aa r X i v : . [ m a t h . AG ] J u l To appear in the volume
Introduction to Lipschitz geometry of singularities ,edited by W. Neumann and A. Pichon, Springer, 2020 or 2021.ULTRAMETRICS AND SURFACE SINGULARITIES
PATRICK POPESCU-PAMPU
Abstract.
The present lecture notes give an introduction to works of Garc´ıa Barroso, Gonz´alezP´erez, Ruggiero and the author. The starting point of those works is a theorem of P loski, statingthat one defines an ultrametric on the set of branches drawn on a smooth surface singularityby associating to any pair of distinct branches the quotient of the product of their multiplic-ities by their intersection number. We show how to construct ultrametrics on certain sets ofbranches drawn on any normal surface singularity from their mutual intersection numbers andhow to interpret the associated rooted trees in terms of the dual graphs of adapted embeddedresolutions. The text begins by recalling basic properties of intersection numbers and multi-plicities on smooth surface singularities and the relation between ultrametrics on finite sets androoted trees. On arbitrary normal surface singularities one has to use Mumford’s definition ofintersection numbers of curve singularities drawn on them, which is also recalled.
Contents
Introduction 11. Multiplicity and intersection numbers for plane curve singularities 32. The statement of P loski’s theorem 73. Ultrametrics and rooted trees 84. A proof of P loski’s theorem using Eggers-Wall trees 135. An ultrametric characterization of arborescent singularities 156. The brick-vertex tree of a connected graph 177. Our strongest generalization of P loski’s theorem 198. Mumford’s intersection theory 209. A reformulation of the ultrametric inequality 2410. A theorem of graph theory 27References 28
Introduction
This paper is an expansion of my notes prepared for the course with the same title given atthe
International school on singularities and Lipschitz geometry , which took place in Cuernavaca(Mexico) from June 11th to 22nd 2018.If S denotes a normal surface singularity , that is, a germ of normal complex analytic surface,a branch on it is an irreducible germ of analytic curve contained in S . In his 1985 paper[21], Arkadiusz P loski proved that if one associates to every pair of distinct branches on the Date : 13 July 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Arborescent singularity, Birational geometry, Block, Cut-vertex, Eggers-Wall tree,Intersection number, Newton-Puiseux series, Normal surface singularity, Resolution of singularities, Tree, Ultra-metric, Valuation. singularity S “ p C , q the quotient A ¨ Bm p A q ¨ m p B q of their intersection number by the productof their multiplicities, then for every triple of pairwise distinct branches, two of those quotientsare equal and the third one is not smaller than them. An equivalent formulation is that theinverses m p A q ¨ m p B q A ¨ B of the previous quotients define an ultrametric on the set of branches on p C , q .Using the facts that the multiplicity of a branch is equal to its intersection number with asmooth branch L transversal to it, and that a given function is an ultrametric on a set if andonly if it is so in restriction to all its finite subsets, one deduces that P loski’s theorem is aconsequence of: Theorem A.
Let L be a smooth branch on the smooth surface singularity S and let F be a finite set of branches on S , transversal to L . Then the function u L : F ˆ F Ñ r , defined by u L p A, B q : “ p L ¨ A q ¨ p L ¨ B q A ¨ B if A ‰ B and u L p A, A q : “ is an ultrametric on F . This may be seen as a property of the pair p S, L q and one may ask whether it extends toother pairs consisting of a normal surface singularity and a branch on it. It turns out thatthis property characterizes the so-called arborescent singularities , that is, the normal surfacesingularities such that the dual graph of every good resolution is a tree. Namely, one has thefollowing theorem, which combines [9, Thm. 85] and [12, Thm. 1.46]: Theorem B.
Let L be a branch on the normal surface singularity S . Then thefunction u L defined as before is an ultrametric on any finite set F of brancheson S distinct from L if and only if S is an arborescent singularity. It is possible to think topologically about ultrametrics on finite sets in terms of certain typesof decorated rooted trees. In particular, any such ultrametric determines a rooted tree. Onemay try to describe this tree directly from the pair p S, F Y t L uq , when S is arborescent andthe ultrametric is the function u L associated to a branch L on it. In order to formulate such adescription, we need the notion of convex hull of a finite set of vertices of a tree: it is the unionof the paths joining those vertices pairwise.The following result was obtained in [9, Thm. 87]: Theorem C.
Let L be a branch on the arborescent singularity S and let F be afinite set of branches on S distinct from L . Then the rooted tree determined bythe ultrametric u L on F is isomorphic to the convex hull of the strict transformof F Y t L u in the dual graph of its preimage by an embedded resolution of it,rooted at the vertex representing the strict transform of L . Even when the singularity S is not arborescent, the function u L becomes an ultrametric inrestriction to suitable sets F of branches on S . Those sets are defined only in terms of convexhulls taken in the so-called brick-vertex tree of the dual graph of an embedded resolution of F Y t L u , and do not depend on any numerical parameter of the exceptional divisor of theresolution, be it a genus or a self-intersection number. The brick-vertex tree of a connectedgraph is obtained canonically by replacing each brick – a maximal inseparable subgraph whichis not an edge – by a star, whose central vertex is called a brick-vertex . One has the followinggeneralization of Theorem C (see [12, Thm. 1.42]): LTRAMETRICS AND SURFACE SINGULARITIES 3
Theorem D.
Let L be a branch on the normal surface singularity S and let F bea finite set of branches on S distinct from L . Consider an embedded resolution of F Y t L u . Assume that the convex hull of its strict transform in the brick-vertextree of the dual graph of its preimage does not contain brick-vertices of valency atleast in the convex hull. Then the function u L is an ultrametric in restrictionto F and the associated rooted tree is isomorphic to the previous convex hull,rooted at the vertex representing the strict transform of L . If S is not arborescent, there may exist other sets of branches on which u L restricts to anultrametric. Unlike the sets described in the previous theorem, in general they do not dependonly on the topology of the dual graph of their preimage on some embedded resolution, butalso on the self-intersection numbers of the components of the exceptional divisor (see [12, Ex.1.44]).The aim of the present notes is to introduce the reader to the previous results. Note that inthe article [12, Part 2], these results were extended to the space of real-valued semivaluations ofthe local ring of S .Let us describe briefly the structure of the paper. In Section 1 are recalled basic facts about multiplicities and intersection numbers of plane curve singularities. In Section 2 are statedtwo equivalent formulations of P loski’s theorem. In Section 3 is explained the relation betweenultrametrics and rooted trees mentioned above, an intermediate concept being that of hierarchy on a finite set. Using this relation, Section 4 presents a proof of Theorem A. This proof uses theso-called Eggers-Wall tree of a plane curve singularity relative to a smooth reference branch L ,constructed using associated Newton-Puiseux series. Section 5 explains the notions used in theformulation of Theorem B, that is, those of good resolution , embedded resolution , associated dualgraph and arborescent singularity . In Section 6 are described the related notions of cut-vertex and brick-vertex tree of a finite connected graph. Section 7 explains and illustrates the statementof Theorem D. In Section 8 is explained Mumford’s intersection theory of divisors on normalsurface singularities, after a proof of a fundamental property of such singularities, stating thatthe intersection form of any of their resolutions is negative definite. In Section 9 the ultrametricinequality concerning the restriction of u L to a triple of branches is reexpressed in terms of thenotion of angular distance on the dual graph of an adapted resolution. A crucial property ofthis distance is stated, which relates it to the cut-vertices of the dual graph. In Section 10 issketched the proof of a theorem of pure graph theory, relating distances satisfying the previouscrucial property and the brick-vertex tree of the graph. This theorem implies Theorem D. Acknowledgements.
This work was partially supported by the French grants ANR-17-CE40-0023-02 LISA and Labex CEMPI (ANR-11-LABX-0007-01). I am grateful to AlexandreFernandes, Adam Parusinski, Anne Pichon, Maria Ruas, Jos´e Seade and Bernard Teissier, whoformed the scientific committee of the
International school on singularities and Lipschitz geom-etry , for having invited me to give a course on the relations between ultrametrics and surfacesingularities. I am very grateful to my co-authors Evelia Garc´ıa Barroso, Pedro Gonz´alez P´erezand Matteo Ruggiero for the collaboration leading to our works [9], [12] presented in this paperand for their remarks on a previous version of it.1.
Multiplicity and intersection numbers for plane curve singularities
In this section we recall the notions of multiplicity of a plane curve singularity and intersectionnumber of two such singularities. One may find more details in [16, Sect. 5.1] or [8, Chap. 8].
PATRICK POPESCU-PAMPU
Let p S, s q be a smooth surface singularity , that is, a germ of smooth complex analyticsurface. Denote by O S,s its local C -algebra and by m S,s its maximal ideal, containing thegerms at s of holomorphic functions vanishing at s .A local coordinate system on S at s is a pair p x, y q P m S,s ˆ m S,s establishing an isomor-phism between a neighborhood of s in S and a neighborhood of the origin in C . Algebraicallyspeaking, this is equivalent to the fact that the pair p x, y q generates the maximal ideal m S,s ,or that it realizes an isomorphism O S,s » C t x, y u . This isomorphism allows to see each germ f P O S,s as a convergent power series in the variables x and y .A curve singularity on p S, s q is a germ p C, s q ã Ñ p
S, s q of not necessarily reduced curve on S , passing through s . As the germ p S, s q is isomorphic to the germ of the affine plane C at anyof its points, one says also that p C, s q is a plane curve singularity . A defining function of p C, s q is a function f P m S,s such that O C,s “ O S,s {p f q , where p f q denotes the principal ideal of O S,s generated by f . We write then C “ Z p f q .The curve singularity p C, s q may also be seen as an effective principal divisor on p S, s q . Thisallows to write C “ ř i P I p i C i , where p i P N ˚ for all i P I and the curve singularities C i arepairwise distinct and irreducible. We say in this case that the C i ’s are the branches of C . A branch on p S, s q is an irreducible curve singularity on p S, s q .Next definition introduces the simplest invariant of a plane curve singularity: Definition 1.1.
Assume that f P O S,s . Its multiplicity is the vanishing order of f at s : m s p f q : “ sup t n P N , f P m nS,s u P N Y t8u . If p C, s q is the curve singularity defined by f , we say also that m s p C q : “ m s p f q is its multi-plicity at s .It is a simple exercise to check that the multiplicity of a curve singularity is independentof the function defining it. If one chooses local coordinates p x, y q on p S, s q , then m s p f q is thesmallest degree of the monomials appearing in the expression of f as a convergent power seriesin the variables x and y . One has m s p f q “ 8 if and only if f “ m s p f q “ f defines a smooth branch on p S, s q .The following definition describes a measure of the way in which two curve singularitiesintersect: Definition 1.2.
Let
C, D ã Ñ p
S, s q be two plane curve singularities defined by f, g P m S,s . Thentheir intersection number is defined by: C ¨ D : “ dim C O S,s p f, g q P N Y t8u , where p f, g q denotes the ideal of O S,s generated by f and g .Note that C ¨ D ă `8 if and only if C and D do not share common branches, which isalso equivalent to the existence of n P N ˚ such that one has the following inclusion of ideals: p f, g q Ě m nS,s . Nevertheless, unlike the multiplicity, the intersection number C ¨ D is not alwaysequal to the smallest exponent n having this property. For instance, if one takes f : “ x and g : “ y , then C ¨ D “ p f, g q Ě p x, y q . We leave the verification of the previous facts as anexercise.The following proposition, which may be proved using Proposition 1.5 below, relates multi-plicities and intersection numbers: LTRAMETRICS AND SURFACE SINGULARITIES 5
Proposition 1.3. If p C, s q ã Ñ p
S, s q is a plane curve singularity, then C ¨ L ě m s p C q for anysmooth branch L through s , with equality if and only if L is transversal to C . More generally, if D is a second curve singularity on p S, s q , then C ¨ D ě m s p C q ¨ m s p D q , with equality if and onlyif C and D are transversal. Let us explain the notion of transversality used in the previous proposition, as it is moregeneral than the standard notion of transversality, which applies only to smooth submanifoldsof a given manifold. If C is a branch on p S, s q and one chooses a local coordinate system p x, y q on p S, s q , as well as a defining function f of C , it may be shown that the lowest degree part of f is a power of a complex linear form in x and y . This linear form defines a line in the tangentplane T s S of S at s , which is by definition the tangent line of C at s . One may show thatit is independent of the choices of local coordinates and defining function of C . If C is nowan arbitrary curve singularity, then its tangent cone is the union of the tangent lines of itsbranches. Given two plane curve singularities on the same smooth surface singularity S , onesays that they are transversal if each line of the tangent cone of one of them is transversal (inthe classical sense) to each line of the tangent cone of the other one.Let us pass now to the question of computation of intersection numbers. A basic methodconsists in breaking the symmetry between the two curve singularities, by working with a definingfunction of one of them and by parametrizing the other one. One has to be cautious and choosea normal parametrization , in the following sense: Definition 1.4. A normal parametrization of the branch p C, s q is a germ of holomorphicmorphism ν : p C , q Ñ p C, s q which is a normalization morphism, that is, it has topologicaldegree one.For instance, if the branch p C, q on p C , q is defined by the function y ´ x , then t Ñ p t , t q is a normal parametrization of C , but u Ñ p u , u q is not. A normal parametrization of a branch p C, s q may be also characterized by asking it to establish a homeomorphism between suitablerepresentatives of the germs p C , q and p C, s q .Normalization morphisms may be defined more generally for reduced germs p X, x q of arbitrarydimension (see [16, Sect. 4.4]), by considering the multi-germ whose multi-local ring is theintegral closure of the local ring O X,x in its total ring of fractions. Except for curve singularities,the source of a normalization morphism is not smooth in general.The following proposition is classical and states the announced expression of intersectionnumbers in terms of a parametrization of one germ and a defining function of the second one(see [2, Prop. II.9.1] or [16, Lemma 5.1.5]):
Proposition 1.5.
Let C be a branch on the smooth surface singularity p S, s q and D be a secondcurve singularity, not necessarily reduced. Let ν : p C , q Ñ p C, s q be a normal parametrizationof C and let g P m S,s be a defining function of D . Then: C ¨ D “ ord t p g ˝ ν p t qq , where ord t denotes the order of a power series in the variable t .Proof. This proof is adapted from that of [16, Lemma 5.1.5].The order of the zero power series is equal to by definition, therefore the statement is truewhen C is a branch of D .Let us assume from now on that C is not a branch of D .Consider a defining function f P m S,s of C . By Definition 1.2:(1) C ¨ D “ dim C O S,s p f, g q “ dim C O S,s {p f qp g C q “ dim C O C,s p g C q , PATRICK POPESCU-PAMPU where we have denoted by g C P O C,s the restriction of g to the branch C .Algebraically, the normal parametrization ν : p C , q Ñ p C, s q corresponds to a morphism oflocal C -algebras O C,s ã Ñ C t t u , isomorphic to the inclusion morphism of O C,s into its integralclosure taken inside its quotient field. In order to distinguish them, denote from now on by g C O C,s the principal ideal generated by g C inside O C,s and by g C C t t u its analog inside C t t u .One has the following equality inside the local C -algebra C t t u : g ˝ ν p t q “ g C . As a consequence: g C C t t u “ t ord t p g ˝ ν p t qq C t t u . Therefore:(2) ord t p g ˝ ν p t qq “ dim C C t t u g C C t t u . By comparing equations (1) and (2), we see that the desired equality is equivalent to:(3) dim C O C,s g C O C,s “ dim C C t t u g C C t t u . The two quotients appearing in (3) are the cokernels of the two injective multiplication maps O C,s ¨ g C ÝÑ O C,s and C t t u ¨ g C ÝÑ C t t u . The associated short exact sequences may be completed intoa commutative diagram in which the first two vertical maps are the inclusion map O C,s ã Ñ C t t u :0 O C,s O C,s O C,s g C O C,s C t t u C t t u C t t u g C C t t u K and cokernel K :0 ÝÑ K ÝÑ O C,s g C O C,s ÝÑ C t t u g C C t t u ÝÑ K ÝÑ . For every finite exact sequence of finite-dimensional vector spaces, the alternating sum of di-mensions vanishes. Therefore:dim C K ´ dim C O C,s g C O C,s ` dim C C t t u g C C t t u ´ dim C K “ . This shows that the desired equality (3) would result from the equality dim C K “ dim C K .This last equality is a consequence of the so-called “snake lemma” (see for instance [1, Prop.2.10]), applied to the previous commutative diagram. Indeed, by this lemma, one has an exactsequence: 0 ÝÑ K ÝÑ C t t u O C,s ÝÑ C t t u O C,s ÝÑ K ÝÑ . Reapplying the previous argument about alternating sums of dimensions, one gets the neededequality dim C K “ dim C K . (cid:3) LTRAMETRICS AND SURFACE SINGULARITIES 7
Note that the previous proof shows in fact that for any abstract branch p C, s q , not necessarilyplanar, one has the equality:(4) dim C O C,s p g q “ ord t p g ˝ ν p t qq , for any g P O C,s and for any normal parametrization ν : p C , q Ñ p C, s q of p C, s q . If the branch p C, s q is contained in an ambient germ p X, s q and H is an effective principal divisor on p X, s q which does not contain the branch, then equality (4) shows that the intersection number of C and H at s may be computed as the order of the series obtained by composing a definingfunction of p H, s q and a normal parametrization of p C, s q . Example 1.6.
Consider the branches: $&% A : “ Z p y ´ x q ,B : “ Z p y ´ x q ,C : “ Z p y ´ x q on the smooth surface singularity p C , q . Denoting by m the multiplicity function at the originof C , we have: m p A q “ , m p B q “ , m p C q “ , as results from Definition 1.1. Using Proposition 1.5 and the fact that whenever m and n arecoprime positive integers, t Ñ p t n , t m q is a normal parametrization of Z p y n ´ x m q , one gets thefollowing values for the intersection numbers of the branches A, B, C : B ¨ C “ , C ¨ A “ , A ¨ B “ . Therefore: $’’’’’’’’’&’’’’’’’’’% B ¨ Cm p B q ¨ m p C q “ ,C ¨ Am p C q ¨ m p A q “ ,A ¨ Bm p A q ¨ m p B q “ . One notices that two of the previous quotients are equal and the third one is greater than them.P loski discovered that this is a general phenomenon for plane branches, as explained in the nextsection. 2.
The statement of P loski’s theorem
In this section we state a theorem of P loski of 1985 and a reformulation of it in terms of thenotion of ultrametric .Denote simply by S the germ of smooth surface p S, s q and by m p A q the multiplicity of abranch p A, s q ã Ñ p
S, s q .In his 1985 paper [21], P loski proved the following theorem: Theorem 2.1. If A, B, C are three pairwise distinct branches on a smooth surface singularity S , then one has the following relations, up to a permutation of the three fractions: A ¨ Bm p A q ¨ m p B q ě B ¨ Cm p B q ¨ m p C q “ C ¨ Am p C q ¨ m p A q . PATRICK POPESCU-PAMPU
Denote by B p S q the infinite set of branches on S . By inverting the fractions appearing inthe statement of Theorem 2.1, it may be reformulated in the following equivalent way: Theorem 2.2.
Let S be a smooth surface singularity. Then the map B p S q ˆ B p S q Ñ r , defined by p A, B q Ñ m p A q ¨ m p B q A ¨ B if A ‰ B, otherwiseis an ultrametric. What does it mean that a function is an ultrametric? We explain this in the next section andwe show how to think topologically about ultrametrics on finite sets in terms of certain kindsof decorated rooted trees. This way of thinking is used then in Section 4 in order to prove thereformulation 2.2 of P loski’s theorem.3.
Ultrametrics and rooted trees
In this section we define the notion of ultrametric and we explain how to think about anultrametric on a finite set in topological terms, as a special kind of rooted and decorated tree.This passes through understanding that the closed balls of an ultrametric form a hierarchy andthat finite hierarchies are equivalent to special types of decorated rooted trees. For more details,one may consult [9, Sect. 3.1].
Definition 3.1.
Let p M, d q be a metric space. It is called ultrametric if one has the followingstrong form of the triangle inequality: d p A, B q ď max t d p A, C q , d p B, C qu , for all A, B, C P M. In this case, one says also that d is an ultrametric on the set M .In any metric space p M, d q , a closed ball is a subset of M of the form: B p A, r q : “ t P P M, d p P, A q ď r u where the center A P M and the radius r P r , are given. As we will see shortly, given aclosed ball, neither its center nor its radius are in general well-defined, contrary to an intuitioneducated only by Euclidean geometry.One has the following characterizations of ultrametrics: Proposition 3.2.
Let p M, d q be a metric space. Then the following properties are equivalent:(1) p M, d q is ultrametric.(2) The triangles are all isosceles with two equal sides not less than the third side.(3) All the points of a closed ball are centers of it.(4) Two closed balls are either disjoint, or one is included in the other.Proof. All the equivalences are elementary but instructive to check. We leave their proofs asexercises. (cid:3)
Example 3.3.
Consider a set M “ t A, B, C, D u and a distance function d on it such that: d p B, C q “ , d p A, B q “ d p A, C q “ , d p A, D q “ d p B, D q “ d p C, D q “
5. Note that one mayembed p M, d q isometrically into a 3-dimensional Euclidean space by choosing an isosceles triangle ABC with the given edge lengths, and by choosing then the point D on the perpendicular tothe plane of the triangle passing through its circumcenter. Let us look for the closed balls ofthis finite metric space. For radii less than 1, they are singletons. For radii in the interval r , q , LTRAMETRICS AND SURFACE SINGULARITIES 9 we get the sets t B, C u , t A u , t D u . Note that both B and C are centers of the ball t B, C u , thatis, B p B, r q “ B p C, r q “ t
B, C u for every r P r , q . Once the radius belongs to the interval r , q ,the balls are t A, B, C u and t D u . Finally, for every radius r P r , , there is only one closedball, the whole set. Figure 1 depicts the set t A, B, C, D u as well as the mutual distances andthe associated set of closed balls. AB C D
12 2 55 5
Figure 1.
The balls of an ultrametric space with four pointsExample 3.3 illustrates the fact that neither the center nor the radius of a closed ball of afinite ultrametric space is well-defined, once the ball has more than one element. Instead, everyclosed ball has a well-defined diameter : Definition 3.4.
The diameter of a closed ball in a finite metric space is the maximal distancebetween pairs of not necessarily distinct points of it.The last characterization of ultrametrics in Proposition 3.2 shows that the set Balls p M, d q of closed balls of an ultrametric space p M, d q is a hierarchy on M , in the following sense: Definition 3.5. A hierarchy on a set M is a subset H of its power set P p M q , satisfying thefollowing properties: ‚ H R H . ‚ The singletons belong to H . ‚ M belongs to H . ‚ Two elements of H are either disjoint, or one is included into the other.If H is a hierarchy on a set M , it may be endowed with the inclusion partial order. We willconsider instead its reverse partial order ĺ H , defined by: A ĺ H B ðñ A Ě B, for all A, B P H . Reversing the inclusion partial order has the advantage of identifying the leaves of the cor-responding rooted tree with the maximal elements of the poset p H , ĺ H q (see Proposition 3.8below).When M is finite , one may represent the poset p H , ĺ H q using its associated Hasse diagram : Definition 3.6.
Let p X, ĺ q be a finite poset. Its Hasse diagram is the directed graph whoseset of vertices is X , two vertices a, b P X being joined by an edge oriented from a to b whenever a ă b and the two points are directly comparable , that is, there is no other element of X lying strictly between them.Hasse diagrams of finite posets are abstract oriented acyclic graphs. This means that theyhave no directed cycles, which is a consequence of the fact that a partial order is antisymmetricand transitive. Hasse diagrams are not necessarily planar, but, as all finite graphs, they maybe always immersed in the plane in such a way that any pair of edges intersect transversely.When drawing a Hasse diagram in the plane as an immersion, we will use the convention toplace the vertex a of the Hasse diagram below the vertex b whenever a ă b . This is alwayspossible because of the absence of directed cycles. This convention makes unnecessary addingarrowheads along the edges in order to indicate their orientations. Example 3.7.
Consider the finite set t , , , , , u of positive divisors of 12, partially orderedby divisibility: a ĺ b if and only if a divides b . Its Hasse diagram is drawn in Figure 2.1242 631 Figure 2.
The Hasse diagram of the set of positive divisors of 12.The Hasse diagrams of finite hierarchies are special kinds of graphs:
Proposition 3.8.
The Hasse diagram of a hierarchy p H , ĺ H q on a finite set M is a tree inwhich the maximal directed paths start from M and terminate at the singletons. Moreover, foreach vertex which is not a singleton, there are at least two edges starting from it.Proof. We sketch a proof, leaving the details to the reader.The first statement results from the fact that the singletons of M are exactly the maximalelements of the poset p H , ĺ H q , that M itself is the unique minimal element and that all theelements of a hierarchy which contain a given element are totally ordered by inclusion.Let us prove the second statement. Consider B P H and assume that it is not a singleton.This means that it is not minimal for inclusion, therefore there exists B P H such that B Ĺ B and B is directly comparable to B . Let A be a point of B z B . Consider B P H whichcontains the point A , is included into B and is directly comparable to it. As A P B z B , thisshows that B is not included in B . We want to show that the two sets B and B are disjoint.Otherwise, by the definition of a hierarchy, we would have B Ĺ B Ĺ B , which contradicts theassumption that B and B are directly comparable. (cid:3) Example 3.9.
Consider the ultrametric space of Example 3.7, represented in Figure 1. Werepeat it on the left of Figure 3. The Hasse diagram of the hierarchy of its closed balls is drawn
LTRAMETRICS AND SURFACE SINGULARITIES 11 on the right of Figure 3. Near each vertex is represented the diameter of the corresponding ball.We have added a root vertex , connected to the vertex representing the whole set. It may bethought as a larger ball, obtained by adding formally to M “ t A, B, C, D u a point ω , infinitelydistant from each point of M . This larger ball is the set M : “ M Y t ω u . AB C D
12 2 55 5 t A u t B u t C u t D ut B, C ut A, B, C u t
A, B, C, D ut A, B, C, D, ω u Figure 3.
The tree of the hierarchy of closed balls of Example 3.9One may formalize in the following way the construction performed in Example 3.9:
Definition 3.10.
The tree of a hierarchy p H , ĺ H q on a finite poset M is its Hasse diagram,completed with a root representing the set M : “ M Y t ω u , joined with the vertex representing M and rooted at M . Here ω is a point distinct from the points of M .The tree of a hierarchy is a rooted tree in the following sense: Definition 3.11. A rooted tree is a tree with a distinguished vertex, called its root . If Θis a rooted tree with root r , then the vertex set of Θ gets partially ordered by declaring that a ĺ r b if and only if the unique segment r r, a s joining r to a in the tree is contained in r r, b s .When Θ is the rooted tree of a hierarchy H on a finite set M , then the partial order ĺ M defined by choosing M as root restricts to the partial order ĺ H if one identifies the set H withthe set of vertices of Θ which are distinct from the root.Proposition 3.8 may be reformulated in the following way as a list of properties of the tree ofthe hierarchy: Proposition 3.12.
Let Θ be the tree of a hierarchy on a finite set, and let r be its root. Then r is a vertex of valency and there are no vertices of valency . This proposition motivates the following definition:
Definition 3.13.
A rooted tree whose root is of valency 1 and which does not possess verticesof valency 2 is a hierarchical tree . The hierarchy of a hierarchical tree p Θ , r q is constructedin the following way: ‚ Define M to be the set of leaves of the rooted tree p Θ , r q , that is, the set of vertices ofvalency 1 which are distinct from the root r . ‚ For each vertex p of Θ different from the root, consider the subset of M consisting ofthe leaves a such that p ĺ r a .We leave as an exercise to prove: Proposition 3.14.
The constructions of Definitions 3.10 and 3.13, which associate a hierar-chical tree to a hierarchy on a finite set and a hierarchy to a hierarchical tree are inverse of eachother.
As a preliminary to the proof, one may test the truth of the proposition on the example ofFigure 3.Let us return to finite ultrametric spaces p M, d q . We saw that the set Balls p M, d q of its closedballs is a hierarchy on M . Proposition 3.14 shows that one may think about this hierarchy asa special kind of rooted tree, namely, a hierarchical tree. This hierarchical tree alone does notallow to get back the distance function d . How to encode it on the tree?The idea is to look at the function defined on Balls p M, d q , which associates to each ball itsdiameter (see Definition 3.4): Proposition 3.15.
Let p M, d q be a finite ultrametric space. Then the map which sends eachclosed ball to its diameter is a strictly decreasing r , -valued function defined on the poset p Balls p M, d q , ĺ q , taking the value exactly on the singletons of M . Equivalently, it is a strictlydecreasing r , -valued function on the set of vertices of the tree of the hierarchy, vanishing onthe set M of leaves and taking the value on the root. As an example, one may look again at Figure 3. The value taken by the previous diameterfunction is written near each vertex of the hierarchical tree.If p Θ , r q is a hierarchical tree, denote by V p Θ q its set of vertices and by a ^ r b the infimumof a and b relative to ĺ r , whenever a, b P V p Θ q . This infimum may be characterized by theproperty that r r, a s X r r, b s “ r r, a ^ r b s . The following is a converse of Proposition 3.15: Proposition 3.16.
Let p Θ , r q be a hierarchical tree and λ : V p Θ q Ñ r , be a strictly decreas-ing function relative to the partial order ĺ r on Θ induced by the root. Assume that λ vanisheson the set M of leaves of Θ and takes the value at r . Then the map d : M ˆ M Ñ r , a, b q Ñ λ p a ^ r b q is an ultrametric on M . Let us introduce a special name for the functions appearing in Proposition 3.16:
Definition 3.17.
Let p Θ , r q be a hierarchical tree. A depth function on it is a function λ : V p Θ q Ñ r , which satisfies the following properties: ‚ it is strictly decreasing relative to the partial order ĺ r on Θ induced by the root r ; ‚ it vanishes on the set of leaves of Θ; ‚ it takes the value at the root r .Note that the first two conditions of Definition 3.17 imply that a depth function vanishes exactly on the set of leaves of the underlying hierarchical tree.One has the following analog of Proposition 3.14: Proposition 3.18.
The constructions of Propositions 3.15 and 3.16 are inverse of each other.That is, giving an ultrametric on a finite set M is equivalent to giving a depth function on ahierarchical tree whose set of leaves is M . It is this proposition which allows to think about an ultrametric as a special kind of rootedand decorated tree. We leave its proof as an exercise (see [3]).
LTRAMETRICS AND SURFACE SINGULARITIES 13 A proof of P loski’s theorem using Eggers-Wall trees
In this section we sketch a proof of P loski’s theorem 2.1 using the equivalence between ul-trametrics on finite sets and certain kinds of rooted trees formulated in Proposition 3.18. Therooted trees used in this proof are the
Eggers-Wall trees of a plane curve singularity relativeto smooth reference branches. The precise definition of Eggers-Wall trees is not given, becausethe proofs of the subsequent generalizations of P loski’s theorem will be of a completely differentspirit.Instead of working both with multiplicities and intersection numbers as in P loski’s originalstatement, we will work only with the latest ones.Let S be a smooth germ of surface and L ã Ñ S be a smooth branch . Define the followingfunction on the set of branches on S which are different from L :(5) u L : p B p S qzt L uq Ñ R ` p A, B q Ñ p L ¨ A q ¨ p L ¨ B q A ¨ B if A ‰ B, . In the remaining part of this section we will sketch a proof of:
Theorem 4.1.
The function u L is an ultrametric. We leave as an exercise to show using Proposition 1.3 that Theorem 4.1 implies the reformu-lation given in Theorem 2.2 of P loski’s Theorem 2.1.Our proof of Theorem 4.1 will pass through the notion of
Eggers-Wall tree associated to aplane curve singularity relative to a smooth branch of reference L (see the proof of Theorem 4.5below). Let us illustrate it by an example. { A Θ L p A q { B Θ L p B q { C Θ L p C q C A B { { { L p A ` B ` C q Figure 4.
The Eggers-Wall tree of the plane curve singularity of Example 4.2
Example 4.2.
Consider again the branches A “ Z p y ´ x q , B “ Z p y ´ x q , C “ Z p y ´ x q on S “ p C , q of Example 1.6. Assume that the branch L is the germ at 0 of the y -axis Z p x q . Thedefining equations of the three branches A, B, C may be considered as polynomial equations in the variable y . As such, they admit the following roots which are fractional powers of x : A : x { ,B : x { ,C : x { . Associate to the root x { a compact segment Θ L p A q identified with the interval r , usingan exponent function e L : Θ L p A q Ñ r , and mark on it the point e ´ L p { q with exponent3 {
2. Define also an index function i L : Θ L p A q Ñ N ˚ , constantly equal to 1 on the interval r e ´ L p q , e ´ L p { qs and to 2 on the interval p e ´ L p { q , e ´ L p8qs (see the left-most segment ofFigure 4). Here the number 2 is to be thought as the minimal positive denominator of theexponent 3 { x { . The segment Θ L p A q endowed with the two functions e L and i L is the Eggers-Wall tree of the branch A relative to the branch L . It is considered as a rootedtree with root e ´ L p q , labeled with the branch L . Its leaf e ´ L p8q is labeled with the branch A .Consider analogously the Eggers-Wall trees Θ L p B q and Θ L p C q , endowed with pairs of exponentand index functions and labeled roots and leaves (see the left part of Figure 4).Look now at the plane curve singularity A ` B ` C . Its Eggers-Wall tree Θ L p A ` B ` C q relative to the branch L is obtained from the individual trees Θ L p A q , Θ L p B q , Θ L p C q by a gluingprocess, which identifies two by two initial segments of those trees.Consider for instance the segments Θ L p A q , Θ L p B q . Look at the order of the difference x { ´ x { of the roots which generated them, seen as a series with fractional exponents. This orderis the fraction 3 {
2, because 3 { ă {
3. Identify then the points with the same exponent ď { L p A q , Θ L p B q . One gets a rooted tree Θ L p A ` B q with root labeled by L andwith two leaves, labeled by the branches A, B . The exponent and index functions of the treesΘ L p A q , Θ L p B q descend to functions with the same name e L , i L defined on Θ L p A ` B q . Endowedwith those functions, Θ L p A ` B q is the Eggers-Wall tree of the curve singularity A ` B .If one considers now the curve singularity A ` B ` C , then one glues analogously the three pairsof trees obtained from Θ L p A q , Θ L p B q , Θ L p C q . The resulting Eggers-Wall tree Θ L p A ` B ` C q is drawn on the right side of Figure 4. It is also endowed with two functions e L , i L , obtained bygluing the exponent and index functions of the trees Θ L p A q , Θ L p B q , Θ L p C q . Its marked pointsare its ends, its bifurcation points and the images of the discontinuity points of the index functionof the Eggers-Wall tree of each branch. Near each marked point is written the correspondingvalue of the exponent function. The index function is constant on each segment p a, b s joiningtwo marked points a and b , where a ă L b . Here ĺ L denotes the partial order on the treeΘ L p A ` B ` C q determined by the root L (see Definition 3.11).One may associate analogously an Eggers-Wall tree Θ L p D q to any plane curve singularity D , relative to a smooth reference branch L . It is a rooted tree endowed with an exponentfunction e L : Θ L p D q Ñ r , and an index function i L : Θ L p D q Ñ N ˚ . The tree and bothfunctions are constructed using Newton-Puiseux series expansions of the roots of a Weierstrasspolynomial f P C rr x ssr y s defining D in a coordinate system p x, y q such that L “ Z p x q . Thetriple p Θ L p D q , e L , i L q is independent of the choices involved in the previous definition (see [9,Proposition 103]). One may find the precise definition and examples of Eggers-Wall trees inSection 4.3 of the previous reference and in [10, Sect. 3]. Historical remarks about this notionmay be found in [10, Rem. 3.18] and [11, Sect. 6.2]. The name, introduced in author’s thesis[22], makes reference to Eggers’ 1982 paper [6] and to Wall’s 2003 paper [27].What allows us to prove Theorem 4.1 using Eggers-Wall trees is that the values u L p A, B q ofthe function u L defined by relation (5) are determined in the following way from the Eggers-Walltree Θ L p D q , for each pair of distinct branches p A, B q of D (recall from the paragraph preceding LTRAMETRICS AND SURFACE SINGULARITIES 15
Proposition 3.16 that A ^ L B denotes the infimum of A and B relative to the partial order ĺ L induced by the root L of Θ L p D q ): Theorem 4.3.
For each pair p A, B q of distinct branches of D and every smooth reference branch L different from the branches of D , one has: u L p A, B q “ ż A ^ L BL de L i L . Example 4.4.
Let us verify the equality stated in Theorem 4.3 on the branches of Example4.2. Looking at the Eggers-Wall tree Θ L p A ` B ` C q on the right side of Figure 4, we see that: ż A ^ L BL de L i L “ ż { de “ . But 1 { u L p A, B q “ p A ¨ B q{ pp L ¨ A qp L ¨ B qq “ p A ¨ B q{ p m p A q ¨ m p B qq “ {
2, as was computed inExample 1.6. The equality is verified. We have used the fact that both A and B are transversalto L , which implies that L ¨ A “ m p A q and L ¨ B “ m p B q .In equivalent formulations which use so-called characteristic exponents , Theorem 4.3 goesback to Smith [23, Section 8], Stolz [24, Section 9] and Max Noether [20]. A modern proof,based on Proposition 1.5, may be found in [28, Thm. 4.1.6].As a consequence of Theorem 4.3, we get the following strengthening of Theorem 4.1: Theorem 4.5.
Let D be a plane curve singularity. Denote by F p D q the set of branches of D .Let L be a reference smooth branch which does not belong to F p D q . Then the function u L is anultrametric in restriction to F p D q and its associated rooted tree is isomorphic as a rooted treewith labeled leaves to the Eggers-Wall tree Θ L p D q .Proof. Consider Θ L p D q as a topological tree with vertex set equal to its set of ends and oframification points. Root it at L . Then it becomes a hierarchical tree in the sense of Definition3.13. The function P Ñ ˆż PL de L i L ˙ ´ is a depth function on it, in the sense of Definition 3.17. Using Theorem 4.3 and Proposition3.18, we get Theorem 4.5. (cid:3) For more details about the proof of P loski’s theorem presented in this section, see [9, Sect.4.3]. 5.
An ultrametric characterization of arborescent singularities
In this section we state a generalization of Theorem 4.1 for all arborescent singularities andthe fact that it characterizes this class of normal surface singularities. We start by recallingthe needed notions of embedded resolution and associated dual graph of a finite set of branchescontained in a normal surface singularity.From now on, S denotes an arbitrary normal surface singularity , that is, a germ of normalcomplex analytic surface. Let us recall the notion of resolution of such a singularity: Definition 5.1.
Let p S, s q be a normal surface singularity. A resolution of it is a properbimeromorphic morphism π : S π Ñ S such that S π is smooth. Its exceptional divisor E π is the reduced preimage π ´ p s q . The resolution is good if its exceptional divisor has normalcrossings and all its irreducible components are smooth. The dual graph Γ p π q of the resolution π is the finite graph whose set of vertices P p π q is the set of irreducible components of E π , twovertices being joined by an edge if and only if the corresponding components intersect.Every normal surface singularity admits resolutions and even good ones. This result, forwhich partial proofs appeared already at the end of the XIXth century, was proved first in theanalytical context by Hirzebruch in his 1953 paper [15]. His proof was inspired by previousworks of Jung [18] and Walker [26], done in an algebraic context.Assume now that F is a finite set of branches on S . It may be also seen as a reduced divisoron S , by thinking about their sum. The notion of embedded resolution of F is an analog of thatof good resolution of S : Definition 5.2.
Let p S, s q be a normal surface singularity and let π : S π Ñ S be a resolutionof S . If A is a branch on S , then its strict transform by π is the closure inside S π of thepreimage π ´ p A z s q . Let F be a finite set of branches on S . Its strict transform by π is theset or, depending on the context, the divisor formed by the strict transforms of the branches of F . The preimage π ´ F of F by π is the sum of its strict transform and of the exceptionaldivisor of π . The morphism π is an embedded resolution of F if it is a good resolution of S and the preimage of F by π is a normal crossings divisor. The dual graph Γ p π ´ F q ofthe preimage π ´ F is defined similarly to the dual graph Γ p π q of π , taking into account all theirreducible components of π ´ F .In the previous definition, the preimage π ´ F of F by π is seen as a reduced divisor. We willsee in Definition 8.4 below that there is also a canonical way, due to Mumford, to define canon-ically a not necessarily reduced rational divisor supported by π ´ F , called the total transform of F by π , and denoted by π ˚ F .The notion of dual graph of a resolution allows to define the following class of arborescentsingularities , whose name was introduced in the paper [9], even if the class had appear before,for instance in Camacho’s work [5]: Definition 5.3.
Let S be a normal surface singularity. It is called arborescent if the dualgraphs of its good resolutions are trees.Remark that in the previous definition we ask nothing about the genera of the irreduciblecomponents of the exceptional divisors.By using the fact that any two resolutions are related by a sequence of blow ups and blowdowns of their total spaces (see [14, Thm. V.5.5]), one sees that the dual graphs of all goodresolutions are trees if and only if this is true for one of them.Consider now an arbitrary branch L on the normal surface singularity S . We may define thefunction u L by the same formula (5) as in the case when both S and L were assumed smooth.Intersection numbers of branches still have a meaning, as was shown by Mumford. We willexplain this in Section 8 below (see Definition 8.5).The following generalization of Theorem 4.1 both gives a characterization of arborescentsingularities and extends Theorem 4.5 to all arborescent singularities S and all – not necessarilysmooth – reference branches L on them (recall that B p S q denotes the set of branches on S ): Theorem 5.4.
Let S be a normal surface singularity and L P B p S q . Then:(1) u L is ultrametric on B p S qzt L u if and only if S is arborescent.(2) In this case, for any finite set F of branches on S not containing L , the rooted tree ofthe restriction of u L to F is isomorphic to the convex hull of F Y t L u in the dual graphof the preimage of F Y t L u by any embedded resolution of F Y t L u , rooted at L . LTRAMETRICS AND SURFACE SINGULARITIES 17
We do not prove in the present notes that if u L is an ultrametric on B p S qzt L u , then S isarborescent. The interested reader may find a proof of this fact in [12, Sect. 1.6]. The remainingimplication of point (1) and point (2) of Theorem 5.4 are, taken together, a consequence ofTheorem 7.1 below. For this reason, we do not give a separate proof of them, the rest of thispaper being dedicated to the statement and a sketch of proof of Theorem 7.1. The notion of brick-vertex tree of a finite connected graph being crucial in this theorem, we dedicate nextsection to it.By combining Theorems 4.5 and 5.4 one gets (see [9, Thm. 112]): Proposition 5.5.
Whenever S and L are both smooth, the Eggers-Wall tree Θ L p D q of a planecurve singularity D ã Ñ S not containing L is isomorphic to the convex hull of the strict transformof F p D q Y t L u in the dual graph of its preimage by any of its embedded resolutions. A prototype of this fact was proved differently in the author’s thesis [22, Thm. 4.4.1], thengeneralized in two different ways by Wall in [28, Thm. 9.4.4] (see also Wall’s comments in [28,Sect. 9.10]) and by Favre and Jonsson in [7, Prop. D.1].6.
The brick-vertex tree of a connected graph
In this section we introduce the notion of brick-vertex tree of a connected graph, which iscrucial in order to state Theorem 7.1 below, the strongest known generalization of P loski’stheorem.
Definition 6.1. A graph is a compact cell complex of dimension ď
1. If Γ is a graph, its setof vertices is denoted V p Γ q and its set of edges is denoted E p Γ q .In the sequel it will be crucial to look at the vertices which disconnect a given graph: Definition 6.2.
Let Γ be a connected graph. A cut-vertex of Γ is a vertex whose removaldisconnects Γ. A bridge of Γ is an edge such that the removal of its interior disconnects Γ.If a, b, c are three not necessarily distinct vertices of Γ, one says that b separates a from c ifeither b P t a, c u or if a and c belong to different connected components of the topological spaceΓ z t b u .Note that an end of a bridge is a cut-vertex if and only if it has valency at least 2 in Γ, thatis, if and only if it is not a leaf of Γ. It will be important to distinguish the class of graphs whichcannot be disconnected by the removal of one vertex, as well as the maximal graphs of this classcontained in a given connected graph: Definition 6.3.
A connected graph is called inseparable if it does not contain cut-vertices. A block of a connected graph Γ is a maximal inseparable subgraph of it. A brick of Γ is a blockwhich is not a bridge.Note that all the bridges of a connected graph are blocks of it.
Example 6.4.
In Figure 5 is represented a connected graph. Its cut-vertices are surrounded inred. Its bridges are represented as black segments. It has three bricks, the edges of each brickbeing colored in the same way.By replacing each brick of a connected graph by a star-shaped graph, one gets canonically atree associated to the given graph:
Definition 6.5.
The brick-vertex tree BV p Γ q of a connected graph Γ is the tree whose set ofvertices is the union of the set of vertices of Γ and of a set of new brick-vertices corresponding Figure 5.
A connected graph, its cut-vertices, its bridges and its bricksbijectively to the bricks of Γ, its edges being either the bridges of Γ or new edges connectingeach brick-vertex to the vertices of the corresponding brick. Formally, this may be written asfollows: ‚ V p BV p Γ qq “ V p Γ q \ t bricks of Γ u . ‚ E p BV p Γ qq “ t bridges of Γ u \ tr v, b s , v P V p Γ q , b is a brick of Γ , v P V p b qu .We denoted by v the vertex v of Γ when it is seen as a vertex of BV p Γ q and b P V p BV p Γ qq the brick-vertex representing the brick b of Γ.The notion of brick-vertex tree was introduced in [12, Def. 1.34]. It is strongly related toother notions introduced before either in general topology or in graph theory, as explained in[12, Rems. 1.35, 2.50].Note that whenever Γ is a tree, BV p Γ q is canonically isomorphic to it, as Γ has no bricks. Example 6.6.
On the left side of Figure 6 is repeated the graph Γ of Figure 5, with its cut-vertices and bricks emphasized. On its right side is represented its associated brick-vertex tree BV p Γ q . Each representative vertex of a brick is drawn with the same color as its correspondingbrick. The edges of BV p Γ q which are not bridges of Γ are represented in magenta and thickerthan the other edges. Γ BV p Γ q Figure 6.
The connected graph of Example 6.6 and its brick-vertex treeThe importance of the brick-vertex tree in our context stems from the following property ofit (see [10, Prop. 1.36]), formulated using the vocabulary introduced in Definition 6.2 and thenotations introduced in Definition 6.5:
Proposition 6.7.
Let Γ be a finite graph and a, b, c P V p Γ q . Then b separates a from c in Γ ifand only if b separates a from c in BV p Γ q . We are ready now to state the strongest known generalization of P loski’s theorem (see Theo-rem 7.1 below).
LTRAMETRICS AND SURFACE SINGULARITIES 19 Our strongest generalization of P loski’s theorem
In this section we formulate Theorem 7.1, which generalizes Theorem 4.5 to all normal surfacesingularities and all branches on them, using the notion of brick-vertex tree introduced in theprevious section.Recall that the notion of brick-vertex tree of a connected graph was introduced in Definition6.5. A fundamental property of normal surface singularities is that the dual graphs of theirresolutions are connected (which is a particular case of the so-called
Zariski’s main theorem ,whose statement may be found in [14, Thm. V.5.2]). This implies that the dual graph of thepreimage (see Definition 5.2) of any finite set of branches on such a singularity is also connected.Therefore, one may speak about its corresponding brick-vertex tree. The convex hull of a finiteset of vertices of it is the union of the segments which join them pairwise.Here is the announced generalization of Theorem 4.5, which is a slight reformulation of [12,Thm. 1.42]:
Theorem 7.1.
Let S be a normal surface singularity. Consider a finite set F of branches onit and an embedded resolution π : S π Ñ S of F . Let Γ be the dual graph of the preimage π ´ F of F by π . Assume that the convex hull Conv BV p Γ q p F q of the strict transform of F by π in thebrick-vertex tree BV p Γ q does not contain brick-vertices of valency at least in Conv BV p Γ q p F q .Then for all L P F , the restriction of u L to F z t L u is an ultrametric and the correspondingrooted tree is isomorphic to Conv BV p Γ q p F q , rooted at L . Example 7.2.
Assume that the dual graph Γ of π ´ F is as shown on the left side of Figure7. The vertices representing the strict transforms of the branches of the set F are drawnarrowheaded. Note that the subgraph which is the dual graph of the exceptional divisor isthe same as the graph of Figure 5. On the right side of Figure 7 is represented using thickred segments the convex hull Conv BV p Γ q p F q . We see that the hypothesis of Theorem 7.1 issatisfied. Indeed, the convex hull contains only two brick-vertices, which are of valency 2 and 3in Conv BV p Γ q p F q . Note that the blue one is of valency 4 in the dual graph Γ, which shows theimportance of looking at the valency in the convex hull Conv BV p Γ q p F q , not in Γ.Γ BV p Γ q Figure 7.
An example where the hypothesis of Theorem 7.1 is satisfiedAs shown in [12, Ex. 1.44], the condition about valency is not necessary in general for u L tobe an ultrametric on F z t L u .Note that we have expressed Theorem 7.1 in a slightly different form than the equivalentTheorem D of the introduction. Namely, we included L in the branches of F . This formulationemphasizes the symmetry of the situation: all the choices of reference branch inside F lead to the same tree, only the root being changed. In fact, we will obtain Theorem 7.1 as a consequenceof Theorem 9.10, in which no branch plays any more a special role.Before that, we will explain in the next section Mumford’s definition of intersection number oftwo curve singularities drawn on an arbitrary normal surface singularity, which allows to definein turn the functions u L appearing in the statement of Theorem 7.1.8. Mumford’s intersection theory
In this section we explain Mumford’s definition of intersection number of Weil divisors on anormal surface singularity, introduced in his 1961 paper [19]. It is based on Theorem 8.1, statingthat the intersection form of any resolution of a normal surface singularity is negative definite.This theorem being fundamental for the study of surface singularities, we present a detailedproof of it.Let π : S π Ñ S be a resolution of the normal surface singularity S . Denote by p E u q u P P p π q thecollection of irreducible components of the exceptional divisor E π of π (see Definition 5.1).Denote by: E p π q R : “ à u P P p π q R E u the real vector space freely generated by those prime divisors, that is, the space of real divisorssupported by E π . It is endowed with a symmetric bilinear form p D , D q Ñ D ¨ D given byintersecting the corresponding compact cycles on S π . We call it the intersection form . Itsfollowing fundamental property was proved by Du Val [25] and Mumford [19]: Theorem 8.1.
The intersection form on E p π q R is negative definite.Proof. The following proof is an expansion of that given by Mumford in [19].The singularity S being normal, the exceptional divisor E π is connected (this is a particularcase of Zariski’s main theorem , see [14, Thm. V.5.2]). Therefore:(6) The dual graph Γ p π q is connected.Consider any germ of holomorphic function f on p S, s q , vanishing at s , and look at the divisorof its lift to the surface S π :(7) p π ˚ f q “ ÿ u P P p π q a u E u ` p π ˚ f q str . Here p π ˚ f q str denotes the strict transform of the divisor defined by f on S . Denote also:(8) $&% e u : “ a u E u P E p π q R , for all u P P p π q ,σ : “ ÿ u P P p π q e u P E p π q R . As f vanishes at the point s , its lift π ˚ f vanishes along each component E u of E π , therefore a u ą u P P p u q . We deduce that p e u q u P P p π q is a basis of E p π q R and that:(9) e u ¨ e v ě , for all u, v P P p π q such that u ‰ v. The divisor p π ˚ f q being principal, its associated line bundle is trivial. Therefore:(10) p π ˚ f q ¨ E u “ u P P p π q , because this intersection number is equal by definition to the degree of the restriction of thisline bundle to the curve E u . By combining the relations (7), (8) and (10), we deduce that:(11) σ ¨ e u “ ´ a u p π ˚ f q str ¨ E u , for every u P P p π q . LTRAMETRICS AND SURFACE SINGULARITIES 21
As the germ of effective divisor p π ˚ f q str along E π has no components of E π in its support,the intersection numbers p π ˚ f q str ¨ E u are all non-negative. Moreover, at least one of them ispositive, because the divisor p π ˚ f q str is non-zero. By combining this fact with relations (11)and with the inequalities a u ą
0, we get:(12) " σ ¨ e u ď , for every u P P p π q , there exists u P P p π q such that σ ¨ e u ă . Consider now an arbitrary element τ P E p π q R zt u . One may develop it in the basis p e u q u P P p π q :(13) τ “ ÿ u P P p π q x u e u . We will show that τ ă
0. As τ was chosen as an arbitrary non-zero vector, this will implythat the intersection form on E p π q R is indeed negative definite. The trick is to express theself-intersection τ using the expansion (13), then to develop it by bilinearity and to replace thevectors e u by σ ´ ř v ‰ u e v in a precise place: τ “ ˜ÿ u x u e u ¸ ““ ÿ u x u e u ` ÿ u ă v x u x v e u ¨ e v ““ ÿ u x u ˜ σ ´ ÿ v ‰ u e v ¸ ¨ e u ` ÿ u ă v x u x v e u ¨ e v ““ ÿ u x u p σ ¨ e u q ´ ÿ u ‰ v x u e u ¨ e v ` ÿ u ă v x u x v e u ¨ e v ““ ÿ u x u p σ ¨ e u q ´ ÿ u ă v p x u ´ x v q e u ¨ e v . We got the equality:(14) τ “ ÿ u x u p σ ¨ e u q ´ ÿ u ă v p x u ´ x v q e u ¨ e v . Using the inequalities (9) and (12), we deduce that its right-hand side is non-positive, thereforethe intersection form is negative semi-definite.It remains to show that τ ă
0. Assume by contradiction that τ “
0. Equality (14) showsthat the following equalities are simultaneously satisfied:(15) ÿ u x u p σ ¨ e u q “ , (16) p x u ´ x v q e u ¨ e v “ , for all u ă v. The relations (16) imply that x u “ x v whenever e u ¨ e v ą
0. As e u “ a u E u with a u ą
0, theinequality e u ¨ e v ą E u ¨ E v ą
0, that is, with the fact that r u, v s is anedge of the dual graph Γ p π q . This dual graph being connected (see (6)), we see that x u “ x v for all u, v P P p π q . Consider now an index u satisfying the second condition of relations (12).Equation (15) implies that x u “
0. Therefore all the coefficients x u vanish, which contradictsthe hypothesis that τ ‰ (cid:3) As a consequence of Theorem 8.1, one may define the dual basis p E _ u q u P P p π q of the basis p E u q u P P p π q by the following relations, in which δ uv denotes Kronecker’s delta-symbol:(17) E _ u ¨ E v “ δ uv , for all p u, v q P P p π q . By associating to each prime divisor E u the corresponding valuation of the local ring O S,s ,computing the orders of vanishing along E u of the pull-backs π ˚ f of the functions f P O S,s ,one injects the set P p π q in the set of real-valued valuations of O S,s . This allows to see theindex u of E u as a valuation. Such valuations are called divisorial . If u denotes a divisorialvaluation, it has a center on any resolution, which is either a point or an irreducible componentof the exceptional divisor. In the second case, one says that the valuation appears in theresolution . The following notion, inspired by approaches of Favre & Jonsson [7, App. A] and[17, Sect. 7.3.6], was introduced in [12, Def. 1.6]: Definition 8.2.
Let u, v be two divisorial valuations of S . Consider a resolution of S in whichboth u and v appear. Then their bracket is defined by: x u, v y : “ ´ E _ u ¨ E _ v . The bracket x u, v y may be interpreted as the intersection number of two Weil divisors on S associated to the divisors E u and E v (see Proposition 8.7 below). As a consequence, it is well-defined. That is, if the divisorial valuations u, v are fixed, then their bracket does not dependon the resolution in which they appear. This fact may be also proved using the property thatany two resolutions of S are related by a sequence of blow ups and blow downs (see [12, Prop.1.5]).It is a consequence of Theorem 8.1 that the brackets are all non-negative (see [12, Prop. 1.4]).Moreover, by the Cauchy-Schwarz inequality applied to the opposite of the intersection form: Lemma 8.3.
For every a, b P P p π q : x a, b y ď x a, a yx b, b y , with equality if and only if a “ b . Let now D be a Weil divisor on S , that is, a formal sum of branches on S . If D is principal,that is, the divisor p f q of a meromorphic germ on S , then one may lift it to a resolution S π asthe principal divisor p π ˚ f q . This divisor decomposes as the sum of an exceptional part p π ˚ D q ex supported by E π and the strict transform of D . The crucial property of the lift p π ˚ f q , alreadyused in the proof of Theorem 8.1 (see relation (10)), is that its intersection numbers with all thecomponents E u of E π vanish. In [19, Sect. II (b)], Mumford imposed this property in order todefine a lift π ˚ D for any Weil divisor D on S : Definition 8.4.
Let D be a Weil divisor on S . Its total transform π ˚ D is the unique sum p π ˚ D q ex ` p π ˚ D q str such that:(1) p π ˚ D q ex P E p π q Q .(2) p π ˚ D q str is the strict transform of D by π .(3) p π ˚ D q ¨ E u “ u P P p π q .The divisor p π ˚ D q ex supported by the exceptional divisor of π is the exceptional transform of D by π .The divisor π ˚ D is well-defined, as results from Theorem 8.1. The point is to show that p π ˚ D q ex exists and is unique with the property (3). Write it as a sum ř v P P p π q x v E v . The last LTRAMETRICS AND SURFACE SINGULARITIES 23 condition of Definition 8.4 may be written as the system: ÿ v P P p π q p E v ¨ E u q x v “ ´p π ˚ D q str ¨ E u , for all u P P p π q . This is a square linear system in the unknowns x v , whose matrix is the matrix of the intersectionform in the basis p E u q u P P p π q . As the intersection form is negative definite, it is non-degenerate,therefore this system has a unique solution. Moreover, all its coefficients being integers, itssolution has rational coordinates, which shows that p π ˚ D q ex P E p π q Q .Using Definition 8.4 and the standard definition of intersection numbers on smooth surfacesrecalled in Section 1, Mumford defined in the following way in [19, Sect. II (b)] the intersectionnumber of two Weil divisors on S : Definition 8.5.
Let
A, B be two Weil divisors on S without common components, and π be aresolution of S . Then the intersection number of A and B is defined by: A ¨ B : “ π ˚ A ¨ π ˚ B. Using the fact that any two resolutions of S are related by a sequence of blow ups and blowdowns (see [14, Thm. V.5.5]), it may be shown that the previous notion is independent ofthe choice of resolution, similarly to that of bracket of two divisorial valuations introduced inDefinition 8.2. In particular, if S is smooth, one may choose π to be the identity. This shows thatin this case Mumford’s definition gives the same intersection number as the standard Definition1.2. Example 8.6.
Let S be the germ at the origin 0 of the quadratic cone Z p x ` y ` z q ã Ñ C (itis the so-called A surface singularity). Let A and B be the germs at 0 of two distinct generatinglines of the cone. One may resolve S by blowing up 0. This morphism π : S π Ñ S separates allthe generators, therefore it is an embedded resolution of t A, B u . The exceptional divisor of π is the projectivisation of the cone, that is, it is a smooth rational curve E . Its self-intersectionnumber is the opposite of the degree of the curve seen embedded in the projectivisation ofthe ambient space C . Therefore, E “ ´
2. Let us compute the total transform π ˚ A “p π ˚ A q str ` xE . The imposed constraint π ˚ A ¨ E “ ´ x “
0, therefore x “ {
2. Wehave used the fact that the strict transform p π ˚ A q str of A by π is smooth and transversal to E ,which implies that p π ˚ A q str ¨ E “ π ˚ A “ p π ˚ A q str ` p { q E and similarly, π ˚ B “ p π ˚ B q str ` p { q E . UsingDefinition 8.5, we get: A ¨ B “ π ˚ A ¨ π ˚ B ““ pp π ˚ A q str ` p { q E q ¨ pp π ˚ B q str ` p { q E q ““ p π ˚ A q str ¨ p π ˚ B q str ` p { qpp π ˚ A q str ` p π ˚ B q str q ¨ E ` p { q E ““ ` p { q ¨ ` p { q ¨ p´ q ““ { . This example shows in particular that the intersection number of two curve singularitiesdepends on the normal surface singularity on which it is computed . Indeed, the branches A and B are also contained in a smooth surface (any two generators of the quadratic cone are obtainedas the intersection of the cone with a plane passing through its vertex). In such a surface, theirintersection number is 1 instead of 1 { Proposition 8.7.
Let
A, B be two distinct branches on S . Consider an embedded resolution π oftheir sum. Denote by E a , E b the components of the exceptional divisor E π which are intersectedby the strict transforms p π ˚ A q str and p π ˚ B q str respectively. Then: A ¨ B “ x a, b y . Proof.
This proof uses directly Definition 8.4.As π is an embedded resolution of A ` B , the strict transforms p π ˚ A q str and p π ˚ B q str aredisjoint. Therefore p π ˚ A q str ¨ p π ˚ B q str “
0. Using the last condition in the Definition 8.4 of thetotal transform of a divisor, we know that p π ˚ A q ¨ p π ˚ B q ex “ p π ˚ A q ex ¨ p π ˚ B q “
0. Combiningboth equalities, we deduce that: A ¨ B “ p π ˚ A q ¨ p π ˚ B q ““ p π ˚ A q ¨ pp π ˚ B q ex ` p π ˚ B q str q ““ p π ˚ A q ¨ p π ˚ B q str ““ pp π ˚ A q ex ` p π ˚ A q str q ¨ p π ˚ B q str ““ p π ˚ A q ex ¨ p π ˚ B q str ““ p π ˚ A q ex ¨ p π ˚ B ´ p π ˚ B q ex q ““ ´p π ˚ A q ex ¨ p π ˚ B q ex ““ ´p´ E _ a q ¨ p´ E _ b q ““ x a, b y . At the end of the computation we have used the equality p π ˚ A q ex “ ´ E _ a , which results fromthe fact that π is an embedded resolution of A . Indeed, this implies that pp π ˚ A q str ` E _ a q¨ E u “ u P P p π q , which shows that one has indeed the stated formula for p π ˚ A q ex . (cid:3) A reformulation of the ultrametric inequality
In this section we explain the notion of angular distance on the set of vertices of the dualgraph of a good resolution of S . Theorem 9.2 states a crucial property of this distance, relatingit to the cut-vertices of the dual graph. Then the ultrametric inequality is reexpressed in termsof the angular distance. This allows to show that Theorem 7.1 is a consequence of Theorem9.10, which is formulated only in terms of the angular distance.Let π : S π Ñ S be a good resolution of the normal surface singularity S . Recall that P p π q denotes the set of irreducible components of its exceptional divisor E π . Using the notion ofbracket from Definition 8.2, one may define (see [13, Sect. 2.7] and [12, Sect. 1.2]): Definition 9.1.
The angular distance is the function ρ : P p π q ˆ P p π q Ñ r , given by: ρ p a, b q : “ $&% ´ log x a, b y x a, a yx b, b y if a ‰ b, a “ b. The fact that the function ρ takes values in the interval r , is a consequence of Lemma8.3. The attribute “angular” was chosen by Gignac and Ruggiero because their definition in [13,Sect. 2.7] was more general, applying to any pair of real-valued semivaluations of the local ring O S,s , and that it depended only on those valuations up to homothety, similarly to the angle oftwo vectors. It is a distance by the following theorem of Gignac and Ruggiero [13, Prop. 1.10](recall that the notion of vertex separating two other vertices was introduced in Definition 6.2):
Theorem 9.2.
The function ρ is a distance on the set P p π q . Moreover, for every a, b, c P P p π q ,the following properties are equivalent: ‚ one has the equality ρ p a, b q ` ρ p b, c q “ ρ p a, c q ; LTRAMETRICS AND SURFACE SINGULARITIES 25 ‚ b separates a and c in the dual graph Γ p π q . This theorem explains the importance of cut-vertices of the dual graph Γ p π q for understandingthe angular distance.Theorem 9.2 is a reformulation of the following theorem, which was first proved by in [9,Prop. 79, Rem. 81] for arborescent singularities, then in [13, Prop. 1.10] for arbitrary normalsurface singularities (see also [12, Prop. 1.18] for a slightly different proof): Theorem 9.3.
Let a, b, c P P p π q . Then: x a, b yx b, c y ď x b, b yx a, c y , with equality if and only if b separates a and c in the dual graph Γ p π q . Theorem 9.3 may be also reformulated in terms of spherical geometry using the sphericalPythagorean theorem (see [12, Prop. 1.19.III]).Using Proposition 8.7 and Definition 9.1 of the angular distance, one may reformulate in thefollowing way the ultrametric inequality for the restriction of the function u L to a set of threebranches: Proposition 9.4.
Let
L, A, B, C be pairwise distinct branches on S . Consider an embeddedresolution of their sum and let E l , E a , E b , E c the irreducible components of its exceptional di-visor which intersect the strict transforms of L, A, B and C respectively. Then the following(in)equalities are equivalent:(1) u L p A, B q ď max t u L p A, C q , u L p B, C qu .(2) p A ¨ B q ¨ p L ¨ C q ě min tp A ¨ C qp L ¨ B q , p B ¨ C qp L ¨ A qu . (3) x a, b yx l, c y ě min tx a, c yx l, b y , x b, c yx l, a yu . (4) ρ p a, b q ` ρ p l, c q ď max t ρ p a, c q ` ρ p l, b q , ρ p b, c q ` ρ p l, a qu . We leave the easy proof of this proposition to the reader. It uses the definitions of the function u L , of the angular distance, as well as Proposition 8.7. Note that excepted the first one, all theinequalities are symmetric in the four branches L, A, B, C . The fourth one is a well-knowncondition in combinatorics, whose name was introduced by Bunemann in his 1974 paper [4]:
Definition 9.5.
Let p X, δ q be a finite metric space. One says that it satisfies the four pointscondition if whenever a, b, c, d P X , one has the following inequality: δ p a, b q ` δ p c, d q ď max t δ p a, c q ` δ p b, d q , δ p a, d q ` δ p b, c qu . In the same way in which a finite ultrametric may be thought as a special kind of decorated rooted tree (see Proposition 3.18), a finite metric space satisfying the four points condition maybe thought as a special kind of decorated unrooted tree (see [3]):
Proposition 9.6.
The metric space p X, δ q satisfies the four points condition if and only if δ is induced by a length function on a tree containing the set X among its set of vertices. If,moreover, one constrains X to contain all the vertices of the tree of valency or , then thistree is unique up to a unique isomorphism fixing X . Let us introduce supplementary vocabulary in order to deal with the special trees appearingin Proposition 9.6:
Definition 9.7.
Let X be a finite set. An X -tree is a tree whose set of vertices contains the set X and such that each vertex of valency at most 2 belongs to X . If p X, δ q is a finite metric spacewhich satisfies the four points condition, then the unique X -tree characterized in Proposition9.6 is called the tree hull of p X, δ q . The basic idea of the proof of Proposition 9.6 is that an X -tree is characterized by the shapesof the convex hulls of the quadruples of points of X , and that those shapes are determined bythe cases of equality in the 12 triangle inequalities and the 3 four points conditions associated toeach quadruple. In Figure 8 are represented the five possible shapes. For instance, the H -shapeis the generic one, characterized by the fact that one has no equality in the previous inequalities. Figure 8.
The possible shapes of an X -tree, when X has four elements.Let us come back to our normal surface singularity S . One has the following property (see[12, Prop. 1.24]): Proposition 9.8.
Let F be a finite set of branches on S . If u L is an ultrametric on F z t L u forone branch L in F , then the same is true for any branch of F . By Proposition 9.4, if u L is an ultrametric on F z t L u for one branch L in F , then one hasthe symmetric relation (2) for every quadruple of branches of F containing L . The subtle pointof the proof of Proposition 9.8 is to deduce from this fact that (2) is satisfied by all quadruples.Given Proposition 9.8, it is natural to try to relate the rooted trees associated to the ultra-metrics obtained by varying L among the branches of F . By looking at quadruples of branchesfrom F , one may prove using Propositions 9.4 and 9.8 that: Proposition 9.9.
Let F be a finite set of branches on S . Consider an embedded resolution of F such that the map associating to each branch A of F the component E a of the exceptional divisorintersected by its strict transform is injective. Denote by F π the set of divisorial valuations a appearing in this way. Then:(1) The function u L is an ultrametric on F z t L u for some branch L P F if and only if theangular distance ρ satisfies the four points condition in restriction to the set F π .(2) Assume that the previous condition is satisfied. Then the rooted tree associated to u L on F z t L u is isomorphic to the tree hull of p F π , ρ q by an isomorphism which sends each endmarked by a branch A of F to the vertex a of the tree hull. Proposition 9.9 implies readily that Theorem 7.1 is a consequence of the following fact (see[12, Cor. 1.40]):
Theorem 9.10.
Let S be a normal surface singularity. Consider a set G of vertices of the dualgraph Γ of a good resolution π : S π Ñ S of S . Assume that the convex hull Conv BV p Γ q p G q of G inthe brick-vertex tree of Γ does not contain brick-vertices of valency at least in Conv BV p Γ q p G q .Then the restriction of the angular distance ρ to G satisfies the four points condition and theassociated tree hull is isomorphic as a G -tree to Conv BV p Γ q p G q . In turn, Theorem 9.10 is a consequence of a graph-theoretic result presented in the nextsection (see Theorem 10.1).
LTRAMETRICS AND SURFACE SINGULARITIES 27
A theorem of graph theory
In this final section we state a pure graph-theoretical theorem, which implies Theorem 9.10of the previous section. As we explained before, that theorem implies in turn our strongestgeneralization of P loski’s theorem, that is, Theorem 7.1.Theorem 9.10 is a consequence of Theorem 9.2 and of the following graph-theoretic result:
Theorem 10.1.
Let Γ be a finite connected graph and δ be a distance on the set V p Γ q of verticesof Γ , such that for every a, b, c P V p Γ q , the following properties are equivalent: ‚ one has the equality δ p a, b q ` δ p b, c q “ δ p a, c q ; ‚ b separates a and c in Γ .Let X be a set of vertices of Γ such that the convex hull Conv BV p Γ q p X q of X in the brick-vertex tree of Γ does not contain brick-vertices of valency at least in Conv BV p Γ q p X q . Then δ satisfies the points condition in restriction to X and the tree hull of p X, δ q is isomorphic to Conv BV p Γ q p X q as an X -tree. The idea of the proof of Theorem 10.1 is to show that, under the given hypotheses, theequalities among the triangle inequalities and four points conditions are as described by thebrick-vertex tree. It is writtend in a detailed way in [12, Thm. 1.38]. a cb v d Γ a cb v d BV p Γ q Figure 9.
A convex hull of four vertices
Example 10.2.
Let us consider again the connected graph Γ of Example 6.6. Look at itsvertices a, b, c, d shown on the left of Figure 9. The corresponding vertices a, b, c, d of the brick-vertex tree BV p Γ q are shown on the right side of the figure. Denoting X : “ t a, b, c, d u , theconvex hull Conv BV p Γ q p X q is also drawn on the right side using thick red segments. We seethat the hypothesis of Theorem 10.1 about the valencies of brick-vertices is satisfied, as the onlybrick-vertex contained in Conv BV p Γ q p X q is of valency 3 in this convex hull.As shown by the F -shape of Conv BV p Γ q p X q , one should have the following equalities andinequalities in the four points conditions concerning X :(18) δ p a, d q ` δ p b, c q “ δ p a, c q ` δ p b, d q ą δ p a, b q ` δ p c, d q . Let us prove that this is indeed the case. Consider the cut vertex v of Γ shown on the left sideof Figure 9. As it separates a from d , we have the equality δ p a, d q “ δ p a, v q ` δ p v, d q . As v does not separate a from b , we have the strict inequality δ p a, v q ` δ p b, v q ą δ p a, b q . Using similar equalities and inequalities, we get: δ p a, d q ` δ p b, c q ““ p δ p a, v q ` δ p v, d qq ` p δ p b, v q ` δ p v, c qq ““ p δ p a, v q ` δ p v, c qq ` p δ p b, v q ` δ p v, d qq ““ δ p a, c q ` δ p b, d q ““ p δ p a, v q ` δ p b, v qq ` p δ p v, d q ` δ p v, c qq “ą δ p a, b q ` δ p c, d q . The (in)equalities (18) are proved.One proves similarly the triangle equalities δ p a, b q ` δ p b, c q “ δ p a, c q , δ p a, b q ` δ p b, d q “ δ p a, d q and the fact that one has no equality among the triangle inequalities concerning the triple t a, c, d u , which shows that the tree hull of p X, δ q has indeed an F -shape, with the vertices a, b, c, d placed as in Conv BV p Γ q p X q . References [1] Atiyah, M. F., Macdonald, I. G.
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