Desingularization: Invariants and Strategy

Application to Dimension 2
1. Auflage 2020
ISBN: 978-3-030-52639-9
Verlag: Springer International Publishing

Buch, Englisch, Band 2270, 258 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 411 g

Reihe: Lecture Notes in Mathematics

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 Hironaka’s 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.

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Cossart, Vincent

Jannsen, Uwe

Saito, Shuji

Weitere Mitwirkende

Schober, Bernd

Fachgebiete

• Mathematik | Informatik Mathematik Algebra
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

- Introduction. - Basic Invariants for Singularities. - Permissible Blow-Ups. B-Permissible Blow-Ups: The Embedded Case. - B-Permissible Blow-Ups: The Non-embedded Case. - Main Theorems and Strategy for Their Proofs. - (u)-standard Bases. - Characteristic Polyhedra of J ? R. - Transformation of Standard Bases Under Blow-Ups. - Termination of the Fundamental Sequences of B-Permissible Blow-Ups, and the Case ex(X) = 1. - Additional Invariants in the Case ex(X) = 2. - Proof in the Case ex(X) = esx(X) = 2, I: Some Key Lemmas. - Proof in the Case ex(X) = ex(X) = 2, II: Separable Residue Extensions. - Proof in the Case ex(X) = ex(X) = 2, III: Inseparable Residue Extensions. - Non-existence of Maximal Contact in Dimension 2. - An Alternative Proof of Theorem 6.17. - Functoriality, Locally Noetherian Schemes, Algebraic Spaces and Stacks. - Appendix by B. Schober: Hironaka’s Characteristic Polyhedron. Notes for Novices.