Introduction to Algebraic Geometry 1st Edition by Brendan Hassett (PDF)

Description

Algebraic geometry, central to pure mathematics, has important applications in such fields as engineering, computer science, statistics and computational biology, which exploit the computational algorithms that the theory provides. Users get the full benefit, however, when they know something of the underlying theory, as well as basic procedures and facts. This book is a systematic introduction to the central concepts of algebraic geometry most useful for computation. Written for advanced undergraduate and graduate students in mathematics and researchers in application areas, it focuses on specific examples and restricts development of formalism to what is needed to address these examples. In particular, it introduces the notion of Gröbner bases early on and develops algorithms for almost everything covered. It is based on courses given over the past five years in a large interdisciplinary programme in computational algebraic geometry at Rice University, spanning mathematics, computer science, biomathematics and bioinformatics.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book seems like a breath of fresh air in a very stalearea of mathematics. He provides an opening that my other bookon the area didn’t:

⭐.If it wasn’t that he uses terms like ‘Ideal’ without any realdefinition, I would have given the book five stars.I shopped really hard before buying this bookand 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 toget more out of it, but it does seem to be an honest teaching effortin print. I say that it well done.

⭐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.

⭐Great book.

⭐This book is definitely not one I would consider good, but I don’t know of another one that presents this subject at the level this one is trying to.The text is aimed at undergraduate students who have already had a course in abstract algebra and does a decent job at presenting the material at the appropriate level for students with that background. However…The author often gives terse explanations, has very few examples and provides little if any explanation as to what was done in the examples. He also leaves most of the development of intuition about the subject to the exercises. If you are planning to use this text for a course hope that you have a good instructor who can make up for the lack of development of geometric intuition in this text.A fair portion of the harder material is presented early and I often felt like I had a hand tired behind my back while working on the problems because the tools one wants to have aren’t developed until half way through the book.About half way through the course I was using this text for I picked up a copy of Ideals, Varieties and Algorithms by Cox, Little, O’shea. That text is written far better than this one, provides many good examples and develops geometric intuition much better than this one does. Unfortunately it is written for students with basically no background in abstract algebra and on account of this goes too slowly for students with that background. I think that using Hassett’s text with Cox, Little, O’Shea’s as a supplement for when Hassett doesn’t make sense or leaves too much to the reader works reasonably well, but it would be nice to have a text at this level which didn’t require a second text to supplement its flaws.

⭐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. MistakesHassett’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 LogicHassett’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 IntuitionThis 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 GraphicsThis 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.

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