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Book DetailsGraduate Studies in MathematicsVolume: 216; 2021; 259 ppMSC: Primary 14
Designed for a one-term introductory course on algebraic varieties over an algebraically closed field, this book prepares students to continue either with a course on schemes and cohomology, or to learn more specialized topics such as toric varieties and moduli spaces of curves. The book balances generality and accessibility by presenting local and global concepts, such as nonsingularity, normality, and completeness using the language of atlases, an approach that is most commonly associated with differential topology. The book concludes with a discussion of the Riemann-Roch theorem, the Brill-Noether theorem, and applications.
The prerequisites for the book are a strong undergraduate algebra course and a working familiarity with basic point-set topology. A course in graduate algebra is helpful but not required. The book includes appendices presenting useful background in complex analytic topology and commutative algebra and provides plentiful examples and exercises that help build intuition and familiarity with algebraic varieties.
ReadershipUndergraduate and graduate students interested in an introduction to fundamentals of algebraic geometry.
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Table of Contents
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Chapters
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Introduction: An overview of algebraic geometry through the lens of plane curves
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Affine algebraic varieties
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Regular functions and morphisms
-
Singularities
-
Abstract varieties via atlases
-
Projective varieties
-
Nonsingular curves and complete varieties
-
Divisors on nonsingular curves
-
Differential forms
-
An invitation to the theory of algebraic curves
-
Complex varieties and the analytic topology
-
A roadmap through algebra
-
-
Additional Material
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Reviews
-
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.
...Comparing this textbook with others on the same level, the result is favorable: The approach is fresh, accessible 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.
Felipe Zaldivar, Universidad Autonoma Metropolitana-I
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RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Reviews
- Requests
Designed for a one-term introductory course on algebraic varieties over an algebraically closed field, this book prepares students to continue either with a course on schemes and cohomology, or to learn more specialized topics such as toric varieties and moduli spaces of curves. The book balances generality and accessibility by presenting local and global concepts, such as nonsingularity, normality, and completeness using the language of atlases, an approach that is most commonly associated with differential topology. The book concludes with a discussion of the Riemann-Roch theorem, the Brill-Noether theorem, and applications.
The prerequisites for the book are a strong undergraduate algebra course and a working familiarity with basic point-set topology. A course in graduate algebra is helpful but not required. The book includes appendices presenting useful background in complex analytic topology and commutative algebra and provides plentiful examples and exercises that help build intuition and familiarity with algebraic varieties.
Undergraduate and graduate students interested in an introduction to fundamentals of algebraic geometry.
-
Chapters
-
Introduction: An overview of algebraic geometry through the lens of plane curves
-
Affine algebraic varieties
-
Regular functions and morphisms
-
Singularities
-
Abstract varieties via atlases
-
Projective varieties
-
Nonsingular curves and complete varieties
-
Divisors on nonsingular curves
-
Differential forms
-
An invitation to the theory of algebraic curves
-
Complex varieties and the analytic topology
-
A roadmap through algebra
-
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.
...Comparing this textbook with others on the same level, the result is favorable: The approach is fresh, accessible 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.
Felipe Zaldivar, Universidad Autonoma Metropolitana-I
