An Invitation to Algebraic Geometry

Rating:
Summary: enjoyable guidance
Review: i'm not a math student, but this book is very readable. it's short(150 pages) but many illustrative examples and exercises cover chief topics and facts, i assume. at first, i tried Eisenbud's "geometry of schemes" but it was too hard and Hartshorne's was somewhat alien to me. then comes this book. it helped me through the Eisenbud's, and convinced me algebraic geometry is an intriguing discipline.

Rating:
Summary: Very good, but understated prerequisites
Review: On the back of the book, it says "Few algebraic prerequisites are presumed beyond a basic course in linear algebra." This is not, in fact, true. It uses a lot of ring theory (and the review of commutative ring theory in ch. 2 is a bit fast for someone unfamiliar with the subject), and a fair amount of topology. When I first got it, I read the first several pages and found them readable, but when I read more (on the car-ride home) I was confronted with the fact that the Zariski topology is coarser than the standard topology on C^n, and is not even Hausdorff. Several questions came to mind (What's a topology? What does for one to be coarser than another? What is a Hausdorff topology?). Still, after I learned more topology, I found the book a delight. Everything is light and interesting, and does a good job of portraying algebraic geometry without technical details. All mathematics looks nicer when you do that, but it takes away from rigor. Hence this should not be your only text on Algebraic Geometry, but I would suggest it as one of them...

Rating:
Summary: Wow!
Review: This could be your only book on algebraic geometry if you just want a sound idea of what algebraic geometry can do. If you actually want to know the field, and you do not already have a lot of expert friends telling you about it, then the advanced books will go much more easily with this expert around. It is a terrific guide to the key ideas--what they mean, how they work, how they look.

The only book like this one in brevity and scope is Reid UNDERGRADUATE ALGEBRAIC GEOMETRY--with its highly informed, highly polemical, final chapter on the state of the art. Both are very good. This one is more advanced. Beyond what Reid covers, Smith sketches Hilbert polynomials, Hironaka's (and very briefly even De Jong's) approach to removing singularities, and ample line bundles. You do need a bit of topology and analysis to follow it. Smith has very many fewer concrete examples than Reid. They are beautifully chosen classics, like Veronese maps and Segre maps, so they teach a lot. And the more you know to start with, the more you will see in each.

The book does geometry over the complex numbers. It is good old conservative material, with terrific graphics of curves and surfaces. The proofs and partial proofs are very clear, intuitive and to the point. But, in fact, just because the proofs are so clear and to the point they usually work in a much broader setting. Long stretches of the book apply just as well over any field or any algebraically complete field. This generality is only mentioned a few times, in passing, but is there if you want it. Smith describes schemes very briefly, and mentions them at each point where they naturally arise. You will not know what schemes "are" at the end of this book. You will know some things they DO. She has no time for fights between "concretely complex" and "abstractly scheming" approaches--for her it is all geometry.