On the Resolution of Higher Singularities of Algebraic Curves Into Ordinary Nodes (Classic Reprint) - Hardcover

B. M. Walker

Hardcover

ISBN 10: 0656038470 ISBN 13: 9780656038473

Publisher: Forgotten Books, 2018

Synopsis

Transforming singularities into ordinary nodes: a detailed, new approach to resolving algebraic curves

This lucid treatment presents a new proof that a plane algebraic curve with ordinary multiple points can be transformed into a space curve and projected back to a plane curve with only ordinary double points. The work builds from a geometric idea attributed to Klein, and methodically develops a concrete path to the resolution process.

The book lays out the transformation in four chapters, explaining how a given curve is moved from the plane to a cubic surface, and how successive projections reduce complex singularities to simple nodes. It discusses triply infinite linear systems of plane cubics, canonical forms, and how bases of cubics relate to the six fundamental points. Readers will see how ordinary points and multiple points behave under the transformation, and how tangents and cycles guide the process toward a curve with only ordinary nodes.

• How to model cubics through six fixed points and its implications for the transformation
• How ordinary and multiple points on the original curve map to the transformed curve
• How to control tangents and cycles to avoid coinciding singularities

Ideal for readers with a background in algebraic geometry, this edition clarifies a classic technique and its modern reformulation, offering concrete steps and careful justification for transforming higher singularities into ordinary nodes.

Edition details

• Publisher: Forgotten Books
• Publication date: 2018
• Language: English
• ISBN 10: 0656038470
• ISBN 13: 9780656038473
• Binding: Hardcover
• Number of pages: 66