eBook ISBN:  9781470412623 
Product Code:  SURV/35.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9781470412623 
Product Code:  SURV/35.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsMathematical Surveys and MonographsVolume: 35; 1990; 295 ppMSC: Primary 14; Secondary 00
This book, based on lectures presented in courses on algebraic geometry taught by the author at Purdue University, is intended for engineers and scientists (especially computer scientists), as well as graduate students and advanced undergraduates in mathematics. In addition to providing a concrete or algorithmic approach to algebraic geometry, the author also attempts to motivate and explain its link to more modern algebraic geometry based on abstract algebra. The book covers various topics in the theory of algebraic curves and surfaces, such as rational and polynomial parametrization, functions and differentials on a curve, branches and valuations, and resolution of singularities. The emphasis is on presenting heuristic ideas and suggestive arguments rather than formal proofs. Readers will gain new insight into the subject of algebraic geometry in a way that should increase appreciation of modern treatments of the subject, as well as enhance its utility in applications in science and industry.

Table of Contents

Chapters

Lecture 1. Rational and polynomial parametrizations

Lecture 2. Fractional linear transformations

Lecture 3. Cubic curves

Lecture 4. Cubic surfaces and general hypersurfaces

Lecture 5. Outline of the theory of plane curves

Lecture 6. Affine plane and projective plane

Lecture 7. Sphere with handles

Lecture 8. Functions and differentials on a curve

Lecture 9. Polynomials and power series

Lecture 10. Review of abstract algebra

Lecture 11. Some commutative algebra

Lecture 12. Hensel’s lemma and Newton’s theorem

Lecture 13. More about Newton’s theorem

Lecture 14. Branches and valuations

Lecture 15. Divisors of functions and differentials

Lecture 16. Weierstrass preparation theorem

Lecture 17. Intersection multiplicity

Lecture 18. Resolution of singularities of plane curves

Lecture 19. Infinitely near singularities

Lecture 20. Parametrizing a quartic with three double points

Lecture 21. Characteristic pairs

Lecture 22. Criterion for one place and Jacobian problem

Lecture 23. Inversion formula and Jacobian problem

Lecture 24. Surfaces

Lecture 25. Hypersurfaces

Lecture 26. Resolution of singularities of algebraic surfaces

Lecture 27. Birational and polyrational transformations

Lecture 28. Valuations and birational correspondence

Lecture 29. Rational cylinders through a variety

Lecture 30. Resultants


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
This book, based on lectures presented in courses on algebraic geometry taught by the author at Purdue University, is intended for engineers and scientists (especially computer scientists), as well as graduate students and advanced undergraduates in mathematics. In addition to providing a concrete or algorithmic approach to algebraic geometry, the author also attempts to motivate and explain its link to more modern algebraic geometry based on abstract algebra. The book covers various topics in the theory of algebraic curves and surfaces, such as rational and polynomial parametrization, functions and differentials on a curve, branches and valuations, and resolution of singularities. The emphasis is on presenting heuristic ideas and suggestive arguments rather than formal proofs. Readers will gain new insight into the subject of algebraic geometry in a way that should increase appreciation of modern treatments of the subject, as well as enhance its utility in applications in science and industry.

Chapters

Lecture 1. Rational and polynomial parametrizations

Lecture 2. Fractional linear transformations

Lecture 3. Cubic curves

Lecture 4. Cubic surfaces and general hypersurfaces

Lecture 5. Outline of the theory of plane curves

Lecture 6. Affine plane and projective plane

Lecture 7. Sphere with handles

Lecture 8. Functions and differentials on a curve

Lecture 9. Polynomials and power series

Lecture 10. Review of abstract algebra

Lecture 11. Some commutative algebra

Lecture 12. Hensel’s lemma and Newton’s theorem

Lecture 13. More about Newton’s theorem

Lecture 14. Branches and valuations

Lecture 15. Divisors of functions and differentials

Lecture 16. Weierstrass preparation theorem

Lecture 17. Intersection multiplicity

Lecture 18. Resolution of singularities of plane curves

Lecture 19. Infinitely near singularities

Lecture 20. Parametrizing a quartic with three double points

Lecture 21. Characteristic pairs

Lecture 22. Criterion for one place and Jacobian problem

Lecture 23. Inversion formula and Jacobian problem

Lecture 24. Surfaces

Lecture 25. Hypersurfaces

Lecture 26. Resolution of singularities of algebraic surfaces

Lecture 27. Birational and polyrational transformations

Lecture 28. Valuations and birational correspondence

Lecture 29. Rational cylinders through a variety

Lecture 30. Resultants