Introduction To Algebraic Geometry

Description

By (author) Cutkosky Steven Dale

Short /annotation
Presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic $0$ and positive characteristic are emphasized.


This book presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic $0$ and positive characteristic are emphasized. The basic tools of classical and modern algebraic geometry are introduced, including varieties, schemes, singularities, sheaves, sheaf cohomology, and intersection theory. Basic classical results on curves and surfaces are proved. More advanced topics such as ramification theory, Zariski”s main theorem, and Bertini”s theorems for general linear systems are presented, with proofs, in the final chapters.

With more than 200 exercises, the book is an excellent resource for teaching and learning introductory algebraic geometry.

Table of contents

• A crash course in commutative algebra
• Affine varieties
• Projective varieties
• Regular and rational maps of quasi-projective varieties
• Products
• The blow-up of an ideal
• Finite maps of quasi-projective varieties
• Dimension of quasi-projective algebraic sets
• Zariski”s main theorem
• Nonsingularity
• Sheaves
• Applications to regular and rational maps
• Divisors
• Differential forms and the canonical divisor
• Schemes
• The degree of a projective variety
• Cohomology
• Curves
• An introduction to intersection theory
• Surfaces
• Ramification and etale maps
• Bertini”s theorem and general fibers of maps
• Bibliography
• Index.

Biographical note
Steven Dale Cutkosky, University of Missouri, Columbia, MO.