Resolution of Singularities,9783764361785

Resolution of Singularities

by ; ; ;
ISBN13: 9783764361785
ISBN10: 3764361786
Format: Hardcover
Pub. Date: 2000-03-01
Publisher(s): Birkhauser
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Summary

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.

Table of Contents

Preface xi
Program of the Working Week on Resolution of Singularities xiii
Participants of the Working Week xv
Oscar Zariski 1899-1986
1(4)
J. Lipman
Resolution of Singularities 1860-1999
5(34)
H. Hauser
Main achievements
5(2)
Some research problems
7(1)
Contributors
8(7)
Dictionary
15(10)
Surveys
25(1)
Miscellanea
26(1)
Selection of references before 1930
26(1)
Selection of references after 1930
27(12)
Part 1: Classes of the Working Week
Alterations and resolution of singularities
39(70)
D. Abramovich
F. Oort
Introduction
39(6)
The alteration theorem
45(1)
Some preliminaries and generalities on varieties
45(3)
Results
48(3)
Some tools
51(7)
Proof of de Jong's main theorem
58(9)
Modifications of the proof for Theorems 2.4 and 2.7
67(2)
Toroidal geometry
69(5)
Weak resolution of singularities I
74(2)
Weak resolution of singularities II
76(2)
Intersection multiplicities
78(2)
Moduli of curves
80(1)
Introduction to moduli of curves
80(6)
Stable reduction and completeness of moduli spaces
86(7)
Construction of moduli spaces
93(3)
Existence of tautological families
96(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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