An Invitation to Algebraic Geometry

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.


Splendid introduction   August 24, 2004
Navin Kadambi (Lakeland, FL)
10 out of 13 found this review helpful

For people just starting on Algebraic Geometry, Robin Hartshorne's book, is very daunting--but it is the ULTIMATE book for professional and advanced readers. But for starters, Karen Smith's "An Invitation to Algebraic Geometry" is simply a SPLENDID way to start working on the basic ideas. The author has some stunning graphs and pictures to help understand material. I loved the book the minute I opened it. BUY it NOW!


enjoyable guidance   February 12, 2002
9 out of 12 found this review helpful

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.


very good   January 10, 2005
Niraj Prasad (Mumbai, India)
2 out of 4 found this review helpful

excellent book ... can make learning algebraic geometry as easy as bedside reading ... highly recommended.