**Graduate Studies in Mathematics**

Volume: 63;
2004;
186 pp;
Hardcover

MSC: Primary 13; 14; 32;

Print ISBN: 978-0-8218-3555-5

Product Code: GSM/63

List Price: $46.00

Individual Member Price: $36.80

**Electronic ISBN: 978-1-4704-2106-9
Product Code: GSM/63.E**

List Price: $46.00

Individual Member Price: $36.80

#### Supplemental Materials

# Resolution of Singularities

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*Steven Dale Cutkosky*

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.

#### Table of Contents

# Table of Contents

## Resolution of Singularities

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents v6 free
- Preface vii8 free
- Chapter 1. Introduction 110 free
- §1.1. Notation 211 free

- Chapter 2. Non-singularity and Resolution of Singularities 312
- Chapter 3. Curve Singularities 1726
- Chapter 4. Resolution Type Theorems 3746
- Chapter 5. Surface Singularities 4554
- Chapter 6. Resolution of Singularities in Characteristic Zero 6170
- §6.1. The operator Δ and other preliminaries 6271
- §6.2. Hypersurfaces of maximal contact and induction in resolution 6675
- §6.3. Pairs and basic objects 7079
- §6.4. Basic objects and hypersurfaces of maximal contact 7584
- §6.5. General basic objects 8190
- §6.6. Functions on a general basic object 8392
- §6.7. Resolution theorems for a general basic object 8998
- §6.8. Resolution of singularities in characteristic zero 99108

- Chapter 7. Resolution of Surfaces in Positive Characteristic 105114
- Chapter 8. Local Uniformization and Resolution of Surfaces 133142
- Chapter 9. Ramification of Valuations and Simultaneous Resolution 155164
- Appendix. Smoothness and Non-singularity II 163172
- Bibliography 179188
- Index 185194
- Back Cover Back Cover1198

#### Readership

Graduate students and research mathematicians interested in algebraic geometry.

#### 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 graduate-level 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.

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