Customer Reviews

Introduction to Algebraic Geometry

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4 star:		(2)
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Hassett tries to give a more concrete approach to classical algebraic geometry by teaching Grobner Basis early and focus on computational algorithms every step of the way. Although this pedagogical idea is good, the book is not. It is rushed and published way too early; it feels very much like lecture notes. It is not surprising then that the book is plagued by many serious problems.

The Good:
1. Introducing Grobner Basis technique early is a good idea. GB provides a powerful tool for experimenting with many of the later concepts in the book.

2. The book offers ample examples. All difficult concepts are immediately followed by a relatively detailed example.

3. There are many exercises, varying in difficulty. Many of the exercises are computational and I highly recommend that the reader use a computer commutative algebra program to help solve them.


The Bad:

1. Mistakes
Hassett's book is riddled with false statements. Some are blatant and minor, some are subtle and occur in statement of major theorems. Luckily, a careful reader with effort can catch all of the mistakes so the book is still read-able.

2. Leaps in Logic
Hassett's proofs and exposition often lack detail. He is especially hand-wavy on the algebra aspect; many k-algebra isomorphisms are simply asserted and not justified in any degree.

3. Lacking in Geometric Intuition
This is specifically referring to chapter 3 and 4. Hassett presents the definition of morphism and rational and related theorems without giving the reader any idea how they relate to the geometry of polynomial maps.

4. Lack of Graphics
This relates to the lack in geometric intuition. The amount of visual aid in this book is very sparse; what little graphics present are almost all useless.


Overall, I felt that this book did a poor job of teaching a mathematics student how to rigorously think about algebraic geometry. Hassett's most grave problem is that it glosses over steps of critical proofs early on and leaves reader confused about how to actually go about proving even rudimentary propositions on their own.

To use an analogy: an introductory algebra student might not know how to prove Lagrange's theorem on their own, but when presented when a possible proof, they should at least be able to tell whether the proof is rigorous and correct. For me, Hassett's text fails in this respect; I have trouble after reading the book to tell whether my proofs are rigorously enough or not. (this is not due to my own lack of mathematical maturity; I have taken a year of algebra)

This book has great potential to be a classic in algebraic geometry but as of now, it falls far far short. I would recommend that readers Wait for the second edition of Hassett's book and use the introductory algebraic geometry text by Joe Harris in the mean time.

There are many introductions to algebraic geometry focusing on similar topics to this one (varieties affine and projective). However this one is especially friendly to newcomers and is the only such book that I think is accessible to an average undergraduate (junior or senior year). It is very well written with excellent examples. If you are interested in algebraic geometry you will definately need to proceed beyond this book, but it is a very nice first step.

This book seems like a breath of fresh air in a very stale
area of mathematics. He provides an opening that my other book
on the area didn't:Algebraic Geometry.
If it wasn't that he uses terms like 'Ideal' without any real
definition, I would have given the book five stars.
I shopped really hard before buying this book
and read the index and table of contents online.
For once I'm not disappointed. I have four pages turned down from my first read through and it may take some more time and study to
get more out of it, but it does seem to be an honest teaching effort
in print. I say that it well done.