Basic Algebraic Geometry 1: Varieties in Projective Space [Paperback]

The central issue with Shafarevich is that it repeatedly blends the algebraic and geometric aspects of the subject together in such a way that you can no longer be sure what is actually a trivial geometric corollary of a deep algebraic result and vice versa. An example occurs in the first few pages, where Shafarevich presents the most confusing treatment of local parameters I've seen. After reading this, I was at a complete loss; only by consulting Fulton's excellent but short Algebraic Curves was I able to cut the issue down to its fundamental algebraic and geometric components.

Shafarevich aims to develop algebraic geometry by starting from the ultra-concrete and slowly moving, over the duration of two volumes, to the ultra-abstract, and his strategy, which is utilized most explicitly in the first part of this book, is to occasionally revise all preceding definitions in light of some new observation, increasing their generality and occasionally abstractness a little bit each time. This is a reasonable idea which I think could have worked marvelously. But as it stands Shafarevich was far too sloppy to have succeeded in this plan. It's never clear when Shafarevich has decided to make a jump to the next level of generality, and when he does it's as if he tries his hardest to make it as difficult as it could be for you to see exactly how the definitions are revised. Most definitions are just dumped into the middles of paragraphs with nothing to set them off in the text. Concepts which are presented in one order early on are revised in a contradictory order later on. The first part will leave you wondering how the word "variety" has any meaning left in Shafarevich's world after having been revised countless times, each revision only applying in certain contexts. Shafarevich's crippling habit of confusing algebra with geometry and vice versa also hinders your ability to understand the underlying philosophy behind some of the revisions.

In fact, Shafarevich is just altogether too wordy, and it's as if he has no idea who his target audience is. On the one hand, he seems desperate to avoid any real commutative algebra, resulting in a terrible presentation in which purely algebraic facts are presented as if they were geometric. Yet he isn't afraid to use some facts from field theory which, while not too hard, are certainly not easier than some of the basic commutative algebra that Shafarevich avoids at all costs during the first part of the book. Shafarevich also wants to use the Zariski topology throughout the book--it plays an absolutely crucial role, for justifiable reasons--and yet he seems afraid to assume the reader actually knows any point-set topology, instead repeating wordy and clunky definitions from elementary topology over and over again throughout the text. You do not study algebraic geometry without knowing some point-set topology--this is a given. So for someone who actually has the proper prerequisites, this book can be incredibly annoying to sift through sometimes, as when Shafarevich takes what is easy and short and makes it hard and long due to his refusal to use a bit of terminology.

In terms of content, Shafarevich is strong and unique. It covers all the concrete properties of varieties that you need to know as a beginning student of this subject. The question you must ask yourself is whether it's worth the supreme trouble it will require for you to actually learn from Shafarevich's consistently sloppy instruction. My answer is that it is certainly not.

My recommendations are Fulton's Algebraic Curves (available for free online in PDF format), Eisenbud's Commutative Algebra with a View Towards Algebraic Geometry (of course, this contains much more than you need, but it's outstanding in all aspects), Harris's Algebraic Geometry: A First Course, and probably also Mumford's Red Book.

In the 1950's algebraic geometry was tedious and hard to grasp because it was mostly commutative algebra, developed by Zariski and Weil and their schools to fill logical gaps in the Italian arguments of the previous half century. The rich geometric texture of the italian school was lost. In the 1960's Serre and Grothendieck introduced homological algebra to the subject and greatly expanded and enhanced it to embrace also arithmetic, but the abstraction level went WAY up, so again it was hard to grasp and relate to geometry. Hartshorne is a member of both Zariski and Grothendieck's schools and appreciates down to earth objects like space curves, but his book has a long beginning section on schemes and cohomology that can definitely throw a beginner off the horse.

Pardon the delay in getting here, but the point is that Shafarevich's book has none of the tediousness of the previous generation, yet benefits from the rigorous foundations via commutative algebra of Zariski's works. I would say Shafarevich's book, is a geometrically oriented explanation of the material that can be explained using Zariski's methods. I.e., it has a rich geometric feel, is very well explained, includes many easy examples, and is rigorous in its use of commutative algebra. This book allowed many of us who were stymied by the huge amount of algebra needed for 1960's Grothendieck style AG, to finally gain admission to the subject. I love this book.

There are three themes one can mention in algebraic geometry, 1) projective varieties, 2) schemes 3) cohomology. The first topic concerns the objects most geometers are interested in. The second one is of more interest to number theorists, but also has value for geometers in understanding limits of varieties. The third is a powerful and abstract technical tool which is valuable to both schools.

So here is the difference between Shafarevich and Hartshorne: Shafarevich focuses mainly on varieties, as the title reveals, has nothing at all in volume I on schemes, and neither volume treats cohomology. Hartshorne quickly reviews varieties then goes straight to schemes and cohomology, presenting schemes before cohomology. Thus Shafarevich is much more elementary than Hartshorne, and is a good introduction to that book. For a book treating cohomology before schemes, try George Kempf's algebraic varieties, but hartshorne will eventually be essential.

A few negative aspects to Shafarevich: it has a large number of typos and outright errors, in spite of numerous editions and reprints. One of the worst mathematical mistakes is the false assertion in the section I.6.3 of the first edition, that for any regular map f:X-->Y the set of points of Y over which the fibers of f have dimension at least r, is closed in Y. One needs the map f to be closed, e.g. proper, for this to hold, and unfortunately this false result is used in the treatment of the tangent bundle of a variety. The proof of the normalization for curves seems also flawed to me. All these mistakes can be fixed, and even in the present flawed form, this book still has no peer in readability and geometric richness with the same scope (although the undergraduate book by Miles Reid, Shafarevich's translator, is excellent for what it covers). I recommend it highly.

I would give it 4 stars for the errors, to leave room for a higher score for a book not only well conceived and well written but also well edited, but there is no other book that approaches this one in its strengths so how can one give it only 4 stars? But how can this book go through so many printings and not repair the most egregious mistakes? (There are other famous books out there with more and worse errors, but we who do not write books should not be too harsh in this regard. What if no one had the courage to attempt to explain this difficult stuff to others?)

I have been a student of AG for the past six years and I have come to the conclusion that Shafarevich is a great place to start. Having said this, one must have the necessary background in algebra and topology. I disagree with the other reviewer about doing this after Hartshorne--start here then do Hartshorne!!! Oh ya, the index refers to both volumes 1 and 2; read the first page of the index!!!