Resolution of Singularities of Embedded Algebraic Surfaces

Besides the description of the geometric part of the author's 1965 proof of desingularization of algebraic surfaces and solids in nonzero characteristic, this new edition provides a self-contained introduction to birational algebraic geometry, based only on commutative algebra. It also gives a short proof of analytic desingularization in characteristic zero for any dimension found in 1996.

Description of the author's proof of desingularization of algebraic surfaces Self-contained introduction to birational algebraic geometry, based only on basic commutative algebra. The unique place where desigularization for solids in characteristic p is done

Klappentext

This new edition describes the geometric part of the author's 1965 proof of desingularization of algebraic surfaces and solids in nonzero characteristic. The book also provides a self-contained introduction to birational algebraic geometry, based only on basic commutative algebra. In addition, it gives a short proof of analytic desingularization in characteristic zero for any dimension found in 1996 and based on a new avatar of an algorithmic trick employed in the original edition of the book. This new edition will inspire further progress in resolution of singularities of algebraic and arithmetical varieties which will be valuable for applications to algebraic geometry and number theory. The book can be used for a second year graduate course. The reference list has been updated.

Inhalt
0 Introduction.- 1. Local Theory.- 1 Terminology and preliminaries.- 2 Resolvers and principalizers.- 3 Dominant character of a normal sequence.- 4 Unramified local extensions.- 5 Main results.- 2. Global Theory.- 6 Terminology and preliminaries.- 7 Global resolvers.- 8 Global principalizers.- 9 Main results.- 3. Some Cases of Three-Dimensional Birational Resolution.- 10 Uniformization of points of small multiplicity.- 11 Three-dimensional birational resolution over a ground field of characteristic zero.- 12 Existence of projective models having only points of small multiplicity.- 13 Three-dimensional birational resolution over an algebraically closed ground field of charateristic ? 2, 3, 5.- Appendix on Analytic Desingularization in Characteristic Zero.- Additional Bibliography.- Index of Notation.- Index of Definitions.- List of Corrections.