Klienditugi: 7440010 (E-R 10-18)

E-raamat: Arc Schemes And Singularities

Edited by Johannes Nicaise (Imperial College London, Uk & Univ Of Leuven, Belgium), Edited by Julien Sebag (Univ Of Rennes 1, France), Edited by David Bourqui (Univ Of Rennes 1, France)

Formaat: 312 pages
Ilmumisaeg: 05-Mar-2020
Kirjastus: World Scientific Europe Ltd
Keel: eng
ISBN-13: 9781786347213

Teised raamatud teemal:

Algebraic geometry

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Formaat: 312 pages
Ilmumisaeg: 05-Mar-2020
Kirjastus: World Scientific Europe Ltd
Keel: eng
ISBN-13: 9781786347213

Teised raamatud teemal:

Algebraic geometry

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This title introduces the theory of arc schemes in algebraic geometry and singularity theory, with special emphasis on recent developments around the Nash problem for surfaces. The main challenges are to understand the global and local structure of arc schemes, and how they relate to the nature of the singularities on the variety. Since the arc scheme is an infinite dimensional object, new tools need to be developed to give a precise meaning to the notion of a singular point of the arc scheme.Other related topics are also explored, including motivic integration and dual intersection complexes of resolutions of singularities. Written by leading international experts, it offers a broad overview of different applications of arc schemes in algebraic geometry, singularity theory and representation theory.

About the Editors

Chap 1 Introduction

(6)

David Bourqui

Johannes Nicaise

Julien Sebag

Bibliography

(3)

Chap 2 Arc Schemes in Geometry and Differential Algebra

(30)

David Bourqui

Johannes Nicaise

Julien Sebag

1 Weil Restrictions

(4)

2 Jet Schemes

(3)

3 Arc Schemes

(2)

4 Some Brief Reminders on Differential Algebra

(5)

5 Adjunction Formulas in Differential Algebra

(7)

6 Algebro-differential Description of Jet/Arc Schemes

(4)

7 The Universal Algebra of Higher Derivations

(3)

Bibliography

(2)

Chap 3 The Grinberg-Kazhdan Formal Arc Theorem and the Newton Groupoids

(20)

Vladimir Drinfeld

1 Introduction

(1)

2 The Grinberg-Kazhdan Theorem

(3)

3 Rephrasing the Proof from Section 2

(2)

4 Introduction to the Newton Groupoids

(5)

5 Newton Groupoids (Details)

(8)

Acknowledgment

(1)

Bibliography

(1)

Chap 4 Non-complete Completions

(12)

Mercedes Haiech

1 Introduction

(2)

2 A Necessary and Sufficient Condition to be Adically Complete

(4)

3 A Not I-adically Complete Completion

(4)

Acknowledgments

(1)

Bibliography

(1)

Chap 5 The Local Structure of Arc Schemes

(30)

David Bourqui

Julien Sebag

1 Introduction

(1)

2 Conventions and Notations

(2)

3 The Drinfeld-Grinberg-Kazhdan Theorem

(12)

4 A Simplification Lemma in Formal Geometry

(1)

5 The Minimal Formal Model of a Rational Non-degenerate Arc

(2)

6 The Case of Degenerate Arcs

(2)

7 Dependency on the Arc

(4)

8 Nilpotency in Formal Neighborhoods

(3)

Bibliography

(3)

Chap 6 Arc Schemes of AfRne Algebraic Plane Curves and Torsion Kahler Differential Forms

(14)

David Bourqui

Julien Sebag

1 Introduction

(2)

2 Conventions and Notations

101

(1)

3 Proof of Theorem 1.1

102

(3)

4 Singular Locus of Torsion Kahler Differential Forms

105

(1)

5 A Structure Statement on Derivation Module of Plane Curves

106

(2)

6 A Consequence on the Schematic Structure of Arc Schemes Associated with Plane Curves

108

(1)

7 A SAGE Code to Compute Nilpotent Kahler Differential Forms of Plane Curves

109

(1)

Bibliography

110

(3)

Chap 7 Models of Affine Curves and G0-actions

113

(8)

Kevin Langlois

1 Introduction

113

(1)

2 Basics

114

(2)

3 Proof of the Main Result

116

(3)

Acknowledgments

119

(1)

Bibliography

119

(2)

Chap 8 Theoremes de Structure sur les Espaces d'Arcs

121

(24)

Alexis Bouthier

1 Introduction

121

(1)

2 Preliminaries

122

(2)

3 Espace D'arcs

124

(7)

4 Morphismes Pro-lisses

131

(3)

5 Sur Certains Espaces Non-noetheriens

134

(4)

6 Enonces Principaux

138

(2)

7 Vers une Theorie des Faisceaux

140

(3)

Bibliographie

143

(2)

Chap 9 Partition Identities and Application to Infinite-Dimensional Grobner Basis and Vice Versa

145

(18)

Pooneh Afsharijoo

Hussein Mourtada

1 Introduction

145

(1)

2 Hilbert Series and Integer Partitions

146

(6)

3 The Lex Grobner Basis of [ Χ12]

152

(4)

4 Two Color Partitions and the Node

156

(4)

Acknowledgments

160

(1)

Bibliography

160

(3)

Chap 10 The Algebraic Answer to the Nash Problem for Normal Surfaces According to de Fernex and Docampo

163

(10)

Monique Lejeune-Jalabert

1 Introduction

163

(1)

2 Arcs, the Nash Map and the Nash Problem

164

(1)

3 Arcs and Wedges

165

(1)

4 Lifting Wedges

166

(6)

Bibliography

172

(1)

Chap 11 The Nash Problem from Geometric and Topological Perspective

173

(24)

J. Fernandez De Bobadilla

M. Pe Pereira

1 Introduction

173

(2)

2 The Idea of the Proof for Surfaces

175

(2)

3 Turning the Problem into a Problem of Convergent Wedges

177

(1)

4 Reduction to an Euler Characteristic Estimate

178

(3)

5 The Euler Characteristic Estimates

181

(5)

6 The Returns of a Wedge and Deformation Theoretic Ideas

186

(1)

7 The Proof by de Fernex and Docampo for the Higher Dimensional Case

187

(4)

8 The Generalized Nash Problem and the Classical Adjacency Problem

191

(1)

9 Holomorphic Arcs

192

(2)

Acknowledgments

194

(1)

Bibliography

194

(3)

Chap 12 Motivic and Analytic Nearby Fibers at Infinity and Bifurcation Sets

197

(24)

Lorenzo Fantini

Michel Raibaut

1 Introduction

197

(4)

2 Motivic Integration and Nearby Cycles

201

(4)

3 Motivic Nearby Cycles at Infinity and the Motivic Bifurcation Set

205

(5)

4 Analytic Nearby Fiber at Infinity and the Serre Bifurcation Set

210

(8)

Acknowledgments

218

(1)

Bibliography

218

(3)

Chap 13 The Neron Multiplicity Sequence of Singularities

221

(10)

Beatriz Pascual-Escudero

Julien Sebag

1 Introduction

221

(1)

2 The Neron Multiplicity Sequence of Singularities

222

(3)

3 Nash and Neron Multiplicity Sequences

225

(2)

4 Cuspidal Plane Curve Singularities

227

(2)

Acknowledgments

229

(1)

Bibliography

229

(2)

Chap 14 The Dual Complex of Singularities After de Fernex, Kollar and Xu

231

(26)

Mirko Mauri

1 Introduction

231

(6)

2 Notation: Birational Dictionary

237

(1)

3 Notation: Simplicial Complexes

238

(1)

4 Dual Complex of a Log Smooth Pair

239

(2)

5 Dual Complex of a Resolution of Singularities and of a dlt Pair

241

(3)

6 Proof of Proposition 5.2 and Main Theorem (1)

244

(4)

7 Running MMP

248

(2)

8 Proof of Main Theorem (2)

250

(2)

9 Proof of Main Theorem (3)

252

(1)

Acknowledgments

253

(1)

Bibliography

253

(4)

Chap 15 Log-Regular Models for Products of Degenerations

257

(22)

Morgan V. Brown

Enrica Mazzon

1 Introduction

257

(3)

2 The Skeleton of a Log-Regular Model

260

(9)

3 The Essential Skeleton of a Product

269

(4)

4 Applications

273

(4)

Acknowledgments

277

(1)

Bibliography

277

(2)

Chap 16 Arc Scheme and Bernstein Operators

279

(18)

Michel Gros

Luis Narvdez Macarro

Julien Sebag

1 Introduction

279

(1)

2 Recollection on Arc Scheme

280

(1)

3 Recollection of Differential Algebra

281

(2)

4 Nilpotent Functions on Arc Scheme and Differential Operators

283

(5)

5 Bernstein Operators

288

(1)

6 Examples and Further Comments

289

(6)

Acknowledgment

295

(1)

Bibliography

295

(2)

Index

297

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