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I am considering reading one of 'Algebraic curves and Riemann Surfaces' by Rick Miranda or 'Lectures on Riemann Surfaces' by Otto Forster. Which one of these is more advanced and comprehensive ? What are the differences in the approaches of these two books ?

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  • $\begingroup$ I've worked through sections of both, and they're both good. I'd say they're about equal in difficulty level and comprehensibility, but you might have a different opinion. Miranda's book is more focused on algebraic curves in general and preparing the reader to go on in algebraic geometry by giving them digestible analytic examples of algebraic constructions they will see in more generality later (I think...). If all you care about is Riemann surfaces, I'd go with Forster's book. $\endgroup$ – Potato May 30 '13 at 19:20
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    $\begingroup$ I know Forster's book quite well, having taught out of a good portion of it a few times. It is extremely well-written, but definitely more analytic in flavor. In particular, it includes pretty much all the analysis to prove finite-dimensionality of sheaf cohomology on a compact Riemann surface. It also deals quite a bit with non-compact Riemann surfaces, but does include standard material on Abel's Theorem, the Abel-Jacobi map, etc. I would also recommend Griffiths's Introduction to Algebraic Curves — a beautiful text based on lectures. $\endgroup$ – Ted Shifrin May 30 '13 at 19:46
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Ted Shifrin notes above:

I know Forster's book quite well, having taught out of a good portion of it a few times. It is extremely well-written, but definitely more analytic in flavor. In particular, it includes pretty much all the analysis to prove finite-dimensionality of sheaf cohomology on a compact Riemann surface. It also deals quite a bit with non-compact Riemann surfaces, but does include standard material on Abel's Theorem, the Abel-Jacobi map, etc. I would also recommend Griffiths's Introduction to Algebraic Curves — a beautiful text based on lectures.

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I think this post is almost a duplicate. See the following:

Perspectives on Riemann Surfaces

I do recommend the recent published book by Donaldson on this subject. It is really interesting to read.

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