Mattias Jonsson

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 MacLanes Method.- 2.1 Sequences of Key Polynomials.- 2.1.1 Key Polynomials.- 2.1.2 From SKPs to Valuations I.- 2.1.3 Proof of Theorem 2.8.- 2.1.4 From SKPs 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 SKPs.- 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 Zariskis 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 Voute etoilee.- 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 Zariskis Terminology for Plane Curve Singularities.- D.2 The Eggers Tree.- E.1 Completeness.- E.2 The Residue Field.- References.

Valuations and multiplier ideals

This article is the third of a series of work on a new approach to the study of singularities of various objects in a local, two-dimensional setting. Our focus in the present paper is on multiplier ideals and singularity exponents. In the discussion below, we fix an equicharacteristic zero, two-dimensional regular local ring (R, m) with algebraically closed residue field. An important example is the ring R = (Do o? holomorphic germs at the origin in C2. In [FJ1], we introduced the space V consisting of all R+U{+00}-valued valuations on R centered at m, and normalized by ^(m) = 1. This space is naturally a tree: it is a union of (uncountably many) real segments patched together in such a way that V remains homotopic to a point. It is also an R-tree in the classical sense for a natural metric. We hence call V the valuative tree. It encodes in a natural way all possible blowups of R centered at m and therefore gives a way of measuring quite precisely singularities of different kinds of objects. The points in V that are not ends form the subtree Vqm of quasimonomial valuations. These valuations, which can alternatively be characterized as Abhyankar valuations of rank 1 or as

Differentiability of volumes of divisors and a problem of Teissier

We give an algebraic construction of the positive intersection products of pseudo-effective classes and use them to prove that the volume function on the Neron-Severi space of a projective variety ...

Valuative analysis of planar plurisubharmonic functions

We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set

Optimal investment with derivative securities

\mathcal{V}

Dynamics on Berkovich Spaces in Low Dimensions

of valuations on R and – by way of a natural Laplace operator defined in terms of the tree structure on

Degree growth of meromorphic surface maps

\mathcal{V}

Optimal investment problems and volatility homogenization approximations

– a positive measure on

Stable manifolds of holomorphic diffeomorphisms

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Ergodic properties of fibered rational maps

. This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampère mass of two plurisubharmonic functions. As a consequence, any generalized Lelong number can be interpreted as an average of valuations. Using our machinery we also show that the singularity of any positive closed (1,1) current T can be attenuated in the following sense: there exists a finite composition of blowups such that the pull-back of T decomposes into two parts, the first associated to a divisor with normal crossing support, the second having small Lelong numbers.