Desingularization: Invariants and Strategy

Produktbeschreibung

This book provides a rigorous and self-contained review of desingularization theory. Focusing on arbitrary dimensional schemes, it discusses the important concepts in full generality, complete with proofs, and includes an introduction to the basis of Hironakäs Theory. The core of the book is a complete proof of desingularization of surfaces; despite being well-known, this result was no more than folklore for many years, with no existing references. Throughout the book there are numerous computations on standard bases, blowing ups and characteristic polyhedra, which will be a source of inspiration for experts exploring bigger dimensions. Beginners will also benefit from a section which presents some easily overlooked pathologies. 

Inhaltsverzeichnis

-ÿIntroduction. -ÿBasic Invariants for Singularities. -ÿPermissible Blow-Ups.ÿB-Permissible Blow-Ups: The Embedded Case. -ÿB-Permissible Blow-Ups: The Non-embedded Case. -ÿMain Theorems and Strategy for Their Proofs. -ÿ(u)-standard Bases. -ÿCharacteristic Polyhedra of J R. -ÿTransformation of Standard Bases Under Blow-Ups. - Termination of the Fundamental Sequences of B-PermissibleÿBlow-Ups, and the Case ex(X) = 1. -ÿAdditional Invariants in the Case ex(X) = 2. -ÿProof in the Case ex(X) = esx(X) = 2, I: Some Key Lemmas. -ÿProof in the Case ex(X) = ex(X) = 2, II: Separable ResidueÿExtensions. -ÿProof in the Case ex(X) = ex(X) = 2, III: Inseparable ResidueÿExtensions. -ÿNon-existence of Maximal Contact in Dimension 2. -ÿAn Alternative Proof of Theorem 6.17. -ÿFunctoriality, Locally Noetherian Schemes, Algebraic Spacesÿand Stacks. -ÿAppendix by B. Schober: Hironaka's CharacteristicÿPolyhedron. Notes for Novices.