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Algebraic Geometry: A Problem Solving Approach
Algebraic geometry has been at the centre of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of exercises, plus some background information and explanations, starting with conics and ending with sheaves and cohomology. The first chapter on conics is appropriate for first-year college students (and many high school students). Chapter 2 leads the reader to an understanding of the basics of cubic curves, while Chapter 3 introduces higher degree curves. Both chapters are appropriate for people who have taken multivariable calculus and linear algebra. Chapters 4 and 5 introduce geometric objects of higher dimension than curves. Abstract algebra now plays a critical role, making a first course in abstract algebra necessary from this point on. The last chapter is on sheaves and cohomology, providing a hint of current work in algebraic geometry.
- GenresMathematicsNonfiction
340 pages, Paperback
First published February 1, 2013
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Displaying 1 - 3 of 3 reviews
April 2, 2021
I adore this book. Every undergraduate textbook needs a book like this as a companion, or maybe a replacement. It's the closest thing to Moore's method in book form I've found: most of the exposition is in the form of problems (with accompanying solutions), and the problems gradually build up the story of algebraic geometry. You *could* read it as a normal theorem-proof style textbook, but you'd be missing out and honestly it wouldn't be very good (I'd pick A Royal Road to Algebraic Geometry for that).
It's partially written as an airplane book for mathematicians, and that does make it a little slow. If I was just starting on the subject I'd want a companion book and I'd skip over some of the sections and exercises. On the other hand if you're not new it really does make for excellent airplane reading.
It's partially written as an airplane book for mathematicians, and that does make it a little slow. If I was just starting on the subject I'd want a companion book and I'd skip over some of the sections and exercises. On the other hand if you're not new it really does make for excellent airplane reading.
March 25, 2022
In an ideal world (har har), I would give this book three stars because of its inconsistencies and its ill-conceived final chapter. We live, however, in an imperfect world and this is the only algebraic geometry book that I have read cover to cover. This isn't due to a lack of effort; I have wanted to learn this material for almost two decades, and this is the first resource I have found that helped me on my journey.
The book has a lovely informal style. The authors don't feel a need to impress the readers with their knowledge or cleverness. They start by using y=x^2 as a motivating example, thereby showing that most of us have already done some algebraic geometry. The book, like Gouvea's p-adic Analysis and Murty's Problems in Analytic Number Theory, is problem-motivated, and most of the work is assigned to the reader as short, do-able exercises. This works brilliantly in chapter one, and less well later on in the book.
As to be expected in a book with eleven authors, not all the pieces fit together seamlessly. The multiple definitions of regular functions, rings of regular functions, coordinate rings, and function fields do not always refer to each other, and it can be tricky to keep them straight. This is especially egregious in chapter six, where the book assumes the reader knows technical terms that are either undefined, or defined multiple ways earlier in the book. Chapter six does not tell the readers which of the earlier definitions to use.
I genuinely enjoyed the first chapter and working through the later sections on divisors and the Riemann-Roch theorem. I wish there had been more motivation for those two topics, but I feel like I am now comfortable with projective space, basic versions of the Zariski topology, Bezout's theorem, and simple divisors. I could not have said that before reading this book.
I feel like every chapter, other than the last, helped me grow as a mathematician. I hope that there is a second edition, and that it simplifies and streamlines the book. The books' problems are almost all due to ambition. I recommend this book, but take heed--do not read chapter 6.
The book has a lovely informal style. The authors don't feel a need to impress the readers with their knowledge or cleverness. They start by using y=x^2 as a motivating example, thereby showing that most of us have already done some algebraic geometry. The book, like Gouvea's p-adic Analysis and Murty's Problems in Analytic Number Theory, is problem-motivated, and most of the work is assigned to the reader as short, do-able exercises. This works brilliantly in chapter one, and less well later on in the book.
As to be expected in a book with eleven authors, not all the pieces fit together seamlessly. The multiple definitions of regular functions, rings of regular functions, coordinate rings, and function fields do not always refer to each other, and it can be tricky to keep them straight. This is especially egregious in chapter six, where the book assumes the reader knows technical terms that are either undefined, or defined multiple ways earlier in the book. Chapter six does not tell the readers which of the earlier definitions to use.
I genuinely enjoyed the first chapter and working through the later sections on divisors and the Riemann-Roch theorem. I wish there had been more motivation for those two topics, but I feel like I am now comfortable with projective space, basic versions of the Zariski topology, Bezout's theorem, and simple divisors. I could not have said that before reading this book.
I feel like every chapter, other than the last, helped me grow as a mathematician. I hope that there is a second edition, and that it simplifies and streamlines the book. The books' problems are almost all due to ambition. I recommend this book, but take heed--do not read chapter 6.
March 25, 2022
In an ideal world (har har), I would give this book three stars because of its inconsistencies and its ill-conceived final chapter. We live, however, in an imperfect world and this is the only algebraic geometry book that I have read cover to cover. This isn't due to a lack of effort; I have wanted to learn this material for almost two decades, and this is the first resource I have found that helped me on my journey.
The book has a lovely informal style. The authors don't feel a need to impress the readers with their knowledge or cleverness. They start by using y=x^2 as a motivating example, thereby showing that most of us have already done some algebraic geometry. The book, like Gouvea's p-adic Analysis and Murty's Problems in Analytic Number Theory, is problem-motivated, and most of the work is assigned to the reader as short, do-able exercises. This works brilliantly in chapter one, and less well later on in the book.
As to be expected in a book with eleven authors, not all the pieces fit together seamlessly. The multiple definitions of regular functions, rings of regular functions, coordinate rings, and function fields do not always refer to each other, and it can be tricky to keep them straight. This is especially egregious in chapter six, where the book assumes the reader knows technical terms that are either undefined, or defined multiple ways earlier in the book. Chapter six does not tell the readers which of the earlier definitions to use.
I genuinely enjoyed the first chapter and working through the later sections on divisors and the Riemann-Roch theorem. I wish there had been more motivation for those two topics, but I feel like I am now comfortable with projective space, basic versions of the Zariski topology, Bezout's theorem, and simple divisors. I could not have said that before reading this book.
I feel like every chapter, other than the last, helped me grow as a mathematician. I hope that there is a second edition, and that it simplifies and streamlines the book. The books' problems are almost all due to ambition. I recommend this book, but take heed--do not read chapter 6.
The book has a lovely informal style. The authors don't feel a need to impress the readers with their knowledge or cleverness. They start by using y=x^2 as a motivating example, thereby showing that most of us have already done some algebraic geometry. The book, like Gouvea's p-adic Analysis and Murty's Problems in Analytic Number Theory, is problem-motivated, and most of the work is assigned to the reader as short, do-able exercises. This works brilliantly in chapter one, and less well later on in the book.
As to be expected in a book with eleven authors, not all the pieces fit together seamlessly. The multiple definitions of regular functions, rings of regular functions, coordinate rings, and function fields do not always refer to each other, and it can be tricky to keep them straight. This is especially egregious in chapter six, where the book assumes the reader knows technical terms that are either undefined, or defined multiple ways earlier in the book. Chapter six does not tell the readers which of the earlier definitions to use.
I genuinely enjoyed the first chapter and working through the later sections on divisors and the Riemann-Roch theorem. I wish there had been more motivation for those two topics, but I feel like I am now comfortable with projective space, basic versions of the Zariski topology, Bezout's theorem, and simple divisors. I could not have said that before reading this book.
I feel like every chapter, other than the last, helped me grow as a mathematician. I hope that there is a second edition, and that it simplifies and streamlines the book. The books' problems are almost all due to ambition. I recommend this book, but take heed--do not read chapter 6.
Displaying 1 - 3 of 3 reviews