Introduction to algebraic geometry

Dear Mr. Kim.

I’ve got an email from the chair, Youngju Choie that you could give some help with algebraic geometry or number theory. I’m quite interested in algebraic geometry even though I don’t know much about it. Would you mind introducing me what algebraic geometry is and some good textbooks for the studies? I’m looking forward to receiving Your advice.
Sincerely yours
Woo-Hyun Cook

Reply:

These days, I’m in the habit of recommending Wikipedia articles for basic introduction to mathematical topics. They are of quite reasonable quality compared to more `popular’ areas, possibly because only somewhat serious people care to comment at all on mathematics. In the article on algebraic geometry, you will find even the `talk’ page to be at a reasonable level. A common difficulty with surveys rather is that they are somewhat too difficult for beginners. However, the ability to sense make of them can serve as one kind of gauge of progress as you study the subject.

When I was an undergraduate, I enjoyed looking occasionally into

A panorama of pure mathematics, by Dieudonne

that contains a nice chapter on algebraic geometry. Of course you should remember that the whole book is one person’s point of view, albeit a mostly well-informed one.

The only introductory textbook I’ve actually read is

Algebraic geometry, by Hartshorne

It gets you going on much of the machinery that you need for modern algebraic geometry, varieties, schemes, sheaves and cohomology, and is methodical to a fault. Even over several years, I have yet to find a single misprint. It definitely smooths the transition to the advanced tools of arithmetic geometry as may be found in

Seminaire de geometrie algebrique

For a gentler ascent to the first summit, look at

Basic algebraic geometry, by Shafarevich

My students have uniformly preferred it to Hartshorne as an introduction.

And then, a nice collection of connections to differential geometry, topology, and classical geometric problems is written-up in

Principles of algebraic geometry, by Griffiths and Harris.

In particular, it really is indispensable at some point for a measure of analytic sophistication.

As I grow older, I feel much less dogmatic about the importance of a very systematic education in algebraic geometry, so I understand if you prefer an informal beginning at the undergraduate level. Two alternatives I can think of are:

(1) A computational approach using, for example,

Ideals, Varieties, Algorithms, by Cox, Little, and O’Shea.

These days it seems a good idea to familiarize oneself early on with computational algebra. But this is good also as a general textbook if you need a review of some basic techniques like polynomial algebra. In the end, you need to have a good sense of how to actually compute things, like the image of an arbitrary polynomial map, at least in principle.

(2) Concentration on a somewhat detailed topic, like the theory of curves. For this, I have a slightly non-standard recommendation:

Rational points on elliptic curves, by Silverman and Tate.

You might be interested in number theory anyway, but even if you aren’t, their specific focus introduces goals that can help to motivate the techniques that turn out to be more generally useful.

There are too many online texts these days of varying quality, which I will avoid mentioning since navigating your way through them can be more confusing than enlightening. However, one solid introduction was written by James Milne and distributed on his website.

One can go on endlessly with tiresome advice on how to study algebraic geometry, but I just jotted down a few lines for now. Of course, you should just start somewhere, and then ask more specific mathematical questions while you study. If you are a complete novice, try out alternative (2) and feel free to ask for more guidance as soon as you begin (which should be today).

MK

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