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Resolution of Singularities
Steven Dale Cutkosky, University of Missouri, Columbia, MO
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2004; 186 pp; hardcover
Volume: 63
ISBN-10: 0-8218-3555-6
ISBN-13: 978-0-8218-3555-5
List Price: US$43 Member Price: US$34.40
Order Code: GSM/63

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.

Reviews

"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

• Introduction
• Non-singularity and resolution of singularities
• Curve singularities
• Resolution type theorems
• Surface singularities
• Resolution of singularities in characteristic zero
• Resolution of surfaces in positive characteristic
• Local uniformization and resolution of surfaces
• Ramification of valuations and simultaneous resolution
• Smoothness and non-singularity
• Bibliography
• Index