Average Reviews:
(More customer reviews)If you want to learn the basic properties of compact Riemann surfaces this is the book to read. If you want to know the "motivations" of modern Algebraic geometry this is again a book to read.
First of all the pace and the style are very casual. You really don't feel overwhelm by a mountain of definitions. The author always favor simplicity and concreteness instead of abstractions and generality. This is really a book that I should have read before taking a class on Schemes. For exemple in the context of Riemann surfaces an "very ample divisor" is simply a linear system without fixed base point that gives rise to an holomorphic embedding. This definition (at least for me) is much much more satisfactory and illuminating than the definition of a very ample sheaf that you can find in Hartshorne (even though his definition is much more general).
There is a very nice chapter on meromorphic differentials which explains how those object can be used to define line integral on any riemann surface. Topics like divisors, Riemann-Roch and curves are treated with a lot of depth. There are not a lot of pictures but having pictures supported by an unclear text is quite useless. Here the writing is so clear (not to say flawless) that on the first reading you really get the idea of what's going on.
There are very few mistakes in this book which is another reason why I like it. I'm really pissed off by those mathematicians
that are rushing to publish their books crowded by mistakes.
But don't get me wrong, I don't have anything against mathematicians that are writing books (this is a learning experience) but don't feel force to publish them unless they are very polished and "innovative".
Finally the last chapters treat of Abel's theorem ( which tells us exactly when a divisor is principal), Sheaves, Cech cohomologies and line bundles. Again the exposition is very well
motivated with a good supply of interesting exemples.
This is the best book that I read on subject and honestly if professor Miranda is writing another book related to my field of research you can be sure that I will have it my collection.
Hugo Chapdelaine,
McGill
Click Here to see more reviews about: Algebraic Curves and Riemann Surfaces (Graduate Studies in Mathematics, Vol 5)
In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of a one semester of complex variable! theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.
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