The Valuative Tree
Published on 16. September 2004
Book
Paperback/Softback
XVI, 244 pages
978-3-540-22984-1 (ISBN)
Description
This volume is devoted to a beautiful object, called the valuative tree and designed as a powerful tool for the study of singularities in two complex dimensions. Its intricate yet manageable structure can be analyzed by both algebraic and geometric means. Many types of singularities, including those of curves, ideals, and plurisubharmonic functions, can be encoded in terms of positive measures on the valuative tree. The construction of these measures uses a natural tree Laplace operator of independent interest.
More details
Series
Edition
2004 ed.
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Professional and scholarly
Research
Illustrations
XVI, 244 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 15 mm
Weight
400 gr
ISBN-13
978-3-540-22984-1 (9783540229841)
DOI
10.1007/b100262
Schweitzer Classification
Other editions
Additional editions
Charles Favre | Mattias Jonsson
The Valuative Tree
E-Book
08/2004
Springer
€53.49
Available for download
Content
1 Generalities.- 1.1 Setup.- 1.2 Valuations.- 1.3 Krull Valuations.- 1.4 Plane Curves.- 1.5 Examples of Valuations.- 1.6 Valuations Versus Krull Valuations.- 1.7 Sequences of Blowups and Krull Valuations.- 2 MacLane's Method.- 2.1 Sequences of Key Polynomials.- 2.2 Classification.- 2.3 Graded Rings and Numerical Invariants.- 2.4 From Valuations to SKP's.- 2.5 A Computation.- 3 Tree Structures.- 3.1 Trees.- 3.2 Nonmetric Tree Structure on V.- 3.3 Parameterization of V by Skewness.- 3.4 Multiplicities.- 3.5 Approximating Sequences.- 3.6 Thinness.- 3.7 Value Semigroups and Approximating Sequences.- 3.8 Balls of Curves.- 3.9 The Relative Tree Structure.- 4 Valuations Through Puiseux Series.- 4.1 Puiseux Series and Valuations.- 4.2 Tree Structure.- 4.3 Galois Action.- 4.4 A Tale of Two Trees.- 4.5 The Berkovich Projective Line.- 4.6 The Bruhat-Tits Metric.- 4.7 Dictionary.- 5 Topologies.- 5.1 The Weak Topology.- 5.2 The Strong Topology on V.- 5.3 The Strong Topology on Vqm.- 5.4 Thin Topologies.- 5.5 The Zariski Topology.- 5.6 The Hausdorff-Zariski Topology.- 5.7 Comparison of Topologies.- 6 The Universal Dual Graph.- 6.1 Nonmetric Tree Structure.- 6.2 Infinitely Near Points.- 6.3 Parameterization and Multiplicity.- 6.4 The Isomorphism.- 6.5 Proof of the Isomorphism.- 6.6 Applications.- 6.7 The Dual Graph of the Minimal Desingularization.- 6.8 The Relative Tree Structure.- 7 Tree Measures.- 7.1 Outline.- 7.2 More on the Weak Topology.- 7.3 Borel Measures.- 7.4 Functions of Bounded Variation.- 7.5 Representation Theorem I.- 7.6 Complex Tree Potentials.- 7.7 Representation Theorem II.- 7.8 Atomic Measures.- 7.9 Positive Tree Potentials.- 7.10 Weak Topologies and Compactness.- 7.11 Restrictions to Subtrees.- 7.12 Inner Products.- 8 Applications of the Tree Analysis.- 8.1Zariski's Theory of Complete Ideals.- 8.2 The Voûte étoilée.- A Infinitely Singular Valuations.- A.1 Characterizations.- A.2 Constructions.- B The Tangent Space at a Divisorial Valuation.- C Classification.- D Combinatorics of Plane Curve Singularities.- D.1 Zariski's Terminology for Plane Curve Singularities.- D.2 The Eggers Tree.- E.1 Completeness.- E.2 The Residue Field.- References.