Title Desingularization : invariants and strategy : application to dimension 2 / Vincent Cossart, Uwe Jannsen, Shuji Saito

Published Cham : Springer, 2020

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Description 1 online resource Series Lecture notes in mathematics ; v. 2270 Lecture notes in mathematics (Springer-Verlag) ; 2270. Contents Intro -- Abstract -- Contents -- 1 Introduction -- 1.1 What Is Desingularization? -- 1.2 Very Short History of Desingularization -- 1.3 How Did we Start? -- 1.4 Summary -- 1.5 Conventions and Concluding Remarks -- 2 Basic Invariants for Singularities -- 2.1 Invariants of Graded Rings and Homogeneous Ideals in Polynomial Rings -- 2.2 Invariants for Local Rings -- 2.3 Invariants for Schemes -- 3 Permissible Blow-Ups -- 4 B-Permissible Blow-Ups: The Embedded Case -- 5 B-Permissible Blow-Ups: The Non-embedded Case -- 6 Main Theorems and Strategy for Their Proofs -- 7 (u)-standard Bases 8 Characteristic Polyhedra of JR -- 9 Transformation of Standard Bases Under Blow-Ups -- 10 Termination of the Fundamental Sequences of B-Permissible Blow-Ups, and the Case e_x(X)=1 -- 11 Additional Invariants in the Case e_x(X)=2 -- 12 Proof in the Case e_x(X)=\overline{e}_x(X)=2, I: Some Key Lemmas -- 13 Proof in the Case e_x(X)=\overline{e}_x(X)=2, II: Separable Residue Extensions -- 14 Proof in the Case e_x(X)=\overline{e}_x(X)=2, III: Inseparable Residue Extensions -- 15 Non-existence of Maximal Contact in Dimension 2 -- 16 An Alternative Proof of Theorem 6.17 17 Functoriality, Locally Noetherian Schemes, Algebraic Spaces and Stacks -- 18 Appendix by B. Schober: Hironaka's Characteristic Polyhedron. Notes for Novices -- 18.1 Introduction -- 18.2 The Newton Polyhedron of an Ideal -- 18.3 The Projected Polyhedron and Its Relation to the Newton Polyhedron -- 18.4 The Directrix and Its Role: Choosing (u) -- 18.5 Determining the Characteristic Polyhedron: Optimizing the Choice of (f -- y) -- 18.6 Invariants from the Polyhedron and the Effect of Blowing Up -- References -- Index Summary 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 Hironakas 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 Bibliography Includes bibliographical references and index Subject Singularities (Mathematics) Invariants. Dimension theory (Algebra) Algebraic geometry. Algebra. Mathematics -- Geometry -- Algebraic. Mathematics -- Algebra -- General. Invariantes Matemáticas -- Geometría -- Funciones algebraicas Dimension theory (Algebra) Invariants Singularities (Mathematics) Form Electronic book Author Jannsen, Uwe. Saitō, Shūji, 1957- ISBN 9783030526405 3030526402

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