(that's me.)

I agree with the other reviewers' comments concerning the phenomenal depth and breadth of the topics covered in this book. Hartshorne builds the soaring edifice of modern algebraic geometry from the ground up. All the way through, the exposition is concise and absolutely clear. The proofs strike an excellent balance between meticulousness and readability.

The approach he takes seems to be to try to acquaint the reader with as much formalism as possible as quickly as possible, and he seems reluctant to offer any sneak previews of vital concepts such as divisors, differentials, and flatness until the reader's brain is "ripe". As a result, Hartshorne is able to state and prove results under extremely general hypotheses. This approach also benefits the kind of reader who wishes to use this as a reference book.

It's important also to note the disadvantages of Hartshorne's approach: Time and again, I found myself utterly baffled by the definitions, because the motivations for them are lacking.

To give a minor example, take the definition (in chapter 1, part 3) of a morphism between two varietes. First, regular functions from a variety over k to k are defined as those that are locally representable as quotients of polynomials (without bothering to give an example of a case of a regular function for which more than one such representation is needed). Then a morphism f: X -> Y is defined as a Zariski-continuous function with the property that whenever you have an open subset V of Y, and a regular function V -> k, then f^-1(V) -> V -> k is regular. There's nothing wrong with this definition, of course, but I found it very difficult to make sense of, initially. A morphism, after all, is supposed to be something that preserves structure, but it's not immediately obvious what "structure" is being preserved in this case (and the full details of this aren't spelt out until much later, after sheaves have been defined). A better didactic approach, I think, would be either (1) to define morphisms of affine varieties simply as functions given by polynomials, and then show that the above definition is the only natural way of generalising this, or (2) to briefly introduce sheaves at the outset, making it clear that the "structure" we wish to define on a variety consists precisely of the sheaf of regular functions.

Another negative effect of Hartshorne's approach is that, if you have to traverse a mire of formalism before meeting an idea, it makes the idea seem more complicated than it actually is.

Certainly there's nothing to stop a dedicated reader just ignoring any temporary befuddlements, secure in the knowledge that eventually everything will make sense, but not all of us have the patience. This book contains an almost ridiculous number of exercises - most of which are supposed to be "formalities", there to flesh out the definitions, but many contain absolutely crucial definitions and lemmas. Attempting to do all the exercises as you go along is very taxing work indeed, and becomes demoralising whenever you get stuck. Perhaps the best strategy is to do only those exercises that are interesting or important for later work.
Also, as others have noted, this book is very tough going on those who don't already have some familiarity with commutative algebra and (later on) homological algebra.

All in all, I think this book will be most useful for people who already know quite a lot of algebraic geometry, commutative/homological algebra etc., and are wishing to consolidate and "modernise" their understanding. For beginners, it's a struggle, but not an unproductive one, especially if assisted by other, less demanding books.

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 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.