Patrick Popescu-Pampu

García Barroso

Ruggiero González Pérez

Adam Fernandes

Anne Parusinski

Maria Pichon

José Ruas

Evelia García Barroso

Pedro González Pérez

ULTRAMETRICS AND SURFACE SINGULARITIES

Keywords: 2010 Mathematics Subject Classification. 14B05 (primary), 14J17, 32S25 Arborescent singularity, Birational geometry, Block, Cut-vertex, Eggers-Wall tree, Intersection number, Newton-Puiseux series, Normal surface singularity, Resolution of singularities, Tree, Ultrametric, Valuation

The starting point of those works is a theorem of P loski, stating that one defines an ultrametric on the set of branches drawn on a smooth surface singularity by associating to any pair of distinct branches the quotient of the product of their multiplicities by their intersection number. We show how to construct ultrametrics on certain sets of branches drawn on any normal surface singularity from their mutual intersection numbers and how to interpret the associated rooted trees in terms of the dual graphs of adapted embedded resolutions. The text begins by recalling basic properties of intersection numbers and multiplicities on smooth surface singularities and the relation between ultrametrics on finite sets and rooted trees. On arbitrary normal surface singularities one has to use Mumford's definition of intersection numbers of curve singularities drawn on them, which is also recalled.

Introduction

This paper is an expansion of my notes prepared for the course with the same title given at the 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 [START_REF]Remarque sur la multiplicité d'intersection des branches planes[END_REF], Arkadiusz P loski proved that if one associates to every pair of distinct branches on the singularity S " pC 2 , 0q the quotient A ¨B mpAq ¨mpBq of their intersection number by the product of their multiplicities, then for every triple of pairwise distinct branches, two of those quotients are equal and the third one is not smaller than them. An equivalent formulation is that the inverses mpAq ¨mpBq A ¨B of the previous quotients define an ultrametric on the set of branches on pC 2 , 0q.

Using the facts that the multiplicity of a branch is equal to its intersection number with a smooth branch L transversal to it, and that a given function is an ultrametric on a set if and only if it is so in restriction to all its finite subsets, one deduces that P loski's theorem is a consequence 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 ˆF Ñ r0, 8q defined by u L pA, Bq :" pL ¨Aq ¨pL ¨Bq A ¨B if A ‰ B and u L pA, Aq :" 0 is an ultrametric on F.

This may be seen as a property of the pair pS, Lq and one may ask whether it extends to other pairs consisting of a normal surface singularity and a branch on it. It turns out that this property characterizes the so-called arborescent singularities, that is, the normal surface singularities such that the dual graph of every good resolution is a tree. Namely, one has the following theorem, which combines [9, Thm. 85] and [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Thm. 1.46]: Theorem B. Let L be a branch on the normal surface singularity S. Then the function u L defined as before is an ultrametric on any finite set F of branches on 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 types of decorated rooted trees. In particular, any such ultrametric determines a rooted tree. One may try to describe this tree directly from the pair pS, F Y tLuq, when S is arborescent and the ultrametric is the function u L associated to a branch L on it. In order to formulate such a description, we need the notion of convex hull of a finite set of vertices of a tree: it is the union of 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 a finite set of branches on S distinct from L. Then the rooted tree determined by the ultrametric u L on F is isomorphic to the convex hull of the strict transform of F Y tLu 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 in restriction to suitable sets F of branches on S. Those sets are defined only in terms of convex hulls taken in the so-called brick-vertex tree of the dual graph of an embedded resolution of F Y tLu, and do not depend on any numerical parameter of the exceptional divisor of the resolution, be it a genus or a self-intersection number. The brick-vertex tree of a connected graph is obtained canonically by replacing each brick -a maximal inseparable subgraph which is not an edge -by a star, whose central vertex is called a brick-vertex. One has the following generalization of Theorem C (see [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Thm. 1.42]):

Multiplicity and intersection numbers for plane curve singularities

In this section we recall the notions of multiplicity of a plane curve singularity and intersection number of two such singularities. One may find more details in [START_REF] De Jong | Local analytic geometry[END_REF]Sect. 5.1] or [START_REF] Fischer | Plane algebraic curves[END_REF]Chap. 8].

Let pS, sq be a smooth surface singularity, that is, a germ of smooth complex analytic surface. Denote by O S,s its local C-algebra and by m S,s its maximal ideal, containing the germs at s of holomorphic functions vanishing at s. A local coordinate system on S at s is a pair px, yq P m S,s ˆmS,s establishing an isomorphism between a neighborhood of s in S and a neighborhood of the origin in C 2 . Algebraically speaking, this is equivalent to the fact that the pair px, yq generates the maximal ideal m S,s , or that it realizes an isomorphism O S,s » Ctx, yu. 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 pS, sq is a germ pC, sq ãÑ pS, sq of not necessarily reduced curve on S, passing through s. As the germ pS, sq is isomorphic to the germ of the affine plane C 2 at any of its points, one says also that pC, sq is a plane curve singularity. A defining function of pC, sq is a function f P m S,s such that O C,s " O S,s {pf q, where pf q denotes the principal ideal of O S,s generated by f . We write then C " Zpf q .

The curve singularity pC, sq may also be seen as an effective principal divisor on pS, sq. This allows to write C " ř iPI p i C i , where p i P N ˚for all i P I and the curve singularities C i are pairwise distinct and irreducible. We say in this case that the C i 's are the branches of C. A branch on pS, sq is an irreducible curve singularity on pS, sq.

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 pf q :" suptn P N, f P m n S,s u P N Y t8u.
If pC, sq is the curve singularity defined by f , we say also that m s pCq :" m s pf q is its multiplicity at s.

It is a simple exercise to check that the multiplicity of a curve singularity is independent of the function defining it. If one chooses local coordinates px, yq on pS, sq, then m s pf q is the smallest degree of the monomials appearing in the expression of f as a convergent power series in the variables x and y. One has m s pf q " 8 if and only if f " 0 and m s pf q " 1 if and only if f defines a smooth branch on pS, sq.

The following definition describes a measure of the way in which two curve singularities intersect: Definition 1.2. Let C, D ãÑ pS, sq be two plane curve singularities defined by f, g P m S,s . Then their intersection number is defined by:

C ¨D :" dim C O S,s pf, gq P N Y t8u,
where pf, gq 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 is also equivalent to the existence of n P N ˚such that one has the following inclusion of ideals: pf, gq Ě m n S,s . Nevertheless, unlike the multiplicity, the intersection number C ¨D is not always equal to the smallest exponent n having this property. For instance, if one takes f :" x 3 and g :" y 2 , then C ¨D " 6 but pf, gq Ě px, yq 5 . We leave the verification of the previous facts as an exercise.

The following proposition, which may be proved using Proposition 1.5 below, relates multiplicities and intersection numbers: Let us explain the notion of transversality used in the previous proposition, as it is more general than the standard notion of transversality, which applies only to smooth submanifolds of a given manifold. If C is a branch on pS, sq and one chooses a local coordinate system px, yq on pS, sq, 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 tangent plane T s S of S at s, which is by definition the tangent line of C at s. One may show that it is independent of the choices of local coordinates and defining function of C. If C is now an arbitrary curve singularity, then its tangent cone is the union of the tangent lines of its branches. Given two plane curve singularities on the same smooth surface singularity S, one says that they are transversal if each line of the tangent cone of one of them is transversal (in the 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 method consists in breaking the symmetry between the two curve singularities, by working with a defining function of one of them and by parametrizing the other one. One has to be cautious and choose a normal parametrization, in the following sense: Definition 1.4. A normal parametrization of the branch pC, sq is a germ of holomorphic morphism ν : pC, 0q Ñ pC, sq which is a normalization morphism, that is, it has topological degree one.

For instance, if the branch pC, 0q on pC 2 , 0q is defined by the function y 2 ´x3 , then t Ñ pt 2 , t 3 q is a normal parametrization of C, but u Ñ pu 4 , u 6 q is not. A normal parametrization of a branch pC, sq may be also characterized by asking it to establish a homeomorphism between suitable representatives of the germs pC, 0q and pC, sq.

Normalization morphisms may be defined more generally for reduced germs pX, xq of arbitrary dimension (see [START_REF] De Jong | Local analytic geometry[END_REF]Sect. 4.4]), by considering the multi-germ whose multi-local ring is the integral 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 intersection numbers 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 pS, sq and D be a second curve singularity, not necessarily reduced. Let ν : pC, 0q Ñ pC, sq be a normal parametrization of C and let g P m S,s be a defining function of D. Then: C ¨D " ord t pg ˝νptqq , where ord t denotes the order of a power series in the variable t.

Proof. This proof is adapted from that of [START_REF] De Jong | Local analytic geometry[END_REF]Lemma 5.1.5].

The order of the zero power series is equal to 8 by definition, therefore the statement is true when 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 pf, gq " dim C O S,s {pf q pg C q " dim C O C,s pg C q ,
where we have denoted by g C P O C,s the restriction of g to the branch C. Algebraically, the normal parametrization ν : pC, 0q Ñ pC, sq corresponds to a morphism of local C-algebras O C,s ãÑ Cttu, isomorphic to the inclusion morphism of O C,s into its integral closure 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 Cttu its analog inside Cttu. One has the following equality inside the local C-algebra Cttu:

g ˝νptq " g C .
As a consequence: g C Cttu " t ordtpg˝νptqq Cttu.

Therefore:

(2) ord t pg ˝νptqq " dim C Cttu g C Cttu .

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 Cttu g C Cttu .
The two quotients appearing in (3) are the cokernels of the two injective multiplication maps O C,s ¨gC ÝÑ O C,s and Cttu ¨gC ÝÑ Cttu. The associated short exact sequences may be completed into a commutative diagram in which the first two vertical maps are the inclusion map O C,s ãÑ Cttu:

0 O C,s O C,s O C,s g C O C,s 0 0 Cttu Cttu Cttu g C Cttu 0
The last vertical map is not necessarily an isomorphism. We want to show that its source and its target have the same dimension. Let us complete it into an exact sequence by considering its kernel K 1 and cokernel K 2 :

0 ÝÑ K 1 ÝÑ O C,s g C O C,s ÝÑ Cttu g C Cttu ÝÑ K 2 ÝÑ 0.
For every finite exact sequence of finite-dimensional vector spaces, the alternating sum of dimensions vanishes. Therefore:

dim C K 1 ´dim C O C,s g C O C,s `dim C Cttu g C Cttu ´dim C K 2 " 0.
This shows that the desired equality (3) would result from the equality dim

C K 1 " dim C K 2 .
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 exact sequence:

0 ÝÑ K 1 ÝÑ Cttu O C,s ÝÑ Cttu O C,s ÝÑ K 2 ÝÑ 0.
Reapplying the previous argument about alternating sums of dimensions, one gets the needed

equality dim C K 1 " dim C K 2 .
Note that the previous proof shows in fact that for any abstract branch pC, sq, not necessarily planar, one has the equality: [START_REF] Buneman | A note on the metric properties of trees[END_REF] dim C O C,s pgq " ord t pg ˝νptqq , for any g P O C,s and for any normal parametrization ν : pC, 0q Ñ pC, sq of pC, sq. If the branch pC, sq is contained in an ambient germ pX, sq and H is an effective principal divisor on pX, sq which does not contain the branch, then equality [START_REF] Buneman | A note on the metric properties of trees[END_REF] shows that the intersection number of C and H at s may be computed as the order of the series obtained by composing a defining function of pH, sq and a normal parametrization of pC, sq.

Example 1.6. Consider the branches:

$ & % A :" Zpy 2 ´x3 q, B :" Zpy 3 ´x5 q, C :" Zpy 6 ´x5 q on the smooth surface singularity pC 2 , 0q. Denoting by m 0 the multiplicity function at the origin of C 2 , we have: m 0 pAq " 2, m 0 pBq " 3, m 0 pCq " 5, as results from Definition 1.1. Using Proposition 1.5 and the fact that whenever m and n are coprime positive integers, t Ñ pt n , t m q is a normal parametrization of Zpy n ´xm q, one gets the following values for the intersection numbers of the branches A, B, C:

B ¨C " 15, C ¨A " 10, A ¨B " 9.

Therefore:

$ ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' % B ¨C m 0 pBq ¨m0 pCq " 1, C ¨A m 0 pCq ¨m0 pAq " 1, A ¨B m 0 pAq ¨m0 pBq " 3 2 .
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 next section.

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 the notion of ultrametric.

Denote simply by S the germ of smooth surface pS, sq and by mpAq the multiplicity of a branch pA, sq ãÑ pS, sq.

In his 1985 paper [START_REF]Remarque sur la multiplicité d'intersection des branches planes[END_REF], 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 ¨B mpAq ¨mpBq ě B ¨C mpBq ¨mpCq " C ¨A mpCq ¨mpAq .
Denote by BpSq the infinite set of branches on S. By inverting the fractions appearing in the 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 BpSq ˆBpSq Ñ r0, 8q defined by

pA, Bq Ñ # mpAq ¨mpBq A ¨B if A ‰ B, 0 otherwise is an ultrametric.
What does it mean that a function is an ultrametric? We explain this in the next section and we show how to think topologically about ultrametrics on finite sets in terms of certain kinds of decorated rooted trees. This way of thinking is used then in Section 4 in order to prove the reformulation 2.2 of P loski's theorem.

Ultrametrics and rooted trees

In this section we define the notion of ultrametric and we explain how to think about an ultrametric 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 and that finite hierarchies are equivalent to special types of decorated rooted trees. For more details, one may consult [9, Sect. In this case, one says also that d is an ultrametric on the set M .

In any metric space pM, dq, a closed ball is a subset of M of the form:

BpA, rq :" tP P M, dpP, Aq ď ru where the center A P M and the radius r P r0, 8q are given. As we will see shortly, given a closed ball, neither its center nor its radius are in general well-defined, contrary to an intuition educated only by Euclidean geometry.

One has the following characterizations of ultrametrics:

Proposition 3.2. Let pM, dq be a metric space. Then the following properties are equivalent:

(1) pM, dq 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 as exercises.

Example 3.3. Consider a set M " tA, B, C, Du and a distance function d on it such that: dpB, Cq " 1, dpA, Bq " dpA, Cq " 2, dpA, Dq " dpB, Dq " dpC, Dq " 5. Note that one may embed pM, dq 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 to the plane of the triangle passing through its circumcenter. Let us look for the closed balls of this finite metric space. For radii less than 1, they are singletons. For radii in the interval r1, 2q, we get the sets tB, Cu, tAu, tDu. Note that both B and C are centers of the ball tB, Cu, that is, BpB, rq " BpC, rq " tB, Cu for every r P r1, 2q. Once the radius belongs to the interval r2, 5q, the balls are tA, B, Cu and tDu. Finally, for every radius r P r5, 8q, there is only one closed ball, the whole set. Example 3.3 illustrates the fact that neither the center nor the radius of a closed ball of a finite ultrametric space is well-defined, once the ball has more than one element. Instead, every closed ball has a well-defined diameter : Definition 3.4. The diameter of a closed ball in a finite metric space is the maximal distance between pairs of not necessarily distinct points of it.

The last characterization of ultrametrics in Proposition 3.2 shows that the set BallspM, dq of closed balls of an ultrametric space pM, dq 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 PpM q, satisfying the following 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 will consider 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 corresponding rooted tree with the maximal elements of the poset pH, ĺ H q (see Proposition 3.8 below).

When M is finite, one may represent the poset pH, ĺ H q using its associated Hasse diagram:

Definition 3.6. Let pX, ĺq be a finite poset. Its Hasse diagram is the directed graph whose set 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 they have no directed cycles, which is a consequence of the fact that a partial order is antisymmetric and transitive. Hasse diagrams are not necessarily planar, but, as all finite graphs, they may be 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 to place the vertex a of the Hasse diagram below the vertex b whenever a ă b. This is always possible because of the absence of directed cycles. This convention makes unnecessary adding arrowheads along the edges in order to indicate their orientations. The Hasse diagrams of finite hierarchies are special kinds of graphs: Proposition 3.8. The Hasse diagram of a hierarchy pH, ĺ H q on a finite set M is a tree in which the maximal directed paths start from M and terminate at the singletons. Moreover, for each 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 maximal elements of the poset pH, ĺ H q, that M itself is the unique minimal element and that all the elements of a hierarchy which contain a given element are totally ordered by inclusion.

Let us prove the second statement. Consider B 1 P H and assume that it is not a singleton. This means that it is not minimal for inclusion, therefore there exists B 2 P H such that B 2 Ĺ B 1 and B 2 is directly comparable to B 1 . Let A be a point of B 1 zB 2 . Consider B 3 P H which contains the point A, is included into B 1 and is directly comparable to it. As A P B 3 z B 2 , this shows that B 3 is not included in B 2 . We want to show that the two sets B 2 and B 3 are disjoint. Otherwise, by the definition of a hierarchy, we would have B 2 Ĺ B 3 Ĺ B 1 , which contradicts the assumption that B 1 and B 2 are directly comparable.

Example 3.9. Consider the ultrametric space of Example 3.7, represented in Figure 1. We repeat it on the left of Figure 3. The Hasse diagram of the hierarchy of its closed balls is drawn 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 be thought as a larger ball, obtained by adding formally to M " tA, B, C, Du a point ω, infinitely distant from each point of M . This larger ball is the set M :" M Y tωu. 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 Θ is 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 rr, as joining r to a in the tree is contained in rr, bs.

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 with the 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 of the 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 1 and there are no vertices of valency 2.

This proposition motivates the following definition: Definition 3.13. A rooted tree whose root is of valency 1 and which does not possess vertices of valency 2 is a hierarchical tree. The hierarchy of a hierarchical tree pΘ, rq is constructed in the following way:

' Define M to be the set of leaves of the rooted tree pΘ, rq, that is, the set of vertices of valency 1 which are distinct from the root r.

' For each vertex p of Θ different from the root, consider the subset of M consisting of the 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 hierarchical tree to a hierarchy on a finite set and a hierarchy to a hierarchical tree are inverse of each other.

As a preliminary to the proof, one may test the truth of the proposition on the example of Figure 3.

Let us return to finite ultrametric spaces pM, dq. We saw that the set BallspM, dq of its closed balls is a hierarchy on M . Proposition 3.14 shows that one may think about this hierarchy as a special kind of rooted tree, namely, a hierarchical tree. This hierarchical tree alone does not allow to get back the distance function d. How to encode it on the tree?

The idea is to look at the function defined on BallspM, dq, which associates to each ball its diameter (see Definition 3.4): Proposition 3.15. Let pM, dq be a finite ultrametric space. Then the map which sends each closed ball to its diameter is a strictly decreasing r0, 8q-valued function defined on the poset pBallspM, dq, ĺq, taking the value 0 exactly on the singletons of M . Equivalently, it is a strictly decreasing r0, 8s-valued function on the set of vertices of the tree of the hierarchy, vanishing on the set M of leaves and taking the value 8 on the root.

As an example, one may look again at Figure 3. The value taken by the previous diameter function is written near each vertex of the hierarchical tree.

If pΘ, rq is a hierarchical tree, denote by V pΘq its set of vertices and by a ^r b the infimum of a and b relative to ĺ r , whenever a, b P V pΘq. This infimum may be characterized by the property that rr, as X rr, bs " rr, a ^r bs. The following is a converse of Proposition 3.15: Proposition 3.16. Let pΘ, rq be a hierarchical tree and λ : V pΘq Ñ r0, 8s be a strictly decreasing function relative to the partial order ĺ r on Θ induced by the root. Assume that λ vanishes on the set M of leaves of Θ and takes the value 8 at r. Then the map

d : M ˆM Ñ r0, 8q pa, bq Ñ λpa ^r bq is an ultrametric on M .
Let us introduce a special name for the functions appearing in Proposition 3.16:

Definition 3.17. Let pΘ, rq be a hierarchical tree. A depth function on it is a function λ : V pΘq Ñ r0, 8s 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 8 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 a hierarchical tree whose set of leaves is M .

It is this proposition which allows to think about an ultrametric as a special kind of rooted and decorated tree. We leave its proof as an exercise (see [3]).

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 ultrametrics on finite sets and certain kinds of rooted trees formulated in Proposition 3. [START_REF] Jung | Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x,y in der Umgebung einer Stelle x=a, y=b[END_REF]. The rooted trees used in this proof are the Eggers-Wall trees of a plane curve singularity relative to smooth reference branches. The precise definition of Eggers-Wall trees is not given, because the proofs of the subsequent generalizations of P loski's theorem will be of a completely different spirit.

Instead of working both with multiplicities and intersection numbers as in P loski's original statement, we will work only with the latest ones.

Let S be a smooth germ of surface and L ãÑ S be a smooth branch. Define the following function on the set of branches on S which are different from L:

(5)

u L : pBpSqztLuq 2 Ñ R pA, Bq Ñ # pL ¨Aq ¨pL ¨Bq A ¨B if A ‰ B, 0 otherwise.
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 reformulation 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 a plane curve singularity relative to a smooth branch of reference L (see the proof of Theorem 4.5 below). Let us illustrate it by an example. Example 4.2. Consider again the branches A " Zpy 2 ´x3 q, B " Zpy 3 ´x5 q, C " Zpy 6 ´x5 q on S " pC 2 , 0q of Example 1.6. Assume that the branch L is the germ at 0 of the y-axis Zpxq. The defining 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 3{2 , B : x 5{3 , C : x 5{6 .
Associate to the root x 3{2 a compact segment Θ L pAq identified with the interval r0, 8s using an exponent function e L : Θ L pAq Ñ r0, 8s and mark on it the point e ´1 L p3{2q with exponent 3{2. Define also an index function i L : Θ L pAq Ñ N ˚, constantly equal to 1 on the interval re ´1 L p0q, e ´1 L p3{2qs and to 2 on the interval pe ´1 L p3{2q, e ´1 L p8qs (see the left-most segment of Figure 4). Here the number 2 is to be thought as the minimal positive denominator of the exponent 3{2 of the monomial x 3{2 . The segment Θ L pAq 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 rooted tree with root e ´1 L p0q, labeled with the branch L. Its leaf e ´1 L p8q is labeled with the branch A. Consider analogously the Eggers-Wall trees Θ L pBq and Θ L pCq, endowed with pairs of exponent and 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 pA `B `Cq relative to the branch L is obtained from the individual trees Θ L pAq, Θ L pBq, Θ L pCq by a gluing process, which identifies two by two initial segments of those trees.

Consider for instance the segments Θ L pAq, Θ L pBq. Look at the order of the difference x 3{2 x5{3 of the roots which generated them, seen as a series with fractional exponents. This order is the fraction 3{2, because 3{2 ă 5{3. Identify then the points with the same exponent ď 3{2 of the segments Θ L pAq, Θ L pBq. One gets a rooted tree Θ L pA `Bq with root labeled by L and with two leaves, labeled by the branches A, B. The exponent and index functions of the trees Θ L pAq, Θ L pBq descend to functions with the same name e L , i L defined on Θ L pA `Bq. Endowed with those functions, Θ L pA `Bq 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 pairs of trees obtained from Θ L pAq, Θ L pBq, Θ L pCq. The resulting Eggers-Wall tree Θ L pA `B `Cq is drawn on the right side of Figure 4. It is also endowed with two functions e L , i L , obtained by gluing the exponent and index functions of the trees Θ L pAq, Θ L pBq, Θ L pCq. Its marked points are its ends, its bifurcation points and the images of the discontinuity points of the index function of the Eggers-Wall tree of each branch. Near each marked point is written the corresponding value of the exponent function. The index function is constant on each segment pa, bs joining two marked points a and b, where a ă L b. Here ĺ L denotes the partial order on the tree Θ L pA `B `Cq determined by the root L (see Definition 3.11).

One may associate analogously an Eggers-Wall tree Θ L pDq to any plane curve singularity D, relative to a smooth reference branch L. It is a rooted tree endowed with an exponent function e L : Θ L pDq Ñ r0, 8s and an index function i L : Θ L pDq Ñ N ˚. The tree and both functions are constructed using Newton-Puiseux series expansions of the roots of a Weierstrass polynomial f P Crrxssrys defining D in a coordinate system px, yq such that L " Zpxq. The triple pΘ L pDq, 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 in Section 4.3 of the previous reference and in [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF]Sect. 3]. Historical remarks about this notion may be found in [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF]Rem. 3.18] and [START_REF] García Barroso | The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses[END_REF]Sect. 6.2]. The name, introduced in author's thesis [START_REF] Popescu-Pampu | Arbres de contact des singularités quasi-ordinaires et graphes d'adjacence pour les 3variétés réelles[END_REF], makes reference to Eggers' 1982 paper [START_REF] Eggers | Polarinvarianten und die Topologie von Kurvensingularitaeten[END_REF] and to Wall's 2003 paper [START_REF] Wall | Chains on the Eggers tree and polar curves[END_REF].

What allows us to prove Theorem 4.1 using Eggers-Wall trees is that the values u L pA, Bq of the function u L defined by relation (5) are determined in the following way from the Eggers-Wall tree Θ L pDq, for each pair of distinct branches pA, Bq of D (recall from the paragraph preceding 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 pDq): Theorem 4.3. For each pair pA, Bq of distinct branches of D and every smooth reference branch L different from the branches of D, one has:

1 u L pA, Bq " ż A^LB L de L i L .
ż A^LB L de L i L " ż 3{2 0 de 1 " 3 2 .
But 1{u L pA, Bq " pA ¨Bq{ ppL ¨AqpL ¨Bqq " pA ¨Bq{ pmpAq ¨mpBqq " 3{2, as was computed in Example 1.6. The equality is verified. We have used the fact that both A and B are transversal to L, which implies that L ¨A " mpAq and L ¨B " mpBq.

In equivalent formulations which use so-called characteristic exponents, Theorem 4.3 goes back to Smith [23, Section 8], Stolz [24, Section 9] and Max Noether [START_REF] Noether | Les combinaisons caractéristiques dans la transformation d'un point singulier[END_REF]. A modern proof, based on Proposition 1.5, may be found in [START_REF] Wall | Singular points of plane curves[END_REF]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 FpDq the set of branches of D.

Let L be a reference smooth branch which does not belong to FpDq. Then the function u L is an ultrametric in restriction to FpDq and its associated rooted tree is isomorphic as a rooted tree with labeled leaves to the Eggers-Wall tree Θ L pDq.

Proof. Consider Θ L pDq as a topological tree with vertex set equal to its set of ends and of ramification points. Root it at L. Then it becomes a hierarchical tree in the sense of Definition 3.13. The function

P Ñ ˆż P L de L i L ˙´1
is a depth function on it, in the sense of Definition 3.17. Using Theorem 4.3 and Proposition 3.18, we get Theorem 4.5.

For more details about the proof of P loski's theorem presented in this section, see [9, Sect. 4.3].

An ultrametric characterization of arborescent singularities

In this section we state a generalization of Theorem 4.1 for all arborescent singularities and the fact that it characterizes this class of normal surface singularities. We start by recalling the needed notions of embedded resolution and associated dual graph of a finite set of branches contained in a normal surface singularity.

From now on, S denotes an arbitrary normal surface singularity, that is, a germ of normal complex analytic surface. Let us recall the notion of resolution of such a singularity: Definition 5.1. Let pS, sq be a normal surface singularity. A resolution of it is a proper bimeromorphic morphism π : S π Ñ S such that S π is smooth. Its exceptional divisor E π is the reduced preimage π ´1psq. The resolution is good if its exceptional divisor has normal crossings and all its irreducible components are smooth. The dual graph Γpπq of the resolution π is the finite graph whose set of vertices Ppπq is the set of irreducible components of E π , two vertices 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, for which partial proofs appeared already at the end of the XIXth century, was proved first in the analytical context by Hirzebruch in his 1953 paper [START_REF] Hirzebruch | Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen[END_REF]. His proof was inspired by previous works of Jung [START_REF] Jung | Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x,y in der Umgebung einer Stelle x=a, y=b[END_REF] and Walker [START_REF] Walker | Reduction of the Singularities of an Algebraic Surface[END_REF], 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 divisor on S, by thinking about their sum. The notion of embedded resolution of F is an analog of that of good resolution of S: Definition 5.2. Let pS, sq be a normal surface singularity and let π : S π Ñ S be a resolution of S. If A is a branch on S, then its strict transform by π is the closure inside S π of the preimage π ´1pAzsq. Let F be a finite set of branches on S. Its strict transform by π is the set or, depending on the context, the divisor formed by the strict transforms of the branches of F. The preimage π ´1F of F by π is the sum of its strict transform and of the exceptional divisor 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 Γpπ ´1F q of the preimage π ´1F is defined similarly to the dual graph Γpπq of π, taking into account all the irreducible components of π ´1F .

In the previous definition, the preimage π ´1F of F by π is seen as a reduced divisor. We will see in Definition 8.4 below that there is also a canonical way, due to Mumford, to define canonically a not necessarily reduced rational divisor supported by π ´1F , 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 arborescent singularities, 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 dual graphs of its good resolutions are trees.

Remark that in the previous definition we ask nothing about the genera of the irreducible components of the exceptional divisors.

By using the fact that any two resolutions are related by a sequence of blow ups and blow downs of their total spaces (see [14, Thm. V.5.5]), one sees that the dual graphs of all good resolutions 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 the function 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 will explain this in Section 8 below (see Definition 8.5).

The following generalization of Theorem 4.1 both gives a characterization of arborescent singularities and extends Theorem 4.5 to all arborescent singularities S and all -not necessarily smooth -reference branches L on them (recall that BpSq denotes the set of branches on S): Theorem 5.4. Let S be a normal surface singularity and L P BpSq. Then:

(1) u L is ultrametric on BpSqztLu 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 of the restriction of u L to F is isomorphic to the convex hull of F Y tLu in the dual graph of the preimage of F Y tLu by any embedded resolution of F Y tLu, rooted at L.

We do not prove in the present notes that if u L is an ultrametric on BpSqztLu, then S is arborescent. The interested reader may find a proof of this fact in [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Sect. 1.6]. The remaining implication of point (1) and point (2) of Theorem 5.4 are, taken together, a consequence of Theorem 7.1 below. For this reason, we do not give a separate proof of them, the rest of this paper 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 next section 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 pDq of a plane curve singularity D ãÑ S not containing L is isomorphic to the convex hull of the strict transform of FpDq Y tLu 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 [START_REF] Popescu-Pampu | Arbres de contact des singularités quasi-ordinaires et graphes d'adjacence pour les 3variétés réelles[END_REF]Thm. 4.4.1], then generalized 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].

The brick-vertex tree of a connected graph

In this section we introduce the notion of brick-vertex tree of a connected graph, which is crucial in order to state Theorem 7.1 below, the strongest known generalization of P loski's theorem. Definition 6.1. A graph is a compact cell complex of dimension ď 1. If Γ is a graph, its set of vertices is denoted V pΓq and its set of edges is denoted EpΓq .

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 removal disconnects 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 if either b P ta, cu or if a and c belong to different connected components of the topological space Γ z tbu.

Note that an end of a bridge is a cut-vertex if and only if it has valency at least 2 in Γ, that is, if and only if it is not a leaf of Γ. It will be important to distinguish the class of graphs which cannot be disconnected by the removal of one vertex, as well as the maximal graphs of this class contained 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 block which 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 in red. Its bridges are represented as black segments. It has three bricks, the edges of each brick being colored in the same way.

By replacing each brick of a connected graph by a star-shaped graph, one gets canonically a tree associated to the given graph: Definition 6.5. The brick-vertex tree BVpΓq of a connected graph Γ is the tree whose set of vertices 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 bricks bijectively to the bricks of Γ, its edges being either the bridges of Γ or new edges connecting each brick-vertex to the vertices of the corresponding brick. Formally, this may be written as follows:

' V pBVpΓqq " V pΓq \ tbricks of Γu. ' EpBVpΓqq " tbridges of Γu \ trv, bs, v P V pΓq, b is a brick of Γ, v P V pbqu.
We denoted by v the vertex v of Γ when it is seen as a vertex of BVpΓq and b P V pBVpΓqq 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 to other notions introduced before either in general topology or in graph theory, as explained in [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Rems. 1.35,2.50].

Note that whenever Γ is a tree, BVpΓq 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 cutvertices and bricks emphasized. On its right side is represented its associated brick-vertex tree BVpΓq. Each representative vertex of a brick is drawn with the same color as its corresponding brick. The edges of BVpΓq which are not bridges of Γ are represented in magenta and thicker than the other edges.

Γ BVpΓq Figure 6. The connected graph of Example 6.6 and its brick-vertex tree

The importance of the brick-vertex tree in our context stems from the following property of it (see [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF]Prop. 1.36]), formulated using the vocabulary introduced in Definition 6.2 and the notations introduced in Definition 6.5: Proposition 6.7. Let Γ be a finite graph and a, b, c P V pΓq. Then b separates a from c in Γ if and only if b separates a from c in BVpΓq.

We are ready now to state the strongest known generalization of P loski's theorem (see Theorem 7.1 below).

Our strongest generalization of P loski's theorem

In this section we formulate Theorem 7.1, which generalizes Theorem 4.5 to all normal surface singularities and all branches on them, using the notion of brick-vertex tree introduced in the previous section.

Recall that the notion of brick-vertex tree of a connected graph was introduced in Definition 6.5. A fundamental property of normal surface singularities is that the dual graphs of their resolutions 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 the preimage (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 finite set 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 on it and an embedded resolution π : S π Ñ S of F. Let Γ be the dual graph of the preimage π ´1F of F by π. Assume that the convex hull Conv BVpΓq pFq of the strict transform of F by π in the brick-vertex tree BVpΓq does not contain brick-vertices of valency at least 4 in Conv BVpΓq pFq. Then for all L P F, the restriction of u L to F z tLu is an ultrametric and the corresponding rooted tree is isomorphic to Conv BVpΓq pFq, rooted at L. Example 7.2. Assume that the dual graph Γ of π ´1F is as shown on the left side of Figure 7. The vertices representing the strict transforms of the branches of the set F are drawn arrowheaded. Note that the subgraph which is the dual graph of the exceptional divisor is the same as the graph of Figure 5. On the right side of Figure 7 is represented using thick red segments the convex hull Conv BVpΓq pFq. We see that the hypothesis of Theorem 7.1 is satisfied. Indeed, the convex hull contains only two brick-vertices, which are of valency 2 and 3 in Conv BVpΓq pFq. Note that the blue one is of valency 4 in the dual graph Γ, which shows the importance of looking at the valency in the convex hull Conv BVpΓq pFq, not in Γ. As shown in [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Ex. 1.44], the condition about valency is not necessary in general for u L to be an ultrametric on F z tLu.

Note that we have expressed Theorem 7.1 in a slightly different form than the equivalent Theorem D of the introduction. Namely, we included L in the branches of F. This formulation emphasizes 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 consequence of 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 of two curve singularities drawn on an arbitrary normal surface singularity, which allows to define in turn the functions u L appearing in the statement of Theorem 7.1.

Mumford's intersection theory

In this section we explain Mumford's definition of intersection number of Weil divisors on a normal surface singularity, introduced in his 1961 paper [START_REF] Mumford | The topology of normal singularities of an algebraic surface and a criterion for simplicity[END_REF]. It is based on Theorem 8.1, stating that 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 detailed proof of it.

Let π : S π Ñ S be a resolution of the normal surface singularity S. Denote by pE u q uPPpπq the collection of irreducible components of the exceptional divisor E π of π (see Definition 5.1).

Denote by: Epπq R :" à uPPpπq RE u the real vector space freely generated by those prime divisors, that is, the space of real divisors supported by E π . It is endowed with a symmetric bilinear form pD 1 , D 2 q Ñ D 1 ¨D2 given by intersecting the corresponding compact cycles on S π . We call it the intersection form. Its following fundamental property was proved by Du Val [25] and Mumford [START_REF] Mumford | The topology of normal singularities of an algebraic surface and a criterion for simplicity[END_REF]:

Theorem 8.1. The intersection form on Epπq R is negative definite.

Proof. The following proof is an expansion of that given by Mumford in [START_REF] Mumford | The topology of normal singularities of an algebraic surface and a criterion for simplicity[END_REF].

The singularity S being normal, the exceptional divisor E π is connected (this is a particular case of Zariski's main theorem, see [14, Thm. V.5.2]). Therefore: [START_REF] Eggers | Polarinvarianten und die Topologie von Kurvensingularitaeten[END_REF] The dual graph Γpπq is connected.

Consider any germ of holomorphic function f on pS, sq, vanishing at s, and look at the divisor of its lift to the surface S π : (7) pπ ˚f q " ÿ uPPpπq a u E u `pπ ˚f q str .

Here pπ ˚f q str denotes the strict transform of the divisor defined by f on S. Denote also:

$ & % e u :" a u E u P Epπq R , for all u P Ppπq, σ :" ÿ uPPpπq e u P Epπq 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 0 for every u P Ppuq. We deduce that pe u q uPPpπq is a basis of Epπq R and that:

(9) e u ¨ev ě 0, for all u, v P Ppπq such that u ‰ v.

The divisor pπ ˚f q being principal, its associated line bundle is trivial. Therefore: [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF] pπ ˚f q ¨Eu " 0 for every u P Ppπq, because this intersection number is equal by definition to the degree of the restriction of this line bundle to the curve E u . By combining the relations ( 7), ( 8) and [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF], we deduce that: [START_REF] García Barroso | The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses[END_REF] σ ¨eu " ´au pπ ˚f q str ¨Eu , for every u P Ppπq.

As the germ of effective divisor pπ ˚f q str along E π has no components of E π in its support, the intersection numbers pπ ˚f q str ¨Eu are all non-negative. Moreover, at least one of them is positive, because the divisor pπ ˚f q str is non-zero. By combining this fact with relations [START_REF] García Barroso | The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses[END_REF] and with the inequalities a u ą 0, we get:

" σ ¨eu ď 0, for every u P Ppπq, there exists u 0 P Ppπq such that σ ¨eu 0 ă 0.

As a consequence of Theorem 8.1, one may define the dual basis p E _ u q uPPpπq of the basis pE u q uPPpπq by the following relations, in which δ uv denotes Kronecker's delta-symbol: [START_REF] Jonsson | Dynamics on Berkovich spaces in low dimensions[END_REF] E _ u ¨Ev " δ uv , for all pu, vq P Ppπq The bracket xu, vy 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 welldefined. That is, if the divisorial valuations u, v are fixed, then their bracket does not depend on the resolution in which they appear. This fact may be also proved using the property that any two resolutions of S are related by a sequence of blow ups and blow downs (see [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Prop. 1.5]).

It is a consequence of Theorem 8.1 that the brackets are all non-negative (see [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Prop. 1.4]). Moreover, by the Cauchy-Schwarz inequality applied to the opposite of the intersection form: 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 pf q of a meromorphic germ on S, then one may lift it to a resolution S π as the principal divisor pπ ˚f q. This divisor decomposes as the sum of an exceptional part pπ ˚Dq ex supported by E π and the strict transform of D. The crucial property of the lift pπ ˚f q, already used in the proof of Theorem 8.1 (see relation [START_REF] García Barroso | The valuative tree is the projective limit of Eggers-Wall trees[END_REF]), is that its intersection numbers with all the components E u of E π vanish. In [19, Sect. II (b)], Mumford imposed this property in order to define 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 pπ ˚Dq ex `pπ ˚Dq str such that:

(1) pπ ˚Dq ex P Epπq Q .

(2) pπ ˚Dq str is the strict transform of D by π.

(3) pπ ˚Dq ¨Eu " 0 for all u P Ppπq. The divisor pπ ˚Dq 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 pπ ˚Dq ex exists and is unique with the property (3). Write it as a sum ř vPPpπq x v E v . The last condition of Definition 8.4 may be written as the system: ÿ vPPpπq pE v ¨Eu qx v " ´pπ ˚Dq str ¨Eu , for all u P Ppπq.

This is a square linear system in the unknowns x v , whose matrix is the matrix of the intersection form in the basis pE u q uPPpπq . As the intersection form is negative definite, it is non-degenerate, therefore this system has a unique solution. Moreover, all its coefficients being integers, its solution has rational coordinates, which shows that pπ ˚Dq ex P Epπq Q . Using Definition 8.4 and the standard definition of intersection numbers on smooth surfaces recalled in Section 1, Mumford defined in the following way in [START_REF] Mumford | The topology of normal singularities of an algebraic surface and a criterion for simplicity[END_REF]Sect. II (b)] the intersection number of two Weil divisors on S: Definition 8.5. Let A, B be two Weil divisors on S without common components, and π be a resolution 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 blow downs (see [14, Thm. V.5.5]), it may be shown that the previous notion is independent of the choice of resolution, similarly to that of bracket of two divisorial valuations introduced in Definition 8.2. In particular, if S is smooth, one may choose π to be the identity. This shows that in this case Mumford's definition gives the same intersection number as the standard Definition 1.2.

Example 8.6. Let S be the germ at the origin 0 of the quadratic cone Zpx 2 `y2 `z2 q ãÑ C 3 (it is the so-called A 1 surface singularity). Let A and B be the germs at 0 of two distinct generating lines of the cone. One may resolve S by blowing up 0. This morphism π : S π Ñ S separates all the generators, therefore it is an embedded resolution of tA, Bu. The exceptional divisor of π is the projectivisation of the cone, that is, it is a smooth rational curve E. Its self-intersection number is the opposite of the degree of the curve seen embedded in the projectivisation of the ambient space C 3 . Therefore, E 2 " ´2. Let us compute the total transform π ˚A " pπ ˚Aq str `xE. The imposed constraint π ˚A ¨E " 0 becomes 1 ´2x " 0, therefore x " 1{2. We have used the fact that the strict transform pπ ˚Aq str of A by π is smooth and transversal to E, which implies that pπ ˚Aq str ¨E " 1.

We obtained π ˚A " pπ ˚Aq str `p1{2qE and similarly, π ˚B " pπ ˚Bq str `p1{2qE. Using Definition 8.5, we get:

A ¨B " π ˚A ¨π˚B " " ppπ ˚Aq str `p1{2qEq ¨ppπ ˚Bq str `p1{2qEq " " pπ ˚Aq str ¨pπ ˚Bq str `p1{2qppπ ˚Aq str `pπ ˚Bq str q ¨E `p1{2q 2 E 2 " " 0 `p1{2q ¨2 `p1{2q 2 ¨p´2q " " 1{2.

This example shows in particular that the intersection number of two curve singularities depends 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 obtained as the intersection of the cone with a plane passing through its vertex). In such a surface, their intersection number is 1 instead of 1{2. Proof. This proof uses directly Definition 8.4.

As π is an embedded resolution of A `B, the strict transforms pπ ˚Aq str and pπ ˚Bq str are disjoint. Therefore pπ ˚Aq str ¨pπ ˚Bq str " 0. Using the last condition in the Definition 8.4 of the total transform of a divisor, we know that pπ ˚Aq ¨pπ ˚Bq ex " pπ ˚Aq ex ¨pπ ˚Bq " 0. Combining both equalities, we deduce that:

A ¨B " pπ ˚Aq ¨pπ ˚Bq " " pπ ˚Aq ¨ppπ ˚Bq ex `pπ ˚Bq str q " " pπ ˚Aq ¨pπ ˚Bq str " " ppπ ˚Aq ex `pπ ˚Aq str q ¨pπ ˚Bq str " " pπ ˚Aq ex ¨pπ ˚Bq str " " pπ ˚Aq ex ¨pπ ˚B ´pπ ˚Bq ex q " " ´pπ ˚Aq ex ¨pπ ˚Bq ex " " ´p´E _ a q ¨p´E _ b q " " xa, by.

At the end of the computation we have used the equality pπ ˚Aq ex " ´E_ a , which results from the fact that π is an embedded resolution of A. Indeed, this implies that ppπ ˚Aq str `E_ a q¨E u " 0 for every u P Ppπq, which shows that one has indeed the stated formula for pπ ˚Aq ex .

A reformulation of the ultrametric inequality

In this section we explain the notion of angular distance on the set of vertices of the dual graph of a good resolution of S. Theorem 9.2 states a crucial property of this distance, relating it to the cut-vertices of the dual graph. Then the ultrametric inequality is reexpressed in terms of the angular distance. This allows to show that Theorem 7.1 is a consequence of Theorem 9.10, 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 Ppπq denotes the set of irreducible components of its exceptional divisor E π . Using the notion of bracket from Definition 8.2, one may define (see [START_REF] Gignac | Local dynamics of non-invertible maps near normal surface singularities[END_REF]Sect. 2.7] and [12, Sect. 1.2]):

Definition 9.1. The angular distance is the function ρ : Ppπq ˆPpπq Ñ r0, 8q given by: ρpa, bq :"

$ & % ´log xa, by 2 xa, ayxb, by if a ‰ b, 0 if a " b.
The fact that the function ρ takes values in the interval r0, 8q is a consequence of Lemma 8.3. 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 of two 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 Ppπq. Moreover, for every a, b, c P Ppπq, the following properties are equivalent:

' one has the equality ρpa, bq `ρpb, cq " ρpa, cq;

The basic idea of the proof of Proposition 9.6 is that an X-tree is characterized by the shapes of the convex hulls of the quadruples of points of X, and that those shapes are determined by the cases of equality in the 12 triangle inequalities and the 3 four points conditions associated to each quadruple. In Figure 8 are represented the five possible shapes. For instance, the H-shape is the generic one, characterized by the fact that one has no equality in the previous inequalities. Let us come back to our normal surface singularity S. One has the following property (see [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]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 tLu for one 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 tLu for one branch L in F, then one has the symmetric relation [START_REF] Barth | Compact complex surfaces[END_REF] for every quadruple of branches of F containing L. The subtle point of 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 ultrametrics obtained by varying L among the branches of F. By looking at quadruples of branches from 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 divisor intersected 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 tLu for some branch L P F if and only if the angular 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 tLu is isomorphic to the tree hull of pF π , ρq by an isomorphism which sends each end marked 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 [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Cor. 1.40]): Theorem 9.10. Let S be a normal surface singularity. Consider a set G of vertices of the dual graph Γ of a good resolution π : S π Ñ S of S. Assume that the convex hull Conv BVpΓq pGq of G in the brick-vertex tree of Γ does not contain brick-vertices of valency at least 4 in Conv BVpΓq pGq. Then the restriction of the angular distance ρ to G satisfies the four points condition and the associated tree hull is isomorphic as a G-tree to Conv BVpΓq pGq.

In turn, Theorem 9.10 is a consequence of a graph-theoretic result presented in the next section (see Theorem 10.1).

A theorem of graph theory

In this final section we state a pure graph-theoretical theorem, which implies Theorem 9.10 of the previous section. As we explained before, that theorem implies in turn our strongest generalization 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 pΓq of vertices of Γ, such that for every a, b, c P V pΓq, the following properties are equivalent:

' one has the equality δpa, bq `δpb, cq " δpa, cq; ' b separates a and c in Γ.

Let X be a set of vertices of Γ such that the convex hull Conv BVpΓq pXq of X in the brickvertex tree of Γ does not contain brick-vertices of valency at least 4 in Conv BVpΓq pXq. Then δ satisfies the 4 points condition in restriction to X and the tree hull of pX, δq is isomorphic to Conv BVpΓq pXq as an X-tree.

The idea of the proof of Theorem 10.1 is to show that, under the given hypotheses, the equalities among the triangle inequalities and four points conditions are as described by the brick-vertex tree. It is writtend in a detailed way in [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Thm. 1.38]. As shown by the F -shape of Conv BVpΓq pXq, one should have the following equalities and inequalities in the four points conditions concerning X: [START_REF] Jung | Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x,y in der Umgebung einer Stelle x=a, y=b[END_REF] δpa, dq `δpb, cq " δpa, cq `δpb, dq ą δpa, bq `δpc, dq.

Let us prove that this is indeed the case. Consider the cut vertex v of Γ shown on the left side of Figure 9. As it separates a from d, we have the equality δpa, dq " δpa, vq `δpv, dq. As v does not separate a from b, we have the strict inequality δpa, vq `δpb, vq ą δpa, bq. Using similar equalities and inequalities, we get: δpa, dq `δpb, cq " " pδpa, vq `δpv, dqq `pδpb, vq `δpv, cqq " " pδpa, vq `δpv, cqq `pδpb, vq `δpv, dqq " " δpa, cq `δpb, dq " " pδpa, vq `δpb, vqq `pδpv, dq `δpv, cqq " ą δpa, bq `δpc, dq.

The (in)equalities [START_REF] Jung | Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x,y in der Umgebung einer Stelle x=a, y=b[END_REF] are proved.

One proves similarly the triangle equalities δpa, bq `δpb, cq " δpa, cq, δpa, bq `δpb, dq " δpa, dq and the fact that one has no equality among the triangle inequalities concerning the triple ta, c, du, which shows that the tree hull of pX, δq has indeed an F -shape, with the vertices a, b, c, d placed as in Conv BVpΓq pXq.

  3.1]. Definition 3.1. Let pM, dq be a metric space. It is called ultrametric if one has the following strong form of the triangle inequality: dpA, Bq ď maxtdpA, Cq, dpB, Cqu, for all A, B, C P M.

Figure 1 Figure 1 .

 11 Figure 1. The balls of an ultrametric space with four points

Example 3. 7 .

 7 Consider the finite set t1, 2, 3, 4, 6, 12u of positive divisors of 12, partially ordered by divisibility: a ĺ b if and only if a divides b. Its Hasse diagram is drawn in Figure 2.

Figure 2 .

 2 Figure 2. The Hasse diagram of the set of positive divisors of 12.

8 Figure 3 .

 83 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 pH, ĺ H q on a finite poset M is its Hasse diagram, completed with a root representing the set M :" M Y tωu, joined with the vertex representing M and rooted at M . Here ω is a point distinct from the points of M .

Figure 4 .

 4 Figure 4. The Eggers-Wall tree of the plane curve singularity of Example 4.2

Example 4 . 4 .

 44 Let us verify the equality stated in Theorem 4.3 on the branches of Example 4.2. Looking at the Eggers-Wall tree Θ L pA `B `Cq on the right side of Figure 4, we see that:

Figure 7 .

 7 Figure 7. An example where the hypothesis of Theorem 7.1 is satisfied

2 .

 2 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 Ppπq in the set of real-valued valuations of O S,s . This allows to see the index u of E u as a valuation. Such valuations are called divisorial. If u denotes a divisorial valuation, it has a center on any resolution, which is either a point or an irreducible component of the exceptional divisor. In the second case, one says that the valuation appears in the resolution. 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 which both u and v appear. Then their bracket is defined by: xu, vy :" ´E_ u ¨E_ v .

Lemma 8 . 3 .

 83 For every a, b P Ppπq: xa, by 2 ď xa, ayxb, by, with equality if and only if a " b.

Definition 8 .

 8 5 allows to give the following interpretation of the notion of bracket introduced in Definition 8.2 (see[START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF] Prop. 1.11]): Proposition 8.7. Let A, B be two distinct branches on S. Consider an embedded resolution π of their sum. Denote by E a , E b the components of the exceptional divisor E π which are intersected by the strict transforms pπ ˚Aq str and pπ ˚Bq str respectively. Then:A ¨B " xa, by.

Figure 8 .

 8 Figure 8. The possible shapes of an X-tree, when X has four elements.

Figure 9 .

 9 Figure 9. A convex hull of four vertices

  Proposition 1.3. If pC, sq ãÑ pS, sq is a plane curve singularity, then C ¨L ě m s pCq for any smooth 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 pS, sq, then C ¨D ě m s pCq ¨ms pDq, with equality if and only if C and D are transversal.

Acknowledgements. This work was partially supported by the French grants ANR-17-CE40-0023-02 LISA and Labex CEMPI (ANR-11-LABX-0007-01).

Consider now an arbitrary element τ P Epπq R zt0u. One may develop it in the basis pe u q uPPpπq : [START_REF] Gignac | Local dynamics of non-invertible maps near normal surface singularities[END_REF] τ " ÿ uPPpπq

x u e u .

We will show that τ 2 ă 0. As τ was chosen as an arbitrary non-zero vector, this will imply that the intersection form on Epπq R is indeed negative definite. The trick is to express the self-intersection τ 2 using the expansion [START_REF] Gignac | Local dynamics of non-invertible maps near normal surface singularities[END_REF], then to develop it by bilinearity and to replace the vectors e u by σ ´řv‰u e v in a precise place:

" ÿ u x 2 u pσ ¨eu q ´ÿ uăv px u ´xv q 2 e u ¨ev .

We got the equality:

u pσ ¨eu q ´ÿ uăv px u ´xv q 2 e u ¨ev .

Using the inequalities (9) and ( 12), we deduce that its right-hand side is non-positive, therefore the intersection form is negative semi-definite. It remains to show that τ 2 ă 0. Assume by contradiction that τ 2 " 0. Equality [START_REF] Hartshorne | Algebraic geometry[END_REF] shows that the following equalities are simultaneously satisfied: [START_REF] Hirzebruch | Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen[END_REF] ÿ u x 2 u pσ ¨eu q " 0, (16) px u ´xv q 2 e u ¨ev " 0, for all u ă v.

The relations [START_REF] De Jong | Local analytic geometry[END_REF] imply that x u " x v whenever e u ¨ev ą 0. As e u " a u E u with a u ą 0, the inequality e u ¨ev ą 0 is equivalent with E u ¨Ev ą 0, that is, with the fact that ru, vs is an edge of the dual graph Γpπq. This dual graph being connected (see ( 6)), we see that x u " x v for all u, v P Ppπq. Consider now an index u 0 satisfying the second condition of relations [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]. Equation [START_REF] Hirzebruch | Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen[END_REF] implies that x u 0 " 0. Therefore all the coefficients x u vanish, which contradicts the hypothesis that τ ‰ 0.

' b separates a and c in the dual graph Γpπq.

This theorem explains the importance of cut-vertices of the dual graph Γpπq for understanding the 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 normal surface singularities (see also [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Prop. 1.18] for a slightly different proof): Theorem 9.3 may be also reformulated in terms of spherical geometry using the spherical Pythagorean theorem (see [START_REF] García Barroso | Ultrametric properties for valuation spaces of normal surface singularities[END_REF]Prop. 1.19.III]).

Using Proposition 8.7 and Definition 9.1 of the angular distance, one may reformulate in the following way the ultrametric inequality for the restriction of the function u L to a set of three branches: Proposition 9.4. Let L, A, B, C be pairwise distinct branches on S. Consider an embedded resolution of their sum and let E l , E a , E b , E c the irreducible components of its exceptional divisor which intersect the strict transforms of L, A, B and C respectively. Then the following (in)equalities are equivalent:

(1) u L pA, Bq ď maxtu L pA, Cq, u L pB, Cqu.

(2) pA ¨Bq ¨pL ¨Cq ě mintpA ¨CqpL ¨Bq, pB ¨CqpL ¨Aqu.

(3) xa, byxl, cy ě mintxa, cyxl, by, xb, cyxl, ayu.

(4) ρpa, bq `ρpl, cq ď maxtρpa, cq `ρpl, bq, ρpb, cq `ρpl, aqu.

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 the inequalities are symmetric in the four branches L, A, B, C. The fourth one is a well-known condition in combinatorics, whose name was introduced by Bunemann in his 1974 paper [START_REF] Buneman | A note on the metric properties of trees[END_REF]: Definition 9.5. Let pX, δq be a finite metric space. One says that it satisfies the four points condition if whenever a, b, c, d P X, one has the following inequality: δpa, bq `δpc, dq ď maxtδpa, cq `δpb, dq, δpa, dq `δpb, cqu.

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 may be thought as a special kind of decorated unrooted tree (see [3]):

Proposition 9.6. The metric space pX, δq 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 1 or 2, then this tree is unique up to a unique isomorphism fixing X.

Let us introduce supplementary vocabulary in order to deal with the special trees appearing in 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 pX, δq is a finite metric space which satisfies the four points condition, then the unique X-tree characterized in Proposition 9.6 is called the tree hull of pX, δq.


Arthur Clini De Souza

S Lanteri

H E Hernandez-Figueroa

M Abbarchi

D Grosso

B Kerzabi

M Elsawy

Deep Learning Inverse Design Methodology for Structural Color Filter Metasurfaces

Surrogate Model. The pivotal point of this work, having multiple functions. Will be used for rapid estimation of spectral responses, generate data that will be used to train the MVANN and operates as a differentiable solver, facilitating efficient design variable optimization through automatic differentiation. An important trick is to bring the wavelength to the input rather than considering a multi-dimensional output.

Schematic diagram of the structure under consideration. We simulated the reflection spectrum of 585 structures. Xpolarized light is injected from top. The motivation behind slanted resonators is to break symmetry in Z, attenuating high order modes. dim= 500 Optimization diagram. We used lorentzians in frequency as targets (special case of Fano resonance). The MVANN is used to predict geometries given a target spectrum. Trained with 7839 predictions. We didn't perform the dataset search for the fixed height and looked at the 5 best designs of the MVANN to find designs with similar H.

Sharp Lorentzian

Spectral Response

targets

We are not limited in finding a pure lorentzian. In this case we search for a broad resonance with the objective of finding a pure red. Comparing to the literature, [1,2], we were able to find good reds with a method also able to find other colors.

Combination of Gaussians

Different Objectives

We introduce a novel technique for designing color filter metasurfaces using a data-driven approach based on deep learning. Our innovative approach employs inverse design principles to identify highly efficient designs that outperform all the configurations in the dataset, which consists of 585 distinct geometries solely. By combining Multi-Valued Artificial Neural Networks and back-propagation optimization, we overcome the limitations of previous approaches, such as poor performance due to extrapolation and undesired local minima. Consequently, we successfully create reliable and highly efficient configurations for metasurface color filters capable of producing exceptionally vivid colors that go beyond the sRGB gamut. Furthermore, our deep learning technique can be extended to design various pixellated metasurface configurations with different functionalities.

Arxiv paper: Broad Lorentzian

Other approaches requires more data to optimize a single objective. In the case of [1,2] the authors focus solely in the red color. In [1] they optimized 13 variables using 360 simulation while in [2] was performed 9600 simulations to optimize 6 variables. Other example is [3] where it is optimized 7 variables considering 5600 simulations.