Algebraic Geometry 3: Further Study of Schemes (Translations of Mathematical Monographs)

Algebraic geometry plays an important role in several branches of science and technology. This book discusses dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion, and Zariski's main theorem. It also presents the theory of algebraic curves and their Jacobians.

This is the third part of the textbook on algebraic geometry by Kenji Ueno (the first two parts were published by the AMS as Volumes 185 and 197 of this series). Here the author presents the theory of schemes and sheaves beyond introductory notions, with the goal of studying properties of schemes and coherent sheaves necessary for full development of modern algebraic geometry. The main topics discussed in the book include dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion and Zariski's main theorem. The author also presents the theory of algebraic curves and their Jacobians and the relation between algebraic and analytic geometry, including Kodaira's Vanishing Theorem. The book contains numerous exercises and problems with solutions, which makes it (together with two previous parts) appropriate for a graduate course on algebraic geometry or for self-study.

Algebraic geometry plays an important role in several branches of science and technology. This is the last of three volumes by Kenji Ueno algebraic geometry. This, in together with Algebraic Geometry 1 and Algebraic Geometry 2, makes an excellent textbook for a course in algebraic geometry. In this volume, the author goes beyond introductory notions and presents the theory of schemes and sheaves with the goal of studying the properties necessary for the full development of modern algebraic geometry. The main topics discussed in the book include dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion, and Zariski's main theorem. Ueno also presents the theory of algebraic curves and their Jacobians and the relation between algebraic and analytic geometry, including Kodaira's Vanishing Theorem. 1. From algebraic varieties to schemes 2. Sheaves and cohomology 3. Further study of schemes.