Algebraic Geometry (Graduate Texts in Mathematics, 52)

This might be one of the most difficult books on the subject matter, and is definitely the most difficult book I read, but if you put in the hard work into it, do all the exercises, you will learn a lot from it. One really cannot blame Hartshorne for the difficulty of this book. Algebraic geometry is a hard topic that requires a large list of prerequistes. If you want to learn algebraic geometry on the level of actual mathematicians then there is no way around the topics in this book. Hartshorne made it possible for the rest of the mathematical community to actually learn this topic, which before him was highly inaccessible. The disadvantage is that much motivation is non-existent. However, if you learn the vocabulary and basic theorems of this topic, then you can try to look for motivations else where. Perhaps, this is a backwarks way of learning the subject but it is very direct and to the point. The advantage is that this book slaps you over the face with all the technical stuff. It is not wordy and to the point. The exercises are helpful and I learned way more from them than reading the actual text. Perhaps a possible compromise to Hartshorne is to learn AG from other sources and then do all of his exercises. I do wish that Hartshorne did a better job on Chapter 1. It is not necessary for the remainder of the text but it helps develop intuition. A problem with Hartshorne's approach is that he defines varieties living in some affine space. This is a bit annoying, he should have defined them with reference to an ambient space (just like the definition of "manifolds" in differential geometry, no reference to an ambient space). I think this would have made the introduction of sheaves more natural. Just a disclaimer. Make sure your basic algebra is solid, especially commutative algebra, and be well-versed in point-set topology.

Hartshorne is a great book. Brutal, but insightful. In my experience, the newer copies of the yellow bois come with varying levels of binding quality (ranging from embarrassing to passable). This copy of Hartshorne has the best binding that I've gotten from one so far (much better than my copy of Conway, which almost survived through a one-semester course!).

Algebraic Geometry Bible

must have

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.

Great book

But "EGA" by Grothendieck and Dieudonne is much easier to read! Even if you don't speak French!

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.