The Valuative Tree / Edition 1

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Overview

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.

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Product Details

  • ISBN-13: 9783540229841
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 11/10/2004
  • Series: Lecture Notes in Mathematics Series , #1853
  • Edition description: 2004
  • Edition number: 1
  • Pages: 244
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.55 (d)

Table of Contents

1 Generalities.- 1.1 Setup.- 1.2 Valuations.- 1.3 Krull Valuations.- 1.4 Plane Curves.- 1.5 Examples of Valuations.- 1.5.1 The Multiplicity Valuation.- 1.5.2 Monomial Valuations.- 1.5.3 Divisorial Valuations.- 1.5.4 Quasimonomial Valuations.- 1.5.5 Curve Valuations.- 1.5.6 Exceptional Curve Valuations.- 1.5.7 Infinitely Singular 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.- 2.1.1 Key Polynomials.- 2.1.2 From SKP’s to Valuations I.- 2.1.3 Proof of Theorem 2.8.- 2.1.4 From SKP’s to Valuations II.- 2.2 Classification.- 2.3 Graded Rings and Numerical Invariants.- 2.3.1 Homogeneous Decomposition I.- 2.3.2 Homogeneous Decomposition II.- 2.3.3 Value Semigroups and Numerical Invariants.- 2.4 From Valuations to SKP’s.- 2.5 A Computation.- 3 Tree Structures.- 3.1 Trees.- 3.1.1 Rooted Nonmetric Trees.- 3.1.2 Nonmetric Trees.- 3.1.3 Parameterized Trees.- 3.1.4 The Weak Topology.- 3.1.5 Metric Trees.- 3.1.6 Trees from Ultrametric Spaces.- 3.1.7 Trees from Simplicial Trees.- 3.1.8 Trees from Q-trees.- 3.2 Nonmetric Tree Structure on V.- 3.2.1 Partial Ordering.- 3.2.2 Dendrology.- 3.2.3 A Model Tree for V.- 3.3 Parameterization of V by Skewness.- 3.3.1 Skewness.- 3.3.2 Parameterization.- 3.3.3 Proofs.- 3.3.4 Tree Metrics.- 3.4 Multiplicities.- 3.5 Approximating Sequences.- 3.6 Thinness.- 3.7 Value Semigroups and Approximating Sequences.- 3.8 Balls of Curves.- 3.8.1 Valuations Through Intersections.- 3.8.2 Balls of Curves.- 3.9 The Relative Tree Structure.- 3.9.1 The Relative Valuative Tree.- 3.9.2 Relative Parameterizations.- 3.9.3 Balls of Curves.- 3.9.4 Homogeneity.- 4 Valuations Through Puiseux Series.- 4.1 Puiseux Series and Valuations.- 4.2 Tree Structure.- 4.2.1 Nonmetric Tree Structure.- 4.2.2 Puiseux Parameterization.- 4.2.3 Multiplicities.- 4.3 Galois Action.- 4.3.1 The Galois Group.- 4.3.2 Action on Vx.- 4.3.3 The Orbit Tree.- 4.4 A Tale of Two Trees.- 4.4.1 Minimal Polynomials.- 4.4.2 The Morphism.- 4.4.3 Proof.- 4.5 The Berkovich Projective Line.- 4.6 The Bruhat-Tits Metric.- 4.7 Dictionary.- 5 Topologies.- 5.1 The Weak Topology.- 5.1.1 The Equivalence.- 5.1.2 Properties.- 5.2 The Strong Topology on V.- 5.2.1 Strong Topology I.- 5.2.2 Strong Topology II.- 5.2.3 The Equivalence.- 5.2.4 Properties.- 5.3 The Strong Topology on Vqm.- 5.4 Thin Topologies.- 5.5 The Zariski Topology.- 5.5.1 Definition.- 5.5.2 Recovering V from VK.- 5.6 The Hausdorff-Zariski Topology.- 5.6.1 Definition.- 5.6.2 The N-tree Structure on VK.- 5.7 Comparison of Topologies.- 5.7.1 Topologies.- 5.7.2 Metrics.- 6 The Universal Dual Graph.- 6.1 Nonmetric Tree Structure.- 6.1.1 Compositions of Blowups.- 6.1.2 Dual Graphs.- 6.1.3 The Q-tree.- 6.1.4 Tangent Spaces.- 6.1.5 The R-tree.- 6.2 Infinitely Near Points.- 6.2.1 Definitions and Main Results.- 6.2.2 Proofs.- 6.3 Parameterization and Multiplicity.- 6.3.1 Farey Weights and Parameters.- 6.3.2 Multiplicities.- 6.4 The Isomorphism.- 6.5 Proof of the Isomorphism.- 6.5.1 Step 1:— :—*— Vdiv is bijective.- 6.5.2 Step 2: A?? = A.- 6.5.3 Step 3:— and—?1 Are Order Preserving.- 6.5.4 Step 4:— Preserves Multiplicity.- 6.6 Applications.- 6.6.1 Curvettes.- 6.6.2 Centers of Valuations and Partitions of V.- 6.6.3 Potpourri on Divisorial Valuations.- 6.6.4 Monomialization.- 6.7 The Dual Graph of the Minimal Desingularization.- 6.7.1 The Embedding of—C* in—*.- 6.7.2 Construction of—C from the Equisingularity Type of C.- 6.8 The Relative Tree Structure.- 6.8.1 The Relative Dual Graph.- 6.8.2 Weights, Parameterization and Multiplicities.- 6.8.3 The Isomorphism.- 6.8.4 The Contraction Map at a Free Point.- 7 Tree Measures.- 7.1 Outline.- 7.1.1 The Unbranched Case.- 7.1.2 The General Case.- 7.1.3 Organization.- 7.2 More on the Weak Topology.- 7.2.1 Definition.- 7.2.2 Basic properties.- 7.2.3 Subtrees.- 7.2.4 Connectedness.- 7.2.5 Compactness.- 7.3 Borel Measures.- 7.3.1 Basic Properties.- 7.3.2 Radon Measures.- 7.3.3 Spaces of Measures.- 7.3.4 The Support of a Measure.- 7.3.5 A Generating Algebra.- 7.3.6 Every Complex Borel Measure is Radon.- 7.4 Functions of Bounded Variation.- 7.4.1 Definitions.- 7.4.2 Decomposition.- 7.4.3 Limits and Continuity.- 7.4.4 The Space N.- 7.4.5 Finite Trees.- 7.4.6 Proofs.- 7.5 Representation Theorem I.- 7.5.1 First Step.- 7.5.2 Second Step: from Functions to Measures.- 7.5.3 Total Variation.- 7.6 Complex Tree Potentials.- 7.6.1 Definition.- 7.6.2 Directional Derivatives.- 7.7 Representation Theorem II.- 7.8 Atomic Measures.- 7.9 Positive Tree Potentials.- 7.9.1 Definition.- 7.9.2 Jordan Decompositions.- 7.10 Weak Topologies and Compactness.- 7.11 Restrictions to Subtrees.- 7.12 Inner Products.- 7.12.1 Hausdorff Measure.- 7.12.2 The Positive Case.- 7.12.3 Properties.- 7.12.4 The Complex Case.- 7.12.5 Topologies and Completeness.- 8 Applications of the Tree Analysis.- 8.1 Zariski’s Theory of Complete Ideals.- 8.1.1 Basic Properties.- 8.1.2 Normalized Blowup.- 8.1.3 Integral Closures.- 8.1.4 Multiplicities.- 8.2 The Voûte étoilée.- 8.2.1 Definition.- 8.2.2 Cohomology.- 8.2.3 Intersection Product.- 8.2.4 Associated Complex Tree Potentials.- 8.2.5 Isometric Embedding.- 8.2.6 Cohomology Groups.- 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.

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