Resolution of Singularities: A research textbook in tribute to Oscar Zariski Based on the courses given at the Working Week in Obergurgl, Austria, September 7–14, 1997

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Herwig Hauser, Joseph Lipman, Frans Oort, Adolfo Quiros
Birkhäuser, Dec 6, 2012 - Mathematics - 598 pages
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.
 

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Contents

J Lipman
1
Contributions
15
A computer program for the resolution of singularities 231
29
Resolving singularities of plane analytic branches
31
Classes of the Working Week
32
Alterations and resolution of singularities
39
The alteration theorem
45
Moduli of curves
80
Transformation of flags under blowup
359
Construction of the induction invariant
364
Transformation of iax under monomial blowup
367
Reduction to monomial blowup
369
Proof of Theorem 1 1
371
References
372
A J de Jong An application of alterations to Dieudonné modules
375
The application of alterations
376

Existence of tautological families
96
Moduli automorphisms and families
100
References
104
J M Aroca Reduction of singularities for differential equations
109
Singular foliations codimension one
111
Reduction of singularities in dimension two
115
Existence of integral curves
123
References
126
J M Aroca Puiseux solutions of singular differential equations
129
The Newton polygon of a differential equation
133
Solutions for first order first degree equations
139
References
144
S Encinas O Villamayor A course on constructive desingularization and equivariance 147
146
First definitions
150
Pairs
153
Constructive desingularization
156
Basic objects
157
Monomial case
165
Compatibility of the function twith induction
170
Equivariant desingularization
179
Change of base field and generalization to the nonhypersurface case
193
Proofs
200
Order of ideals and uppersemicontinuity
208
Blowups
210
Desingularization
212
On basic objects
213
A doityourself help guide of theorem 7 10
219
Nonembedded constructive desingularization Compatibility with group actions and formal isomorphisms
221
References
225
Cossart
239
D
259
B van Geemen F Oort
285
T Geisser
299
with one toric morphism
315
Puiseux expansion and the semigroup of a curve
317
Deforming curves
319
Resolution using toric morphisms
321
Existence of a toric resolution
325
Simultaneous resolution
331
The transforms of plane curves
334
An example
335
References
339
H Hauser Excellent surfaces and their taut resolution
341
Preliminaries
346
Definition of the centers of blowup
348
Transformation of equimultiple locus under blowup
353
Essential surjectivity up to isogeny
378
References
380
F V Kuhlmann Valuation theoretic and model theoretic aspects of local uniformization
381
Local uniformization and the Implicit Function Theorem
388
Hensels Lemma
389
A crash course in ramification theory
394
A valuation theoretical interpretation of local uniformization
399
Inertial generation and Abhyankar places
401
The defect
404
Maximal immediate extensions
408
A quick look at Puiseux Series fields
410
The tame and the wild valuation theory
412
Some notions and tools from model theoretic algebra
415
Saturation and embedding lemmas
421
The Generalized GrauertRemmert Stability Theorem
424
Relative local uniformization
428
Local uniformization for Abhyankar places
432
NonAbhyankar places and the Henselian Rationality of immediate function fields
433
Bad places
437
The role of the transcendence basis and the dimension
439
The space of all places of FK
443
Fet
446
Local uniformization vs AxKochenErshov
448
Back to local uniformization in positive characteristic
451
References
452
T Lé Les singularités Sandwich
457
Normalisation et éclatements
458
Singularités Sandwich
460
Approche combinatoire
462
Singularités primitives singularités minimales
472
Courbes polaires des singularités rationnelles
475
Transformation de Nash des singularités minimales
477
La résolution des singularités des surfaces via la transformée de Nash
479
Références
481
J Lipman
482
Equisingularity and simultaneous resolution of singularities 485
484
Stratifying conditions
488
The Zariski stratification
494
Equiresolvable stratifications
498
Simultaneous resolution of quasiordinary singularities
501
References
503
Resolution of weighted homogeneous surface singularities
507
F
518
H Reitberger
533
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