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.



Recommended books/lecture notes for vector bundle on algebraic curve
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](http://doi.org/10.2307/2372388). 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](https://math.berkeley.edu/~nadler/atiyah.classification.pdf) 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](https://gallica.bnf.fr/ark:/12148/bpt6k6459126x/f59.image)*.


**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](http://books.google.fr/books/about/Lectures_on_introduction_to_moduli_probl.html?id=1yAiRAAACAAJ&redir_esc=y) 


[Here](http://www.mimuw.edu.pl/~jarekw/postscript/Lukecin-Newstead.ps) is a great survey by Newstead, one of the creators of the theory. 


[Scharlau's history](https://ivv5hpp.uni-muenster.de/u/scharla/scharlau/grothendieck/Grothendieck.pdf) of Grothendieck's classification. And a proof of that classification can be found on page 23 of [Montserrat Teixidor's](https://pdfs.semanticscholar.org/dd79/abcec738d08efe948e4a9aa5ad74dacf16e2.pdf) survey (which by the way is one of the best texts I can recommend as an answer to your question).  

[An elementary survey](http://www.math.ubc.ca/~cautis/classes/notes-bundles.pdf) by Cautis, emphasising the comparison of holomorphic, and topological vector bundles on Riemann surfaces. 


And finally, [another survey](https://www.cimat.mx/Eventos/c_vectorbundles/ramanan_notes.pdf) by one of the historical masters of the field.


