AFFINE ALGEBRAIC GEOMETRY: GEOMETRY OF POLYNOMIAL RINGS

Algebraic geometry is more advanced with the completeness condition for projective or complete varieties. Many geometric properties are well described by the finiteness or the vanishing of sheaf cohomologies on such varieties. For non-complete varieties like affine algebraic varieties, sheaf cohomology does not work well and research progress used to be slow, although affine spaces and polynomial rings are fundamental building blocks of algebraic geometry. Progress was rapid since the Abhyankar–Moh–Suzuki Theorem of embedded affine line was proved, and logarithmic geometry was introduced by Iitaka and Kawamata.

Readers will find the book covers vast basic material on an extremely rigorous level:

Contents:

• Preface
• Introduction to Algebraic Geometry
• Geometry on Affine Surfaces
• Geometry and Topology of Polynomial Rings — Motivated by the Jacobian Problem
• Postscript
• Bibliography
• Index

Readership: Mathematics students, both undergraduate and graduate, where knowledge of group, ring and linear algebra is required, and researchers. If the book is used as a textbook, it is for students in the beginning class of algebraic geometry and commutative algebra.

Key Features:

• One of the first textbooks ever to explain polynomial rings and affine spaces
• Description of contents is simple, concrete and easier to understand so that readers have as less obstacles as possible
• Almost all results are proved when they are used or referred to, except for the knowledge of algebra or topology to be acquired in the undergraduate study
• At the end of each chapter, there are many problems attached, which the author hope to guide the readers to advance into this area of research