Algebraic Geometry (Graduate Texts in Mathematics)

Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. After receiving his Ph.D. from Princeton in 1963, Hartshorne became a Junior Fellow at Harvard, then taught there for several years. In 1972 he moved to California where he is now Professor at the University of California at Berkeley. He is the author of "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles. His current research interest is the geometry of projective varieties and vector bundles. He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively. Professor Hartshorne is married to Edie Churchill, educator and psychotherapist, and has two sons. He has travelled widely, speaks several foreign languages, and is an experienced mountain climber. He is also an accomplished amateur musician: he has played the flute for many years, and during his last visit to Kyoto he began studying the shakuhachi.

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.

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.

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".

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.

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.