Introduction to Singularities and Deformations (eBook)

Gert-Martin Greuel, Christoph Lossen, Eugenii I. Shustin (Autoren)

eBook Download: PDF

2007 | 1. Auflage
472 Seiten
Springer-Verlag
978-3-540-28419-2 (ISBN)

Infos

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences.

This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. Plane curve singularities are a classical object of study, rich of ideas and applications, which still is in the center of current research and as such provides an ideal introduction to the general theory. Deformation theory is an important technique in many branches of contemporary algebraic geometry and complex analysis. This introductory text provides the general framework of the theory while still remaining concrete.

In the first part of the book the authors develop the relevant techniques, including the Weierstraß preparation theorem, the finite coherence theorem etc., and then treat isolated hypersurface singularities, notably the finite determinacy, classification of simple singularities and topological and analytic invariants. In local deformation theory, emphasis is laid on the issues of versality, obstructions, and equisingular deformations. The book moreover contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum which is based on deformations of the parameterization. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained.

The material, which can be found partly in other books and partly in research articles, is presented from a unified point of view for the first time. It is given with complete proofs, new in many cases. The book thuscan serve as source for special courses in singularity theory and local algebraic and analytic geometry.

Inhalt

Preface 7
Contents 11
I Singularity Theory 13
1 Basic Properties of Complex Spaces and Germs 20
2 Hypersurface Singularities 122
3 Plane Curve Singularities 173
II Local Deformation Theory 233
1 Deformations of Complex Space Germs 234
2 Equisingular Deformations of Plane Curve Singularities 278
A Sheaves 391
A.1 Presheaves and Sheaves 391
A.2 Gluing Sheaves 393
A.3 Sheaves of Rings and Modules 394
A.4 Image and Preimage Sheaf 395
A.5 Algebraic Operations on Sheaves 396
A.6 Ringed Spaces 397
A.7 Coherent Sheaves 399
A.8 Sheaf Cohomology 400
A.9 Cech Cohomology and Comparison 404
B Commutative Algebra 409
B.1 Associated Primes and Primary Decomposition 409
B.2 Dimension Theory 411
B.3 Tensor Product and Flatness 414
B.4 Artin-Rees and Krull Intersection Theorem 418
B.5 The Local Criterion of Flatness 419
B.6 The Koszul Complex 422
B.7 Regular Sequences and Depth 426
B.8 Cohen-Macaulay, Flatness and Fibres 428
B.9 Auslander-Buchsbaum Formula 434
C Formal Deformation Theory 437
C.1 Functors of Artin Rings 437
C.2 Obstructions 443
C.3 The Cotangent Complex 448
C.4 Cotangent Cohomology 453
C.5 Relation to Deformation Theory 455
References 459
Index 467

Produkt-Details

• Erscheint lt. Verlag: 23.2.2007
• Reihe/Serie: Springer Monographs in Mathematics
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: XII, 472 p. 54 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• Schlagworte: Algebra • Complex space germs • deformation theory • Equisingularity • Local and semi-local rings • Plane curve singularities • singularity theory • Versal deformation
• ISBN-10: 3-540-28419-2 / 3540284192
• ISBN-13: 978-3-540-28419-2 / 9783540284192