lectures on vector bundles (1997)

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