Resolution of Singularities

Front Cover
American Mathematical Soc., 2004 - Mathematics - 186 pages
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.
 

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Contents

Chapter 1 Introduction
1
Chapter 2 Nonsingularity and Resolution of Singularities
3
Chapter 3 Curve Singularities
17
Chapter 4 Resolution Type Theorems
37
Chapter 5 Surface Singularities
45
Chapter 6 Resolution of Singularities in Characteristic Zero
61
Chapter 7 Resolution of Surfaces in Positive Characteristic
105
Chapter 8 Local Uniformization and Resolution of Surfaces
133
Chapter 9 Ramification of Valuations and Simultaneous Resolution
155
Appendix Smoothness and Nonsingularity II
163
Bibliography
179
Index
185
Back Cover
189
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