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.
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Marc
Bon dosage de difficulté pour une introduction
Reviewed in France on January 3, 2016