The coverage of these first three chapters is roughly comparable to that of A Guide to Plane Algebraic Curves by Keith Kendig, another (very good) book that eschews technical details in favor of intuition and examples (although in that book the reader is not expected to work through problems to get the details). In addition, parts of chapter 3 require some background in complex analysis. While the term “group” is defined from scratch, the authors do seem to expect some prior knowledge of rings and fields, at least to the extent, for example, that the reader would understand phrases like “field of characteristic p”. Since groups, rings and fields are all mentioned here (for example: the group law on a cubic, the ring of regular functions, and function fields), at least some prior exposure to these ideas would be valuable. The advertising material on the back cover of the book, repeating a statement made in the preface, says that these chapters are “appropriate for people who have taken multivariable calculus and linear algebra”, but by this point in the book that is an overly optimistic statement. The next two chapters are also about curves, but of higher degree: chapter 2 takes the discussion from conics to cubics (including the group law on a smooth cubic) and chapter 3 looks at curves of higher degree (culminating in the Riemann-Roch theorem). By the end of the chapter the reader has not only seen that the complex projective line is a sphere and that non-degenerate conics are equivalent to spheres, but has also seen connections between Pythagorean triples and the geometry of conics. ![]() Assuming, as far as I could tell, only the rudiments of matrix language and notation, the authors discuss conics over the real numbers and then over the complex numbers, and then introduce projective spaces. To be more specific: chapter 1 is on conic sections. The book starts with elementary material that should be comprehensible to people with only a modest mathematics background and gradually works its way to considerably more sophisticated mathematics. For one thing, the organization of the material is superb. There is a caveat, however: this is, as the title makes very clear, a problem book rather than a textbook, and therefore the extent to which this book will succeed for any particular reader depends on that reader’s willingness to buy into a “Moore method”-style program. The book under review is one such, and is certainly one that should be looked at carefully by anybody contemplating teaching a course in the subject or wanting to learn it by self-study. ![]() Algebraic geometry is a notoriously difficult subject for a novice to get the hang of, and therefore any book that is intended to make this subject accessible to beginners deserves serious consideration.
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