This guide is an informal and accessible introduction to plane algebraic curves. It also serves as an entry point to algebraic geometry, which is playing an ever-expanding role in areas ranging from biology and chemistry to robotics and cryptology.
The guide's unifying theme is: Give curves enough living space and beautiful theorems will follow.
By keeping the exposition simple and readily understandable, and by introducing abstract concepts with concrete examples and pictures, the book offers readers a lucid overview of the subject. It can also be used as the text in an undergraduate course on plane algebraic curves, or as a companion to algebraic geometry at the graduate level.
Table of Contents
1. A Gallery of Algebraic Curves
2. Points at Infinity
3. From Real to Complex
4. Topology of Algebraic Curves in P2 (C)
6. The Big Three: C, K, S
Excerpt: Preface (p.X)
...there exist curves--many with very simple defining polynomials--that bend, twist and contort so much that in order to fit in the plane, they must have self-intersections and/or kinks. Such points are rare (accounting for their name “singularities”), but rare or not, questions arise:
What do curves look like around singularities?
Are some singularities easily understood, while others are more complicated?
How is their number and type related to the amount of twisting and contorting of the curve?
For a curve with singularities, what happens to Bézout's theorem?
For a curve with singularities, what happens to that remarkably simple genus formula?
Can you transform a curve with singularities into a curve without singularities?
About the Author
Keith Kendig (Cleveland State University) is the author of two other MAA books: Conics and Sink or Float? Thought Problems in Math and Physics. He serves on the editorial boards of Mathematics Magazine and the Spectrum series of MAA books.
I have admired Keith Kendig’s expository skills for about 35 years now, ever since I picked up, as a graduate student, his Graduate Texts in Mathematics volume (#44) on algebraic geometry, which I thought at the time was the clearest and most accessible available entrée into the subject. Unfortunately, in the decades since I first read that book, I have not had a great deal of time to spend on algebraic geometry, and so, while I once knew, for example, what the Riemann-Roch theorem said and meant, I had long since forgotten that. If, a month ago, somebody had asked me to state the theorem, I could only have mumbled something about “divisors” and hope I would not be asked just what a divisor was.
What a pleasure, therefore, for me to, after all these years, once again be reminded by Kendig of the joys of algebraic geometry, this time at a much more expository level than in his earlier book — exactly what I, as a mathematically-trained person with no plans to do research in the area but with a desire to at least have an overview of the subject, needed. Continued...