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THE VALUATIVE TREE Contents Introduction 3 Part 1. Generalities 7 THE VALUATIVE TREE CHARLES FAVRE AND MATTIAS JONSSON Abstract. We describe the set V of all R [ f1g valued valuations ν on the ring C[[x; y]] normalized by minfν(x); ν(y)g = 1. We show it has a natural structure of R-tree, induced by the order relation ν1 ≤ ν2 iff ν1(φ) ≤ ν2(φ) for all φ. The space V can also be metrized, endowing it with a metric tree struc- ture. We explain how to obtain these structures directly from the Riemann- Zariski variety of C[[x; y]]. The tree structure on V also provides an identifi- cation of valuations with balls of irreducible curves in a natural ultrametric. Finally, we show that the dual graphs of all sequences of blow-ups patch to- gether, yielding an R-tree naturally isomorphic to V. Contents Introduction 3 Part 1. Generalities 7 Part 2. MacLane's method 13 2. Sequences of toroidal key polynomials. 13 3. Graded rings and numerical invariants 19 4. From valuations to STKP's 23 5. A computation 24 Part 3. Tree structures 27 6. Trees 27 7. Tree structure on 30 8. Metric tree structureV on 33 V Part 4. Topologies 37 9. The weak topology 37 10. The strong topology 38 11. The Zariski topology. 40 12. The Hausdorff-Zariski topology 43 13. Comparison of topologies 44 Part 5. Valuations and curves 46 Date: October 3, 2002. 1991 Mathematics Subject Classification. Primary: 14H20, Secondary: 13A18, 54F50. Key words and phrases. Valuations, trees, Riemann-Zariski variety. 1 2 CHARLES FAVRE AND MATTIAS JONSSON 14. Valuations through intersections 46 15. Balls of curves 46 16. Multiplicities 47 17. Classification 48 Appendix 49 A. Thinness 49 B. Valuations through Puiseux series 50 C. The universal dual tree 54 References 58 THE VALUATIVE TREE 3 Introduction In this paper we study valuations on the regular local complete ring R = C[[x; y]] of formal power series in two variables. Our study is motivated by local, two-dimensional, questions in dynamical systems and complex analysis and even though the present work makes no explicit reference to any of these fields, some of our findings bear their mark. The earliest systematic study of valuations in two dimensions was done in the fundamental work by Zariski [Z], who, among other things, identified the set VK of (not necessarily real valued) valuations on R, vanishing on C∗ and positive on the maximal ideal m = (x; y), with sequences of infinitely nearby points. The space K carries a natural topology (the Zariski topology) and is known as the Riemann-ZariskiV variety. It is a non-Hausdorff quasi-compact space. The obstruction for K being Hausdorff comes from the fact that certain valuations, the divisorial ones,V do not define closed points. Namely, their associated valuation rings strictly contain other valuation rings. As a remedy to this, we may consider only valuations whose valuation rings are maximal under inclusion. This produces a new space which is now Hausdorff and compact. We call it valuation space; our aim is toVdescribe its structure in detail. We identify the elements of with valuations ν : R R+ = [0; ], normalized by ν(m) = 1 (see Theorem 11.6).V The quotient topology! on induced1 from V VK is the topology of pointwise convergence: νn ν iff νn(φ) ν(φ) for all φ R. We call it the weak topology. A strong top!ology can similarly! be defined2 by (normalized) uniform convergence. This topology is metrizable. Our main result is that these two topologies endow with the structure of a tree modeled on the real line. Thus we also refer to Vas the valuative tree. We distinguish between two different types of tree structures.V A tree is a partially ordered set having a unique minimal element (its root) in which segments are isomorphic to real intervals. A metric tree is a metric space in which two points are joined by a unique path isometric to a real interval (a metric tree is often called R-tree in the literature.) The non-metric tree structure arises as follows. For ν ; ν , we say ν ν 1 2 2 V 1 ≤ 2 when ν1(φ) ν2(φ) for all φ R. The valuation νm sending φ to its multiplicity m(φ) at the≤origin is then dominated2 by any other valuation. This natural order defines a (non-metric) tree structure on , rooted at νm (Theorem 7.1). An interesting consequence of the tree structureV is that any subset of valuations in admits an infimum. Moreover, any tree carries a natural weak tree topology. OnV it coincides with the weak topology (Theorem 9.1). AsV for the metric tree structure, any irreducible φ m defines an irreducible local formal curve as well as a curve valuation ν 2. A curve valuation is a φ 2 V maximal element under and the segment [νm; νφ] is isomorphic, as a totally ordered set, to the interv≤al [1; ]. We construct a natural parameterization 1 [1; ] α ν [νm; ν ] which is invariant in the following sense: if f : 1 3 7! φ,α 2 φ R R is a ring automorphism, then the induced map on takes νφ,α, to ! V 1 νf(φ),α. The number α is hence an invariant of a valuation called its skewness . 1the skewness is the inverse of the volume of a valuation as defined in [ELS] (see remark 8.5) 4 CHARLES FAVRE AND MATTIAS JONSSON It can be computed directly through the formula α(ν) = supφ ν(φ)=m(φ). We use skewness to define a metric on valuation space , giving it the structure of a metric tree (Theorem 8.8) and show that this treeV metric induces the strong topology (Theorem 10.1). There are four kinds of points in the valuative tree . The interior points are valuations that become monomial (i.e. determined byVtheir values on x and y) after a finite sequence of blowups. We call them toroidal. They include all diviso- rial valuations but also all irrational valuations such as the monomial valuation defined by ν(x) = 1, ν(y) = p2. The ends of are curve valuations and infinitely singular valuations, which are valuations withV infinite multiplicity. In going from K to we discarded certain valuations, namely the ones with nonmaximal valuationV rings.V These valuations nevertheless do have a natural place in the valuative tree. They are tree tangent vectors at points correspond- ing to divisorial valuations (see Theorem 11.7). Geometrically they are curve valuations where the curve is defined by an exceptional divisor. The valuative tree is a beautiful object which may be viewed in a number of different ways. Each corresponds to a particular interpretation of a valua- tion, and each gives a new insight into it. Some of them will hopefully lead to generalizations in a broader context. Let us describe three such points of views. The first way consists of identifying valuations with balls of curves. For any two irreducible curves defined by φ ; φ m, set d(φ ; φ ) = m(φ )m(φ )=(φ φ ) 1 2 2 1 2 1 2 1 · 2 where φ1 φ2 is the intersection multiplicity of φ1 and φ2. It is a nontrivial fact that d defines· a ultrametric on the set of all irreducible formal curves (c.f. [G]). This fact allows us to associate to ( ; d) a metric tree by declaringCa point in to be a closed ball in , the partialC order given byTrevC erse inclusion of balls, andTC the metric derived fromC radii of balls. Theorem 15.2 states that the tree is isometric to the valuative tree with its end points removed (i.e. to the setTC V tor of toroidal valuations). We use this result to extend intersection products andV multiplicities to any valuation. The second way uses Zariski's identification of valuations with sequences of infinitely nearby points. We let Γπ be the dual graph of a finite sequence of blow-ups π. It is a simplicial tree and as such defines a nonmetric tree. When one sequence π0 contains another π, the dual graph Γπ is naturally a subtree of Γπ0 . This allows to take the union Γ of Γπ over all sequences π of blow-ups above the origin. The result, called the universal dual graph, is a nonmetric tree that we show is isomorphic to the set of toroidal valuations. Moreover, Vtor we may recursively and consistently attach to any vertex in any Γπ a vector (a; b) (N )2, called its Farey weight and the associated Farey parameter a=b 2 ∗ (see also [HP]). The Farey parameter induces on tor a new invariant of valuation called the thinness which is a modified version ofVskewness using the multiplicity of valuations. We may use the thinness to parameterize : this induces a new Vtor metric tree structure on tor, the thin metric. See Theorem C.2. A third way is to view Vthe valuative tree in the context of Berkovich spaces and Bruhat-Tits buildings. As a nonmetric tree, embeds as the closure of a disk in the Berkovich projective line P1(k) over theV local field k = C((x)). The thin THE VALUATIVE TREE 5 metric on then also arises from an identification of a subset of the Berkovich V projective line with the Bruhat-Tits building of PGL2. The applications of the tree structure of to dynamics and analysis will be explored in forthcoming papers [FJ1, FJ2]. V We have divided this paper into five parts and an appendix. In the first part we give basic definitions, examples and results on valuations.
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