Elementary Algebraic Geometry (Student Mathematical Library, Vol. 20) (Student Mathematical Library, V. 20) Review

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Elementary Algebraic Geometry (Student Mathematical Library, Vol. 20) (Student Mathematical Library, V. 20) ReviewFirst, a calibration: I am a total neophyte to algebraic geometry, and haven't taken a university algebra course since a few decades ago when I was a physics major. This book is one of several on the subject (along with some books on commutative algebra) that I'm using to get an amateur's orientation.
As so often happens, this book looked great in the bookstore. It is thin, reasonably well-illustrated compared to other books in the field, and even helps you gets your toes wet in sheaves, category theory and some other neat topics.
That said, I believe the prerequisites in the preface (university algebra, with a complex variables course optional) are understated; e.g. it helps to know something about fibres, lifts and other topics from geometry. It might be relevant that these notes were prepared at a German university; you should consider that "undergraduates" there are heading toward the equivalent of a US M.S. degree, not B.S./B.A.
More detrimental is that the presentation slogs from one proof to another and too rarely pauses for breath to consider the "big picture" significance of what you're proving. Notwithstanding that Joe Harris's "Algebraic Geometry: A First Course" is even less of a piece of cake for me than it might be for you, his style is a breath of fresh air when it comes to enlightening you as to some geometric context and payoff for all this effort. Other supplements I found helpful include Reid and Schenck.
PS in 2008: I very belatedly found the terrific "An Invitation to Algebraic Geometry," by Karen E. Smith &al. (Springer 2000, corrected printing 2004). This is the hands-down best introduction to the subject, IMHO.Elementary Algebraic Geometry (Student Mathematical Library, Vol. 20) (Student Mathematical Library, V. 20) Overview

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