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as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be nice if he could proceed studying parabolics on elliptic curve. Do you know a specific reference? If not, what would you suggest?

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  • $\begingroup$ arxiv.org/abs/math/0006174 $\endgroup$ – Francesco Polizzi Apr 19 '14 at 9:27
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    $\begingroup$ I would certainly not give that paper to a first year graduate student! It is highly technical and difficult. You won't find a specific reference for parabolic bundles on elliptic curves; for a gentle introduction to parabolic bundles I would suggest Seshadri's book "Fibrés vectoriels sur les courbes algébriques", Astérisque 96. $\endgroup$ – abx Apr 19 '14 at 11:41
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    $\begingroup$ It might be easier to begin with parabolic bundles on $\mathbb{P}^1$. $\endgroup$ – ulrich Apr 19 '14 at 12:14
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What about Mehta and Seshadri (Math. Ann. 248, 1980)?

Their paper handles the genus 2 or more cases (with a focus on one marked point), but my understanding is that their result and proof (generically) generalizes to genus 1 and 0 (and arbitrary many marked points). Perhaps a good project for your student would be to read their paper, and explain how the genus 1 case can be deduced from their work.

The main point of my response is not to just offer a reference. It is to suggest that a good reading assignment for a graduate student might involve reading and modifying an argument. I think this puts the student in a position to need to actively understand the ideas, but just passively read them.

I recall reading in a different paper (of H. Boden) that a detailed account of the genus 1 and 0 cases is in Seshadri (Asterisque, 96,1982); already mentioned in a comment. So perhaps it is best to have the student not look at that one (for the exercise), but it is good for the mentor to have resources in case one gets stuck. For the record, Hans Boden's papers on parabolic bundles also make good papers for graduate students to read (since he writes well); in particular, his paper Variations of moduli of parabolic bundles is rather nice.

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