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Conics and Cubics: A Concrete Introduction to Algebraic Curves

Robert Bix
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Donald L. Vestal
, on

A professor once told me, in class, that algebraic geometry is a beautiful subject. He also said, outside of class, that algebraic geometry is a hard subject. So it’s no surprise that this is a topic generally reserved for graduate students. But could it, or at least some of it, be presented, at the undergraduate level? This book attempts to do that. It introduces the subject at a very concrete level. The polynomial equations studied are at most degree three, so the curves are all lines, conics, or cubics.

In the first chapter, projective geometry is introduced to study how curves intersect (not just in the Euclidean plane, but also at infinity). The intersection of lines and curves is studied as a precursor to Bezout’s Theorem. The second chapter focuses on conics and the idea of transforming these curves into a given conic. More work is done with the intersection of curves, including the notion of “peeling off a conic” from the intersection of two curves. At the end, projective geometry is used to develop the duality of a conic and its envelope. In the third chapter, the author proves that all irreducible cubics can be reduced an elliptic curve y2 = x3 + fx2 + gx + h. The group structure of rational points on this elliptic curve is also (briefly) explored. Complex numbers are introduced and Bezout’s Theorem and the Fundamental Theorem of Algebra are proven. The final chapter uses parameterization to determine intersection multiplicities of curves, and the duality of conics and their envelopes is extended to higher dimensions.

At the beginning of each of the four chapters the author provides a synopsis of the historical development of the subject. And within each section, many exercises are provided for further discussion and illumination. They tend to be one of three kinds: a very concrete problem involving specific polynomials, a concrete problem involving geometry, or a problem asking for a proof. Below are three examples.

Prove that the cubic C: x2y = x3 + 1 is irreducible. Prove that C is singular at the point Q at infinity on vertical lines. Determine how many lines through Q intersect C three times there.

Consider the following theorem: In the projective plane, let A, C, D, E, F be five points on a conic. Then the points Q =  tan A  DE, R = AC  ∩ EF, and S = CD ∩ FA are collinear. State the version of this theorem that holds in the Euclidean plane when A is the only point at infinity named.

Let L = 0 be the tangent line to a cubic C = 0 at a flex P. If C is reducible, prove that L is a factor of C.

How accessible is the material to an undergraduate? The material in this book requires a decent background in algebra and geometry. (A little intuition is helpful too.) Very little knowledge of calculus is needed: just the concept of a tangent line. And the author manages to keep things concrete (especially in the exercises). So the end result is a book which is accessible to a motivated undergrad.

Donald L. Vestal is Assistant Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)

 Intersection of Curves.- Conics.- Cubics.- Intersection Properties.- References.- Index.