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Basic Algebraic Geometry 2: Schemes and Complex Manifolds

Springer Verlag
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So, what in the world is a probability PhD student doing reviewing Shafarevich’s two-volume Basic Algebraic Geometry? Good question. The answer is I really don’t know. For some reason, I found myself attracted to the field, wanting to know what it was all about. I have no plans to switch from probability to algebraic geometry, but I wanted more than an introduction to the subject. Basic Algebraic Geometry fills that role beautifully.

There are many excellent introductions on the subject at a range of levels. I am familiar with Reid’s Undergraduate Algebraic Geometry, which is nice and entertaining. Smith’s An Invitation to Algebraic Geometry is a tasty introduction — if after looking through it you are not interested in algebraic geometry, I don’t think you ever will be. There are several papers in the Princeton Companion to Mathematics that are excellent, in particular, Ellenberg’s article on Arithmetic Geometry has a great introduction to what schemes are and why they are useful (and it is available on line).

To start reading Basic Algebraic Geometry, the reader should be comfortable with algebra at the level of Artin’s Algebra, or Dummit and Foote’s Abstract Algebra. In fact, both these books have nice introductory sections on algebraic geometry. Depending on your background, you might also need a topology text nearby, but nothing too fancy (as Basic Algebraic Geometry does not cover cohomology). Other than that, not much else is required, other than lots of paper and pencils.

Shafarevich’s Basic Algebraic Geometry is now in two volumes, the first dealing with classical topics, namely varieties in projective space; the second volume discussing schemes and complex manifolds. The complete tables of contents can be found on Google Books.

The book has a nice pace; there are many examples and pictures that help solidify the ideas in the reader’s mind. The prose is excellent; it is hard to believe it is a translation. It feels like the author is right beside you, explaining things as you go. In particular, the subject of schemes has a fearsome reputation, but at least at this level of generality, the explanations are clear and well motivated.

There are many exercises, most of them interesting and doable. The first book has an appendix on various algebraic concepts, and the second concludes with a nice historical sketch of a dozen or so pages.

Perhaps it is just a manifestation of the maxim that the best math book on any topic is the third one that you read, but I found these books to be the best ones I’ve read on algebraic geometry.

After reading Basic Algebraic Geometry, next steps might include Hartshorne’s book Algebraic Geometry and/or Ravi Vakil’s notes. In fact, to somebody trying to learn algebraic geometry, the best advice I can offer is that the key word in the previous sentence is “and”. There is such a huge range of material to cover, so many different viewpoints, and so many books and notes available on the web that choosing just one to follow would be like eating only one item at an all you can eat buffet.

Peter Rabinovitch is a Systems Architect at Research in Motion, and a PhD student in probability. When not working, he likes to eat spicy food, and does not enjoy losing money at casinos. He enjoys vacationing at Disneyworld and visiting the land of algebraic geometry, but plans to remain a probabilist.

Date Received: 
Wednesday, October 4, 2006
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Igor R. Shafarevich
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Peter Rabinovitch
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Thursday, November 11, 2010