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Algebraic Geometry (Graduate Texts in Mathematics)

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This book is one of the most used in graduate courses in algebraic geometry and one that causes most beginning students the most trouble. But it is a subject that is now a "must-learn" for those interested in its many applications, such as cryptography, coding theory, physics, computer graphics, and engineering. That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic. That being said, most of the books on the subject, including this one, are written from a very formal point of view. Those interested in applications will have to face up to this when attempting to learn the subject. To read this book productively one should gain a thorough knowledge of commutative algebra, a good start being Eisenbud's book on this subject. Also, it is important to dig into the original literature on algebraic geometry, with the goal of gaining insight into the constructions and problems involved. The author of this book does not make an attempt to motivate the subject with historical examples, and so such a perusal of the literature is mandatory for a deeper appreciation of algebraic geometry. The study of algebraic geometry is well worth the time however, since it is one that is marked by brilliant developments, and one that will no doubt find even more applications in this century.

Varieties, both affine and projective, are introduced in chapter 1. The discussion is purely formal, with the examples given unfortunately in the exercises. The Zariski topology is introduced by first defining algebraic sets, which are zero sets of collections of polynomials. The algebraic sets are closed under intersection and under finite unions. Therefore their complements form a topology which is the Zariski topology. The properties of varieties are discussed, along with morphisms between them. "Functionals" on varieties, called regular functions in algebraic geometry, are introduced to define these morphisms. Rational and birational maps, so important in "classical" algebraic geometry are introduced here also. Blowing up is discussed as an example of a birational map. A very interesting way, due to Zariski, of defining a nonsingular variety intrinsically in terms of local rings is given. The more specialized case of nonsingular curves is treated, and the reader gets a small taste of elliptic curves in the exercises. A very condensed treatment of intersection theory in projective space is given. The discussion is primarily from an algebraic point of view. It would have been nice if the author would have given more motivation of why graded modules are necessary in the definition of intersection multiplicity.

The theory of schemes follows in chapter 2, and to that end sheaf theory is developed very quickly and with no motivation (such as could be obtained from a discussion of analytic continuation in complex analysis). Needless to say scheme theory is very abstract and requires much dedication on the reader's part to gain an in-depth understanding. I have found the best way to learn this material is via many examples: try to experiment and invent some of your own. The author's discussion on divisors in this chapter is fairly concrete however.

The reader is introduced to the cohomology of sheaves in chapter 3, and the reader should review a book on homological algebra before taking on this chapter. Derived functors are used to construct sheaf cohomology which is then applied to a Noetherian affine scheme, and shown to be the same as the Cech cohomology for Noetherian separated schemes. A very detailed discussion is given of the Serre duality theorem.

Things get much more concrete in the next chapter on curves. After a short proof o the Riemann-Roch theorem, the author studies morphisms of curves via Hurwitz's theorem. The author then treats embeddings in projective space, and shows that any curve can be embedded in P(3), and that any curve can be mapped birationally into P(2) if one allows nodes as singularities in the image. And then the author treats the most fascinating objects in all of mathematics: elliptic curves. Although short, the author does a fine job of introducing most important results.

This is followed in the next chapter by a discussion of algebraic surfaces in the last chapter of the book. The treatment is again much more concrete than the earlier chapters of the book, and the author details modern formulations of classical constructions in algebraic geometry. Ruled surfaces, and nonsingular cubic surfaces in P(3) are discussed, as well as intersection theory. A short overview of the classification of surfaces is given. The reader interested in more of the details of algebraic surfaces should consult some of the early works on the subject, particularly ones dealing with Riemann surfaces. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. A perusal of the works of some of the Italian geometers could also be of benefit as it will give a greater appreciation of the methods of modern algebraic geometry to put their results on a rigorous foundation.

This is THE book to use if you're interested in learning algebraic geometry via the language of schemes. Certainly, this is a difficult book; even more so because many important results are left as exercises. But reading through this book and completing all the exercises will give you most of the background you need to get into the cutting edge of AG. This is exactly how my advisor prepares his students, and how his advisor prepared him, and it seems to work.

Some helpful suggestions from my experience with this book:
1) if you want more concrete examples of schemes, take a look at Eisenbud and Harris, The Geometry of Schemes;
2) if you prefer a more analytic approach (via Riemann surfaces), Griffiths and Harris is worth checking out, though it lacks exercises.

This book hardly needs a review on Amazon, because if you have as much math background as it needs, then you must already know it is indispensible for learning about schemes in algebraic geometry. The book is clear, concise, very well organized, and very long. If you do not already know the Noether normalization theorem, and the Hilbert Nullstellensatz, then you do not want this book yet--you want an introduction to commutative algebra.

Robin Hartshorne is a master of Grothendieck's general machinery for generalizing the tools of classical algebraic geometry to apply to families of varieties, and more broadly to number theory. A fundamental difficulty is to grapple with algebro geometric objects such as doubled lines, or surfaces with embedded curves and points in them, that arise as "limits" of simpler varieties. Here the algebra is essential as the naive set of points does not reveal the antecedents of the limiting object. Even more in number theory, when the rings of coefficients used may not admit solutions, the structure of the rings themselves is all you have to go on. For the most basic invariants, when we leave the complex numbers and Riemann's topological and integration techniques are not available, sheaf cohomology is the abstract substitute.

These esoteric developments did not arise spontaneously, but out of classical problems that should be approached first in order to motivate and appreciate the power of the tools in chapters 2,3 of this book. Professor Hartshorne says himself that he taught the chapters out of order when he first was writing the book. The average reader should probably read the chapters in the order he taught them in, not the order they appear in this book. Thus first read chapters 4 and 5 on curves and surfaces, or possibly read 1,4,5, to get first a general introduction, then study curves and surfaces. Only then delve into chapters 2 and 3 for the sophisticated stuff.

If you really want to start with the classical roots, begin instead with Rick Miranda's book on Algebraic curves and Riemann surfaces. Of course there are hardy souls who can wade right through Hartshorne's book in order, but for many that is a prescription for losing heart and losing interest in the subject. When all is said and done, there are very valuable ideas and tools in this book that are not available as easily anywhere else. You just have to learn how to get at them. You might want to read in whatever order appeals to you. But do not feel obligated to just plow from page 1 on. Or try the first volume of Shafarevich and then this, or bounce back and forth as the spirit moves you. Kempf also has a book on Algebraic varieties with sheaf cohomology but not schemes, which may ease the abstraction level, and there is also Serre's original paper FAC in that vein.

The motivation is nonexistent, and the examples are trivial. If you want to learn anything you have to trudge through exercises which require techniques that are not addressed in the text. I don't mind working to learn a subject, but spending two hours trying to understand what a question is asking is a bit much.

The best, and most concise review I have ever heard was, "Hartshorne is the worst book on Algebraic Geometry, except for all the others".

Other reviewers have said a lot worth saying. This is a very good, five star book, but it has flaws. It is fine for an ambitious beginner (and I think it is actually good for that purpose), but even he or she must find other sources, and expect an exciting time, but very rough waters. There are things that this book simply does not teach well. Once you finally figure them out, you will usually find them somewhere in here, tucked away, usually crammed in what looks like a ridiculously pithy statement in a little tiny corner of the page. Then after you are over your initial white hot anger you will be happy that now you know where to find it in Hartshorne. And after awhile you might even start liking the way Hartshorne wrote it. And so it goes!

Algebraic Geometry is the first textbook on scheme-theoretic algebraic geometry. Scheme theory was created in the 1960's by Alexander Grothendieck. Grothendieck also co-authored an extremely well-written, 1800-page reference manuscript on scheme theory called "Éléments de Géométrie Algébrique" (EGA). However, EGA is unsuitable as a textbook because it had no examples or motivation and proved every theorem in great detail and maximal generality.

Algebraic Geometry has 5 chapters. The first chapter summarizes algebraic geometry before schemes. The next two chapters compress EGA to 230 pages(!). The last two chapters show how well scheme theory can solve classical problems from algebraic geometry.

That should be a hint that Algebraic Geometry is one of the most dense and difficult math textbooks ever written.

To achieve that kind of compression, Hartshorne's writing is extremely terse. He assumes a solid understanding of commutative algebra and point-set topology. He often gives one or two-sentence proofs and explanations that, when fleshed out and made complete, would need both many pages and new techniques that are never mentioned in the text. He also gives almost no motivation throughout Chapters II and III, because Chapters IV and V fill this role. When he does give motivation, it is usually relegated to the exercises, many of which, again, require techniques that are never mentioned in the text. Finally, he assigns the proofs of many essential and extremely difficult theorems as exercises.

There are other, much more user-friendly introductions to scheme theory than Algebraic Geometry---For example, The Red Book of Varieties and Schemes, The Geometry of Schemes, and Algebraic Geometry and Arithmetic Curves. These books, along with EGA, can also serve as complements to Algebraic Geometry when Hartshorne's writing becomes too dense to learn from.

However, Algebraic Geometry is unique in that no other textbook on scheme theory covers nearly as much material as it does. Also, for all of its density, Algebraic Geometry is very well-written and an excellent reference, especially considering how much it covers and the length and complexity of its source material. Because of this, I cannot foresee any significantly better replacement for it being written in the near future. Algebraic Geometry will probably continue to be a necessary and useful pain to learners of scheme theory, just as it has been for the past 30 years.

Excelent and useful text, indispensable for graduate students and research ,athematicians working on algebraic geometry. Hartshorne walks the fine line between commutative algebra and their geometrical counterparts with elegance. The book is also rich in references, providing many directions for further study.

Here's my impression after doing the first 30 pages: What makes this a really good book is the exercises. Not too hard, always interesting. If you are new to the subject you need to look up results from commutative algebra somewhere else. It can be a little strange getting used to working with the Zariski topology. All open sets are dense, so you don't have the notion of a small neighborhood of a point. For instance any bijection between two curves is a homeomorphism.