Algebraic Geometry has been at the center 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.
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Format:Paperback
Over the last twenty-five years there have been many excellent books written on algebraic geometry -- off the top of my head, Kunz, Perrin, Gibson, Shafarevich, Brieskorn and Knorrer, Fischer, and Miranda come to mind. These books are pitched at various levels -- Gibson being the most elementary and Shafarevich, Perrin, and Miranda being pitched at a grad level. But none of these books take one from an elementary level to things like Riemann-Roch, sheaves, and cohomology. As far as I know the book by Garrity et al is the first that does so. As such, it's ideal for an upper division undergrad course or as a supplementary text for a grad course.
The book starts with establishing the equivalence of conics in the complex projective plane and then moves on smoothly to discussing tangents and singularities, elliptic curves, Bezout's theorem, Riemann-Roch, affine and projective varieties, and -- finally -- a brief intro to sheaves and cohomology. Motivation for generalisation is provided each step of the way -- no abstract axiomatic development here. After finishing the book, a student would be prepared to tackle Perrin, Shafarevich, or even Hartshorne.
I admire the smooth and seamless way the authors move from topic to topic, building the subject from simple and intuitive beginnings to things like Riemann-Roch. It's difficult to find books that take one from the elementary and classical geometry of algebraic curves to modern algebraic geometry.
The book develops mostly through problems -- but the problems aren't difficult, largely computational, and what I consider to be almost routine in character. Anyone who has completed an undergrad course in abstract algebra should have no problem.
For these reasons I argue that the book is the best introduction to algebraic geometry currently on the market.
The book starts with establishing the equivalence of conics in the complex projective plane and then moves on smoothly to discussing tangents and singularities, elliptic curves, Bezout's theorem, Riemann-Roch, affine and projective varieties, and -- finally -- a brief intro to sheaves and cohomology. Motivation for generalisation is provided each step of the way -- no abstract axiomatic development here. After finishing the book, a student would be prepared to tackle Perrin, Shafarevich, or even Hartshorne.
I admire the smooth and seamless way the authors move from topic to topic, building the subject from simple and intuitive beginnings to things like Riemann-Roch. It's difficult to find books that take one from the elementary and classical geometry of algebraic curves to modern algebraic geometry.
The book develops mostly through problems -- but the problems aren't difficult, largely computational, and what I consider to be almost routine in character. Anyone who has completed an undergrad course in abstract algebra should have no problem.
For these reasons I argue that the book is the best introduction to algebraic geometry currently on the market.
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