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An Invitation to Algebraic Geometry
This is a description of the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra.
- GenresMathematicsGeometry
180 pages, Paperback
First published October 31, 2000
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70 people want to read
About the author
Karen E. Smith
3 booksLibrarian Note: There is more than one author by this name in the Goodreads database.
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Displaying 1 - 5 of 5 reviews
March 10, 2019
One of the clearest introductions to the subject. The book is beautifully written and the pictures are greatly elucidating, causing one to wonder why illustrations are not more common in books on geometry (one might correctly guess 'elitism'). The book is entertaining and has the right balancing between rigour and intuition. Some exercises are easy, and some are a bit more challenging, but almost never too difficult. Rigorous proofs of hard statements are sometimes deferred to other references, and rightly so, but nevertheless the theorems are explained well within the context.
My only gripe is chapter 5 - I feel that some more emphasis on the important examples of Segre, Veronese and Grassmannian varieties would have been welcome, and in the last part on the Hilbert polynomial, I felt it skipped too much detail for me to follow or do the exercises.
All throughout the book, the author is constantly reminding us of the relationship and/or benefit with the modern language of schemes, which reinforced my motivation to follow through to their study. Indeed, especially in the later chapters (6-8), the repeated (yet artful) references to Hartshorne made me quite eager to quickly finish the book so that I could quench my thirst on the beautiful geometric theorems presented there!
My only gripe is chapter 5 - I feel that some more emphasis on the important examples of Segre, Veronese and Grassmannian varieties would have been welcome, and in the last part on the Hilbert polynomial, I felt it skipped too much detail for me to follow or do the exercises.
All throughout the book, the author is constantly reminding us of the relationship and/or benefit with the modern language of schemes, which reinforced my motivation to follow through to their study. Indeed, especially in the later chapters (6-8), the repeated (yet artful) references to Hartshorne made me quite eager to quickly finish the book so that I could quench my thirst on the beautiful geometric theorems presented there!
February 18, 2020
This would be my go-to recommendation as a first exposure to algebraic geometry. It's nothing more than a presentation of some basic ideas and motivating questions of the subject, with minimal technical machinery, and plenty of examples. This book is almost entirely aimed at giving a high-level picture of what the subject is about, with strong geometric intuition, and almost no technical details. There are exercises, but they are inessential. As such, this book lends itself very well to being read for pleasure or as a source of intuition, to supplement a more thorough and technical account from some other text. The book is also not afraid to provide occasional glimpses of the modern generalizations of the classical theory, for example by giving an idea of how scheme theory generalizes a geometric concept, without going into detail. Anyone with a background in basic topology and elementary ring theory could pick up this book.
October 3, 2024
the wild Amazone
For the Advanced Mathematician
4/10
The claim made by the advertisement for this book is that all the prerequisite necessary is a course in Linear Algebra. I just finished reading 4 textbooks on that subject and have a couple of university courses in it too. I found the present book unintelligible. There is so much symbolism that isn't defined. It may be a great book for the advanced mathematician, but not as an introductory text for someone wanting to teach themselves the Algebraic Geometry. If you are already an expert, this might be for you. If not, stay away. The format is much smaller than I anticipated as well.
Paul T. Cusack
//////
Very good, but understated prerequisites
On the back of the book, it says "Few algebraic prerequisites are presumed beyond a basic course in linear algebra." This is not, in fact, true.
It uses a lot of ring theory (and the review of commutative ring theory in ch. 2 is a bit fast for someone unfamiliar with the subject), and a fair amount of topology.
When I first got it, I read the first several pages and found them readable, but when I read more (on the car-ride home) I was confronted with the fact that the Zariski topology is coarser than the standard topology on C^n, and is not even Hausdorff.
Several questions came to mind
What's a topology?
What does for one to be coarser than another?
What is a Hausdorff topology?
Still, after I learned more topology, I found the book a delight.
Everything is light and interesting, and does a good job of portraying algebraic geometry without technical details. All mathematics looks nicer when you do that, but it takes away from rigor. Hence this should not be your only text on Algebraic Geometry, but I would suggest it as one of them...
asdfasdfasdf
//////
Wow!
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.
Colin McLarty
//////
Brilliant introduction to the subject
This book sets aside a lot of the tedious background required to get started in algebraic geometry and lets you get acquainted with the actual ideas of the subject. The section on vector bundles at the end could stand to be a little longer, but it's the best book on algebraic geometry that I know of.
Daniel McLaury
/////
Splendid introduction
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.
/////
Very good for my level - entry level
I have only some knowledge on differential geometry and algebraic topology. This is a very good starting material for my level. I can understand the content easily. This book focuses on the key ideas of the topic, without being bothered by the algebraic details.
Xipan
/////
For the Advanced Mathematician
4/10
The claim made by the advertisement for this book is that all the prerequisite necessary is a course in Linear Algebra. I just finished reading 4 textbooks on that subject and have a couple of university courses in it too. I found the present book unintelligible. There is so much symbolism that isn't defined. It may be a great book for the advanced mathematician, but not as an introductory text for someone wanting to teach themselves the Algebraic Geometry. If you are already an expert, this might be for you. If not, stay away. The format is much smaller than I anticipated as well.
Paul T. Cusack
//////
Very good, but understated prerequisites
On the back of the book, it says "Few algebraic prerequisites are presumed beyond a basic course in linear algebra." This is not, in fact, true.
It uses a lot of ring theory (and the review of commutative ring theory in ch. 2 is a bit fast for someone unfamiliar with the subject), and a fair amount of topology.
When I first got it, I read the first several pages and found them readable, but when I read more (on the car-ride home) I was confronted with the fact that the Zariski topology is coarser than the standard topology on C^n, and is not even Hausdorff.
Several questions came to mind
What's a topology?
What does for one to be coarser than another?
What is a Hausdorff topology?
Still, after I learned more topology, I found the book a delight.
Everything is light and interesting, and does a good job of portraying algebraic geometry without technical details. All mathematics looks nicer when you do that, but it takes away from rigor. Hence this should not be your only text on Algebraic Geometry, but I would suggest it as one of them...
asdfasdfasdf
//////
Wow!
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.
Colin McLarty
//////
Brilliant introduction to the subject
This book sets aside a lot of the tedious background required to get started in algebraic geometry and lets you get acquainted with the actual ideas of the subject. The section on vector bundles at the end could stand to be a little longer, but it's the best book on algebraic geometry that I know of.
Daniel McLaury
/////
Splendid introduction
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.
/////
Very good for my level - entry level
I have only some knowledge on differential geometry and algebraic topology. This is a very good starting material for my level. I can understand the content easily. This book focuses on the key ideas of the topic, without being bothered by the algebraic details.
Xipan
/////
January 28, 2019
This is one of the clearest introductions to the topic that I have encountered. As a physicist, I found this to be highly accessible and would definitely recommend it to others interested in learning the subject.
November 5, 2023
One of those books that restores your faith in mathematical exposition.
Displaying 1 - 5 of 5 reviews