A Concise Introduction to Algebraic Varieties

For a quick and reader-friendly short introduction to algebraic varieties this new book could be a nice choice for a one-semester course. With a minimum of commutative algebra background, summarized in one appendix, and a bare minimum of point-set topology, this textbook covers the basics of abstract algebraic varieties over an algebraically closed field. From affine varieties and regular functions and morphisms on them, vanishing ideals, rational maps, dimension, tangent spaces and singularities, to abstract algebraic varieties, including the important example of projective varieties with its homogeneous ideals and coordinate rings, the book systematically develops the theory with plenty of examples and many exercises interspersed throughout the text.

 

A variation in the exposition is the definition of abstract algebraic varieties using atlases instead of sheaves of regular functions, following a path close to the one used in differential geometry or topology. Thus, the exposition is shorter, allowing the reader to get familiarized with some geometric aspects of algebraic varieties, leaving the more advanced themes for a more advanced course.

 

The last four chapters of the book are devoted to algebraic curves, with several goals in the development. First, this is a family of varieties with a rich theory that motivates the introduction of several important concepts and properties in a simpler case. At the same time, the theory of algebraic curves serves as a testing ground for methods and ideas that can be used for higher dimensional varieties, opening a landscape to be explored by an interested reader in more advanced textbooks.

 

Comparing this textbook with others on the same level, the result is favorable: The approach is fresh, accesible with a basic background on commutative algebra, and with a good mixture of local and global aspects with motivations, examples and well-chosen proofs that balance the algebraic

and geometric sides of a given topic.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx