Hardcover ISBN:  9780821835555 
Product Code:  GSM/63 
186 pp 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $39.20 
Electronic ISBN:  9781470421069 
Product Code:  GSM/63.E 
186 pp 
List Price:  $46.00 
MAA Member Price:  $41.40 
AMS Member Price:  $36.80 

Book DetailsGraduate Studies in MathematicsVolume: 63; 2004MSC: Primary 13; 14; 32;
The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic.
The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of \(\mathcal{D}\)modules, topology, and mathematical physics.
This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic.
Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text.
The book is suitable for a second course on an exciting topic in algebraic geometry. A core course on resolutions is contained in Chapters 2 through 6. Additional topics are covered in the final chapters. The prerequisite is a course covering the basic notions of schemes and sheaves.ReadershipGraduate students and research mathematicians interested in algebraic geometry.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Nonsingularity and resolution of singularities

Chapter 3. Curve singularities

Chapter 4. Resolution type theorems

Chapter 5. Surface singularities

Chapter 6. Resolution of singularities in characteristic zero

Chapter 7. Resolution of surfaces in positive characteristic

Chapter 8. Local uniformization and resolution of surfaces

Chapter 9. Ramification of valuations and simultaneous resolution

Appendix. Smoothness and nonsingularity II


Additional Material

Reviews

This book, based on the author's lectures at the University of Missouri and the Chennai Mathematics Institute, presents a purely algebraic approach to the resolution of singularities...requires the level of knowledge of algebraic geometry and commutative algebra usually covered in an introductory graduatelevel course. ... It is suitable for anyone who wants to learn about the algebraic theory of resolution of singularities and read a reasonably short proof of the existence of resolutions in characteristic zero.
Bulletin of the London Mathematical Society 
It has been a pleasure for the reviewer to read this beautiful book, which is a must for graduate students interested in the subject. It fills a gap in graduate texts, covering the most important results in resolution of singularities in an elegant and didactic style.
Mathematical Reviews


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 Request Review Copy
 Get Permissions
The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic.
The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of \(\mathcal{D}\)modules, topology, and mathematical physics.
This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic.
Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text.
The book is suitable for a second course on an exciting topic in algebraic geometry. A core course on resolutions is contained in Chapters 2 through 6. Additional topics are covered in the final chapters. The prerequisite is a course covering the basic notions of schemes and sheaves.
Graduate students and research mathematicians interested in algebraic geometry.

Chapters

Chapter 1. Introduction

Chapter 2. Nonsingularity and resolution of singularities

Chapter 3. Curve singularities

Chapter 4. Resolution type theorems

Chapter 5. Surface singularities

Chapter 6. Resolution of singularities in characteristic zero

Chapter 7. Resolution of surfaces in positive characteristic

Chapter 8. Local uniformization and resolution of surfaces

Chapter 9. Ramification of valuations and simultaneous resolution

Appendix. Smoothness and nonsingularity II

This book, based on the author's lectures at the University of Missouri and the Chennai Mathematics Institute, presents a purely algebraic approach to the resolution of singularities...requires the level of knowledge of algebraic geometry and commutative algebra usually covered in an introductory graduatelevel course. ... It is suitable for anyone who wants to learn about the algebraic theory of resolution of singularities and read a reasonably short proof of the existence of resolutions in characteristic zero.
Bulletin of the London Mathematical Society 
It has been a pleasure for the reviewer to read this beautiful book, which is a must for graduate students interested in the subject. It fills a gap in graduate texts, covering the most important results in resolution of singularities in an elegant and didactic style.
Mathematical Reviews