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Algebraic Curves and Riemann Surfaces (Graduate Studies in Mathematics, Vol 5)

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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

This book gives a very readable account of Riemann Surfaces-- a good course in Complex Analysis is all that's required as a prereq. The proofs are very clear, the material is presented beautifully, and (most of) the exercises are fairly straight forward and supplement the book very well. The notion of divisors, proof of the Riemann Roch theorem and Abel's theorem are explained very nicely. It serves as the perfect transition into more advanced books in algebraic geometry and on complex manifolds.

I am teaching the course this semester from this book and really enjoying it. The book was obviously written with the insight obtained from teaching the course several times and revising and perfecting the notes. A bonus for the teacher is that Rick has preserved the organization of a course in the book. That is, each section is roughly what you can cover in one lecture, so you can pace your 40 lecture course to try to cover 40 sections. And if you cannot lecture as fast as Rick writes, no worries, his explanations are so clear you can honestly assign the rest as reading.

I am also getting a lesson in pedagogy from the book. I am in the habit of proving everything in detail at the greatest possible depth, which helps me maybe, but leaves many students behind. Also I usually never cover a lot of material because everything takes me too long to treat. But Rick has intelligently chosen to cover everything at a uniform depth. if some proof is too complicated to explain fully he assumes it. But nothing is lost since he includes instead a detailed explanation of a simple and very illustrative example, which as everyone knows but me, is more instructive than an abstract general argument.

Even so, his explicit arguments and explanations are so clear they illuminate even those topics which he omits. Today for example, we covered his section on covering spaces, and enlarging them to branched covering spaces, such as non constant holomorphic functions give. Rick's discussion was so clear, I was led to expand it slightly to prove the existence of a Riemann surface for a general irreducible plane curve. His own treatment never proves this, choosing instead to give many beautiful and very helpful examples of how to fill in singularities of plane curves by smooth points. But his explanation of the relation between branched and unbranched coverings was so clear I saw it clearly myself and could not resist.

Time and again he makes clear how one obtains a better picture of what is going on from several well treated examples than one abstract argument. Of course he also has many very excellent abstract proofs too. This book is written for a well prepared upper level undergraduate audience, and for that reason it is super useful for graduate students and even old professors like me.

Let me observe for beginners that thesE ideas were introduced by RIemann in the analytic context and only later translated into algebraic language. There is a reason no one thought of these ideas algebraically before Riemann. The concepts are inherently analytic and topological. Hence it is almost impossible to understand how their algebraic versions were thought of unless you learn the analytic versions first. Hence people starting from Walker or Hartshorne, or even Shafarevich are handicapped by trying to understand the motivation for algebraic concepts which were introduced to mimic analytic ones that are not mentioned. How are you going to appreciate the genus of a curve if you think it is the smallest possible integer g such that for every divisor D of degree d on the curve, we have l(D) is greater than or equal to 1-g + deg(D), where l(D) is the vector dimension of the space of rational functions f with divisor greater than or equal to -D? This is the kind of unilluminating definition given in purely algebraic treatments of the subject. Riemann explains it as the number of handles in a surface which is topologically equivalent to a sphere with a finite number of handles. I.e. for a sphere it is zero, and for a doughnut it is one, etc...

One remark of a personal nature. On page 19 of the first edition there was a problem F to show a certain space curve is a non singular complete intersection, whereas in fact it was highly singular. I assigned it since I like problems where the instructions are wrong and the student has to find the right answer himself. (They are easier to grade, since they are wrong unless the student finds the mistake.) I wondered if Rick felt the same and had actually intended this error to be present. Apparently not, since the second edition had a corrected version of the problem. So I was fooled, my students were working a corrected problem and I was expecting them to have to deal with the flawed one.

The proof of the Riemann Roch theorem given here is that of Weil, as made clear by Serre, not the original one of Riemann, so in my course we will discuss both and contrast them when the time comes. After giving the analytic treatment of the whole subject, Rick gives a gentle introduction to the algebraic way of treating the same ideas, highly useful to new student I would expect. i only regret i will not have time to cover the entire book. This is the place to start if you really want to understand this subject from the original analytic viewpoint, and also see how it transitions into todays version. This book is written from the perspective I myself recommend and would have used in a book of my own, but it is written by someone with a greater pedagogical gift and greater grasp of the subject than I or most people have. He also took a lot of time and care with it. This is a real find, as this is not otherwise an easy subject to learn. Rick has made a big contribution to the accessibility to this fundamentally important topic.

This book is great for self-study. It is easy to follow, a perfect amount of examples, and the proofs arent weighed down by unnecessary examples. The author is also surprisingly funny. It is a very good non-trivial introduction. I would definitely recommend it to someone who doesn't have much experience in algebraic topology.

I went to graduate school in mathematics more than 30 years ago and my present job is not related to mathematical research. Recently I have started reading mathematics books at the level similar to some of my old graduate courses just for pure enjoyment of learning. Then I realized that I did not take any course on Riemann surfaces. So I looked at parts of Fulton's introductory text book on algebraic topology (Chapter on Riemann-Roch Therorem) and then to Miranda's book on Riemann surfaces. All I wanted to know is what Riemann-Roch Theorem looks like or how the proof looks like. (I was not hoping to master the subject at the level of passing a qualifying exams on the subject.)

For now, I have only looked at Miranda's book up to Chapter VI Theorem 3.11 (Riemann Roch Theorem II). What I did may be called a very crude glancing rather than a careful reading. But even with these limitations on my part, I would like to recommend this book to advanced undergraduate or graduate students who want to have an introduction to Riemann surfaces. From pedagogiacal view point, the author writes very carefully, kindly, and friendly to the reader. The pace is unhurried and I like the style.

For example, after giving the definition of the space of memomorphic function with poles bounded by D, L(D) using mathematical expressions, he writes the following sentence:
"Hence the conditions imposed on a meromorphic function f to get into a space L(D) are one of two types: either poles are being allowed (to specified order and no worse) or zeros are being requried (to at least some specified oder) at discrete set of points of X."

To a non-smart reader (or not so alert person) like me, it was a pleasure and a great help to read such friendly explanation righ after a formal definition. May be many smart readers do not require these extra comments, but in my case, they helped me a lot. So I really thank the author for giving these nice guiding posts here and there. This book seems to be full of these helpful comments. Another such example is in Chapter III Section 4, on the covering spaces and fundamental group. He writes "There is a lot to check here, but the bottom line is that ..." and without necessarily giving all proofs, give a concise summary of topology facts needed.

I realize I am not touching upon the later parts of the book which is supposed to have wonderful content for more advanced topics. I hope to be able to reach there someday.

If I may say one final slightly critical thing about this wonderful book-this book does not contain figures. I wish they had at least a few to help us visualize, though it can be argued that it may not worth the space and the average reader will not suffer much from the lack of them.