The Valuative Tree

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