Resolution of Singularities

, by Hauser, H.; Lipman, J.; Oort, F.; Quiros, A.

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ISBN: 9783764361785 | 3764361786
Cover: Hardcover
Copyright: 3/1/2000

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In September 1997, the Working Week on Resolution of Singularities was held at Obergurgl in the Tyrolean Alps. Its objective was to manifest the state of the art in the field and to formulate major questions for future research. The four courses given during this week were written up by the speakers and make up part I of this volume. They are complemented in part II by fifteen selected contributions on specific topics and resolution theories. The volume is intended to provide a broad and accessible introduction to resolution of singularities leading the reader directly to concrete research problems.

Preface

Program of the Working Week on Resolution of Singularities

xiii

Participants of the Working Week

Oscar Zariski 1899-1986

(4)

J. Lipman

Resolution of Singularities 1860-1999

(34)

H. Hauser

Main achievements

(2)

Some research problems

(1)

Contributors

(7)

Dictionary

(10)

Surveys

(1)

Miscellanea

(1)

Selection of references before 1930

(1)

Selection of references after 1930

(12)

Part 1: Classes of the Working Week

Alterations and resolution of singularities

(70)

D. Abramovich

F. Oort

Introduction

(6)

The alteration theorem

(1)

Some preliminaries and generalities on varieties

(3)

Results

(3)

Some tools

(7)

Proof of de Jong's main theorem

(9)

Modifications of the proof for Theorems 2.4 and 2.7

(2)

Toroidal geometry

(5)

Weak resolution of singularities I

(2)

Weak resolution of singularities II

(2)

Intersection multiplicities

(2)

Moduli of curves

(1)

Introduction to moduli of curves

(6)

Stable reduction and completeness of moduli spaces

(7)

Construction of moduli spaces

(3)

Existence of tautological families

(4)

Moduli, automorphisms, and families

100

(9)

References

104

(5)

Reduction of singularities for differential equations

109

(20)

J.-M. Aroca

Introduction

109

(2)

Singular foliations (codimension one)

111

(4)

Reduction of singularities in dimension two

115

(8)

Existence of integral curves

123

(6)

References

126

(3)

Puiseux solutions of singular differential equations

129

(18)

J.-M. Aroca

Introduction

129

(4)

The Newton polygon of a differential equation

133

(6)

Solutions for first order, first degree equations

139

(8)

References

144

(3)

A course on constructive desingularization and equivariance

147

(84)

Introduction

147

(3)

First definitions

150

(3)

Pairs

153

(3)

Constructive desingularization

156

(1)

Basic objects

157

(8)

Monomial case

165

(5)

Compatibility of the function t with induction

170

(9)

Equivariant desingularization

179

(14)

Change of base field and generalization to the non-hypersurface case

193

(7)

Proofs

200

(8)

Exercise: Order of ideals and upper-semi-continuity

208

(2)

Exercise: Blow-ups

210

(2)

Exercise: Desingularization

212

(1)

Exercise: On basic objects

213

(6)

Exercise: A do-it-yourself help guide of theorem 7.10

219

(2)

Exercise: Non-embedded constructive desingularization. Compatibility with group actions and formal isomorphisms

221

(10)

References

225

(6)

Part 2: Contributions

A computer program for the resolution of singularities

231

(8)

G. Bodnar

J. Schicho

Introduction

231

(1)

How to blow up a circle

232

(4)

How to compute the centers of the blowups

236

(3)

References

238

(1)

Uniformisation et desingularisation des surfaces d'apres Zariski

239

(20)

V. Cossart

Introduction

239

(1)

Uniformasation et recollement

240

(11)

Desingularisation par eclatements normalises

251

(8)

References

257

(2)

Toric varieties and toric resolution

259

(26)

D. Cox

Introduction

259

(2)

Cones, fans, and toric varieties

261

(5)

Divisors and support functions

266

(4)

Other constructions of toric varieties

270

(6)

Toric blow-ups and resolution of singularities

276

(7)

Conclusion

283

(1)

Acknowledgements

283

(2)

References

283

(2)

A compactification of a fine moduli space of curves

285

(14)

B. van Geemen

F. Oort

Introduction

285

(1)

Notation

286

(2)

Construction of a tautological family

288

(6)

Local structure of the moduli space

294

(5)

References

297

(2)

Applications of de Jong's theorem on alterations

299

(16)

T. Geisser

Introduction

299

(1)

de Jong's theorem and Serre's conjecture

300

(1)

Grothendieck topologies for which alterations are coverings

301

(6)

Applications using trace maps

307

(8)

References

313

(2)

Resolving singularities of plane analytic branches with one toric morphism

315

(26)

R. Goldin

B. Teissier

Introduction

315

(2)

Puiseux expansion and the semigroup of a curve

317

(2)

Deforming curves

319

(2)

Resolution using toric morphisms

321

(4)

Existence of a toric resolution

325

(6)

Simulataneous resolution

331

(3)

The transforms of plane curves

334

(1)

Appendix: An example

335

(6)

References

339

(2)

Excellent surfaces and their taut resolution

341

(34)

H. Hauser

Introduction

341

(5)

Preliminaries

346

(2)

Definition of the centers of blowup

348

(5)

Transformation of equimultiple locus under blowup

353

(6)

Transformation of flags under blowup

359

(5)

Construction of the induction invariant

364

(3)

Transformation of iax under monomial blowup

367

(2)

Reduction to monomial blowup

369

(2)

Proof of Theorem 1.1

371

(4)

References

372

(3)

An application of alterations to Dieudonne modules

375

(6)

A.J. de Jong

Introduction

375

(1)

The application of alterations

376

(1)

Full faithfulness up to isogeny

376

(2)

Essential surjectivity up to isogeny

378

(3)

References

380

(1)

Valuation theoretic and model theoretic aspects of local uniformization

381

(76)

F.-V. Kuhlmann

Introduction

381

(1)

What does local uniformization mean?

381

(7)

Local uniformization and the Implicit Function Theorem

388

(1)

Hensel's Lemma

389

(5)

A crash course in ramification theory

394

(5)

A valuation theoretical interpretation of local uniformization

399

(2)

Intertial generation and Abhyankar places

401

(3)

The defect

404

(4)

Maximal immediate extensions

408

(2)

A quick look at Puisex series fields

410

(2)

The tame and the wild valuation theory

412

(3)

Some notions and tools from model theoretic algebra

415

(6)

Saturation and embedding lemmas

421

(3)

The Generalized Grauert-Remmert Stability Theorem

424

(4)

Relative local uniformization

428

(4)

Local uniformization for Abhyankar places

432

(1)

Non-Abhyankar places and the Henselian Rationality of immediate function fields

433

(4)

Bad places

437

(2)

The role of the transcendence basis and the dimension

439

(4)

The space of all places of F/K

443

(3)

Fp((t))

446

(2)

Local uniformization vs. Ax-Kochen-Ershov

448

(3)

Back to local uniformization in positive characteristic

451

(6)

References

452

(5)

Les singularites Sandwich

457

(28)

D.T. Le

Introduction

457

(1)

Normalisation et eclatements

458

(2)

Singularites Sandwich

460

(2)

Approche combinatoire

462

(10)

Singularites primitives, singularites minimales

472

(3)

Courbes polaires des singularites rationnelles

475

(2)

Transformation de Nash des singularites minimales

477

(2)

La resolution des singularites des surfaces via la transformee de Nash

479

(6)

References

481

(4)

Equisingularity and simultaneous resolution of singularities

485

(22)

J. Lipman

Introduction - equisingular stratifications

485

(3)

Stratifying conditions

488

(6)

The Zariski stratification

494

(4)

Equiresolvable stratifications

498

(3)

Simultaneous resolution of quasi-ordinary singularities

501

(6)

References

503

(4)

Resolution of weighted homogeneous surface singularities

507

(12)

G. Muller

Weighted homogeneous singularities

507

(1)

Singular line bundles and resolution

508

(3)

Automorphy factors

511

(2)

The hypersurface case

513

(2)

Automorphisms

515

(4)

References

516

(3)

Alterations and birational anabelian geometry

519

(14)

F. Pop

Introduction

519

(4)

Outline of the proof of the Birational Conjecture

523

(4)

Alterations and geometric sets of Zariski prime divisors

527

(6)

References

530

(3)

The turbulent fifties in resolution of singularities

533

(6)

H. Reitberger

Introduction

533

(1)

L. Derwidue's ``resolution'' and O. Zariski's review

533

(1)

B. Segre's ``scioglimento'' and J.G. Semple's remarks

534

(1)

W. Grobner's ``Auflosung'' and P. Abellanas' review

535

(2)

References

537

(2)

Valuations

539

(52)

M. Vaquie

Introduction

539

(1)

Anneau de valuation

540

(3)

Valuation

543

(2)

Hauteur d'une valuation

545

(6)

Valuations composees

551

(3)

Prolongement d'une valuation

554

(8)

Centre d'une valuation

562

(3)

Variete de Riemann

565

(10)

Uniformisation et resolution des singularites

575

(1)

Valuation sur un anneau noetherien

576

(7)

Exemples de valuation

583

(8)

References

589

(2)

Index

591

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Resolution of Singularities

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Table of Contents

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