Walter D Neumann
email: neumann@math.columbia.edu

AND Helge Møller Pedersen

Anne Pichon
email: anne.pichon@univ-amu.fr

MINIMAL SURFACE SINGULARITIES ARE LIPSCHITZ NORMALLY EMBEDDED

Keywords: 1991 Mathematics Subject Classification. 14B05, 32S25, 32S05, 57M99 bilipschitz, Lipschitz geometry, minimal surface singularity, Lipschitz normal embeddings

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We show that minimal surface singularities are Lipschitz normally embedded, i.e., their outer and inner metrics are bilipschitz equivalent, and that they are the only rational surface singularities with this property. The proof is based on a preliminary result which gives a general characterization of Lipschitz normally embedded normal surface singularities.

Introduction

If pX, 0q is a germ of an analytic space of pure dimension dimpX, 0q, we denote by mpX, 0q its multiplicity and by edimpX, 0q its embedding dimension.

Minimal singularities were introduced by J. Kollár in [START_REF] Kollár | Toward moduli of singular varieties[END_REF] as the germs of analytic spaces pX, 0q of pure dimension which are reduced, Cohen-Macaulay, whose tangent cone is reduced and whose multiplicity is minimal in the sense that Abhyankar's inequality mpX, 0q ě edimpX, 0q ´dimpX, 0q `1

is an equality (see [START_REF] Kollár | Toward moduli of singular varieties[END_REF]Section 3.4] or [START_REF] Bondil | Résolution des singularités de surfaces par éclatements normalisés[END_REF]Section 5]).

In this paper, we only deal with normal surfaces. In this case, minimality can be defined as follows [START_REF] Kollár | Toward moduli of singular varieties[END_REF]Remark 3.4.10]): a normal surface singularity pX, 0q is minimal if it is rational with a reduced minimal (also called fundamental) cycle.

Minimal surface singularities play a key role in resolution theory of normal complex surfaces since they appear as central objects in the two main resolution algorithms: the resolution obtained as a finite sequence of normalized Nash transformations ( [START_REF] Spivakovsky | Sandwiched singularities and desingularization of surfaces by normalized Nash transformations[END_REF]), and the one obtained by a sequence of normalized blow-up of points ( [START_REF] Zariski | The reduction of the singularities of an algebraic surface[END_REF]). The question of the existence of a duality between these two algorithms, asserted by D. T. Lê in [START_REF] Dũng | Geometry of surface singularities[END_REF]Section 4.3] (see also [4, Section 8]) remains open, and the fact that minimal singularities seem to be the common denominator between them suggests the need of a better understanding of this class of surface germs.

In this paper, we study minimal surface singularities from the point of view of their Lipschitz geometries, and we show that they are characterized by a remarkable metric property: they are Lipschitz normally embedded. Let us explain what this means.

If pX, 0q is a germ of a complex variety, then any embedding φ : pX, 0q ãÑ pC n , 0q determines two metrics on pX, 0q: the outer metric d out px 1 , x 2 q :" ||φpx 1 q ´φpx 2 q|| (i.e., distance in C n ) and the inner metric d inn px 1 , x 2 q :" inftlengthpφ ˝γq : γ is a rectifyable path in X from x 1 to x 2 u , using the riemannian metric on X t0u induced by the hermitian metric on C n . For all x, y P X, d inn px, yq ě d out px, yq, and the outer metric determines the inner metric. Up to bilipschitz equivalence both these metrics are independent of the choice of complex embedding. We speak of the (inner or outer) Lipschitz geometry of pX, 0q when considering these metrics up to bilipschitz equivalence. Definition 1.1. A germ of a complex normal variety pX, 0q is Lipschitz normally embedded if inner and outer metrics coincide up to bilipschitz equivalence, i.e., there exists a neighbourhood U of 0 in X and a constant K ě 1 such that for all x, y P U

1 K d inn px, yq ď d out px, yq.
It is a classical fact that the topology of a germ of a complex variety pX, 0q Ă pC n , 0q is locally homeomorphic to the cone over its link X pǫq " S 2n´1 ǫ X X, where S 2n´1 ǫ denotes the sphere with small radius ǫ centered at the origin in C n . If pX, 0q is a curve germ then it is in fact bilipschitz equivalent to the metric cone over its link with respect to the inner metric, while the data of its Lipschitz outer geometry is equivalent to that of the embedded topology of a generic plane projection (see [START_REF] Pham | Fractions Lipschitziennes d'une algebre analytique complexe et saturation de Zariski[END_REF][START_REF] Fernandes | Topological equivalence of complex curves and bi-Lipschitz maps[END_REF][START_REF] Neumann | Lipschitz geometry of complex curves[END_REF]). Therefore, an irreducible complex curve is Lipschitz normally embedded if and only if it is smooth. Our main result shows that this is not true in higher dimension: any minimal surface singularity is Lipschitz normally embedded. In section 8 we also prove a converse to this among rational singularities, so: Theorem 1.2. A rational surface singularity is Lipschitz normally embedded if and only if is minimal.

The paper is organized as follows. In Section 2, we give basic definitions about generic projections of a normal surface germ and their polar curves and discriminants. In Section 3, we recall the geometric decomposition with rates of a normal surface germ given in [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF], which completely describes the inner Lipschitz geometry and an important part of the outer geometry. The proof of theorem 1.2 is based on two preliminary results. The first one is a characterization of Lipschitz normally embedded surface singularities (Theorem 4.5). The second one is a complete description of the geometric decomposition of a minimal singularity given in Section 5 by using results of [START_REF] Birbrair | The thick-thin decomposition and the bilipschitz classification of normal surface singularities[END_REF] and the explicit description of the polar and discriminant curves of minimal surface singularities given in [START_REF] Spivakovsky | Sandwiched singularities and desingularization of surfaces by normalized Nash transformations[END_REF] and [START_REF] Bondil | Discriminant of a generic projection of a minimal normal surface singularity[END_REF][START_REF] Bondil | Fine polar invariants of minimal singularities of surfaces[END_REF]. Finally, one direction of Theorem 1.2 (minimal singularities are normally embedded) is proved in Section 6 and illustrated through an example in Section 7, and the other direction is proved in Section 8.

Acknowledgments. Neumann was supported by NSF grant DMS-1206760. Pichon was supported by ANR-12-JS01-0002-01 SUSI. We are also grateful for the hospitality and support of the following institutions: Columbia University, Institut de Mathématiques de Marseille and Aix Marseille Université, and IAS Princeton.

Generic projections

We denote by Gpk, nq the grassmannian of k-dimensional subspaces of C n . For D P Gpn ´2, nq let ℓ D : C n Ñ C 2 be the linear projection with kernel D. Definition 2.1. Let pγ, 0q Ă pC n , 0q be a complex curve germ. Let V Ă Gpn´2, nq be the open dense subset such that for each D P V , D does not contain any limit of bisecant lines to γ. The projection ℓ D is generic for pγ, 0q if D P V . (See [START_REF] Briançon | Déformations équisingulières des germes de courbes gauches réduites[END_REF] for the definition of the cone of limits of bisecants.) Definition 2.2. ([16, (2.2.2)] and [23, V. (1.2.2)].) Let pX, 0q Ă pC n , 0q be a normal surface germ. We assume that the restriction ℓ D | X is a finite morphism (this is true for D in an open dense set of Gpn ´2, nq). Let Π D Ă X be the polar curve of this projection, i.e., the closure in pX, 0q of the singular locus of the restriction of ℓ D to X t0u, and let ∆ D " ℓ D pΠ D q be the discriminant curve.

There exists an open dense subset Ω Ă Gpn ´2, nq such that (1) for each D in Ω, the projection ℓ D is generic for its polar curve Π D ;

(2) tpΠ D , 0q : D P Ωu forms an equisingular family of curve germs in terms of strong simultaneous resolution.

We say the projection ℓ D :

C n Ñ C 2 is generic for pX, 0q if D P Ω.
Remark 2.3. For each D P Ω the restriction ℓ D | X : pX, 0q Ñ pC 2 , 0q is a finite cover whose degree equals the multiplicity of pX, 0q and D X C X,0 " t0u where C X,0 denotes the tangent cone to pX, 0q. In fact, for any D, these two properties are equivalent ([4, Remarque 2.2]).

Lipschitz geometry and geometric decomposition of a normal surface singularity

In this section, we describe the geometric decomposition of pX, 0q introduced in [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF], which completely determines the inner geometry and is an invariant of the outer geometry. We give first the description through resolution as presented in [20, Section 9] and we give an alternative but equivalent description through carrousel decomposition at the end of the section. See also [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF] for more details.

We need to define the contact exponent between two germs of curves. Let pC, 0q and pD, 0q be two irreducible plane curve germs intersecting only at 0. Let us choose coordinates x and y in C 2 so that C and D admit Puiseux parametrizations respectively y " αpx 1{m q " ÿ iěm a i x i{m and y " βpx 1{n q " ÿ jěn b j x j{n , where m and n are the multiplicities of C and D. Replacing x by ωx in α (resp. β) where ω m " 1 (resp. ω n " 1), we get all the Puiseux parametrizations of C (resp. D).

Denote by α i px 1{m q ( m i"1 the set of Puiseux parametrizations of C and β j px 1{n q ( n j"1 that of D.

Definition 3.1. The contact exponent between C and D is the rational number defined by: q C,D " max 1ďiďm,1ďjďn ord x pα i px 1{m q ´βj px 1{n qq ( More generally, if C, D are two germs of curves in pC n , 0q intersecting only at 0, we define the contact exponent q C,D between C and D as the contact exponent between ℓpCq and ℓpDq where ℓ :

C n Ñ C 2 is a generic projection for C Y D.
Notice that q C,D depends neither on the choice of ℓ nor on that of the coordinates x and y. Notice also that the contact exponent between two smooth curves is an integer. It is in fact the minimal number of blow-ups of points necessary to separate their strict transforms.

In order to define the geometric decomposition of pX, 0q, we consider the minimal good resolution π 0 : p r X 0 , Eq Ñ pX, 0q with the following two properties: (1) it resolves the basepoints of a general linear system of hyperplane sections of pX, 0q (i.e., it factors through the normalized blow-up of the maximal ideal of X); (2) it resolves the basepoints of the family of polar curves of generic plane projections (i.e., it factors through the Nash modification of X). This resolution is obtained from the minimal good resolution of pX, 0q by blowing up further until the basepoints of the two kinds are resolved. We denote by Γ 0 the dual resolution graph of π 0 . Definition 3.2. An L-curve is an exceptional curve in π ´1 0 p0q which intersects the strict transform of a generic hyperplane section. The vertex of Γ 0 representing an L-curve is an L-node.

A P-curve (P for "polar") will be an exceptional curve in π ´1 0 p0q which intersects the strict transform of the polar curve of any generic linear projection. The vertex of Γ 0 representing this curve is a P-node.

A vertex of Γ 0 is called a node if it is an Lor P-node or has valency ě 3 or represents an exceptional curve of genus ą 0.

A string of a resolution graph is a connected subgraph whose vertices have valency 2 and are not nodes, and a bamboo is a non-node vertex of valency 1 union a string attached to it. Now, consider the resolution π : r X Ñ X obtained from r X 0 by blowing up each intersection point of pairs of curves of π ´1p0q which correspond to nodes of Γ 0 . We then obtain a resolution satisfying (1) and (2) and such that there are no adjacent nodes in its resolution graph. Let Γ be the resolution graph of π. Denote by E 1 , . . . , E r the exceptional curves in E " π ´1p0q and by v k the vertex of Γ corresponding to E k .

For each k " 1, . . . , r, let N pE k q be a small closed tubular neighbourhood of E k and let

N pE k q " N pE k q ď k 1 ‰k N pE k 1 q.
For any subgraph Γ 1 of Γ define:

N pΓ 1 q :" ď v k PΓ 1 N pE k q and N pΓ 1 q :" N pΓq ď v k RΓ 1 N pE k q .
We now describe the geometric decomposition of pX, 0q. It is a decomposition of pX, 0q as a union of semi-algebraic pieces of three types: Bp1q, Bpqq with q ą 1 and Apq, q 1 q where the q ă q 1 are rational numbers ě 1. The pieces Bp1q are metrically conical, i.e., bilipschitz equivalent to a strict metric cone in the inner metric. For each piece Bpqq with q ą 1, Bpqq t0u fibers over a punctured disk D 2 t0u with 2-manifold fibers having diameter of order Opt q q at distance t from the origin. We call q the rate of Bpqq. Each Apq, q 1 q is an intermediate piece between a Bpqq and a Bpq 1 q piece and is topologically the cone on a toral annulus T 2 ˆI. For a more precise definition of pieces see [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF]Section 2].

If v j is a vertex of Γ, we denote by Γ j the subgraph of Γ consisting of v j union any attached bamboos. Proposition 3.3. [20, Proposition 9.3] The pieces of the geometric decomposition of pX, 0q are as follows:

(1) the Bp1q-pieces are the sets π 1 pN pΓ j qq where v j is an L-node;

(2) each Bpqq-piece for q ą 1 is a set π 1 pN pΓ j qq where v j is a node which is not an L-node;

(3) the Apq, q 1 q-pieces (which have 1 ď q ă q 1 ) are the π 1 pN pσqq where σ is a maximal string between two nodes. In both cases ( 1) and ( 2), the rate q is the contact exponent between the π 1 -images of two curvettes of E j meeting E j at distinct points.

If E 1 is an irreducible component of a normal crossing divisor E in a complex smooth surface S, we call curvette of E 1 a small smooth curve on S transversal to E 1 at a smooth point of E.

Remark 3.4. The geometric decomposition of pX, 0q is a refinement of the thickthin decomposition of pX, 0q introduced in [START_REF] Birbrair | The thick-thin decomposition and the bilipschitz classification of normal surface singularities[END_REF]. Namely the thick part of pX, 0q is the union of Bp1q pieces plus adjacent Ap1, qq-pieces.

The geometric decomposition can be encoded in the dual resolution graph Γ of π decorated with a rate q at each node. See the left graph in Example 5. [START_REF] Briançon | Déformations équisingulières des germes de courbes gauches réduites[END_REF].

In [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF]Definition 8.6], we define an equivalence relation between pieces by saying that two pieces with same rate q are equivalent, if they can be made equal by attaching a "q-collar" (a Bpqq piece which is topologically the cone on T 2 ˆI) at each boundary component. Similarly, two Apq, q 1 q-pieces are equivalent if they can be made equal by removing a q-collars at the outer boundaries and removing q 1 -collars at the inner boundaries. Proposition 3.5. [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF]Proposition 8.7] The geometric decomposition is unique up to equivalence of the pieces and it is an invariant of the outer Lipschitz geometry.

Let us now explain how the geometric decomposition is related (and even built) from the geometry of the discriminant of a generic plane projection.

We first construct a decomposition of the germ pC 2 , 0q into Bpqq-and Apq, q 1 qpieces based on a resolution of ∆.

Let ρ : Y Ñ C 2 be the minimal sequence of blow-ups starting with the blowup of 0 P C 2 which resolves the basepoints of the family of images ℓpΠ D q by ℓ of the polar curves of generic plane projections and let ∆ be some ℓpΠ D q. We set ρ ´1p0q " Ť m k"1 C k , where C 1 is the exceptional curve of the first blow-up. Denote by R the dual graph of ρ, so v 1 is its root vertex. We call a ∆-curve an exceptional curve in ρ ´1p0q intersecting the strict transform of ∆, and a ∆-node a vertex of R which represents a ∆-curve. We call any vertex of R which is either v 1 or a ∆-node or a vertex with valency ě 3 a node of R. A string is, as in Definition 3.2, a connected subgraph whose vertices have valency 2 and are not nodes, and a bamboo is again a non-node vertex of valency 1 union a string attached to it.

If two nodes are adjacent, we blow up the intersection points of the two corresponding curves in order to create a string between them. Denote ρ 1 : Y 1 Ñ C 2 the obtained resolution.

The decomposition of pC 2 , 0q is as follows:

(1) the single Bp1q-piece is the set ρ 1 pN pC 1 qq;

(2) the Bpqq-pieces for q ą 1 are the sets ρ 1 pN pR k qq where R k is a subgraph of R consisting of a node v k which is not the root v 1 plus any attached bamboo. The rate q is the contact exponent between the ρ 1 -images of two curvettes of the exceptional curve corresponding to v k . (3) the Apq, q 1 q-pieces are the sets ρ 1 pN pσqq where σ is a maximal string between two nodes.

Definition 3.6. We call this decomposition of pC 2 , 0q into B-and A-pieces the carrousel decomposition of pC 2 , 0q with respect to ∆.

The pieces of the carrousel decomposition can also described in terms of truncated Puiseux parametrizations of the components of ∆. See [START_REF] Birbrair | The thick-thin decomposition and the bilipschitz classification of normal surface singularities[END_REF]Section 12] or [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF]Sections 3,[START_REF] Briançon | Déformations équisingulières des germes de courbes gauches réduites[END_REF] for details.

Let us now describe how the carrousel decomposition of pC 2 , 0q with respect to ∆ and the geometric decomposition of pX, 0q are related. Definition 3.7. Let ℓ : pX, 0q Ñ pC 2 , 0q be a generic plane projection of pX, 0q as defined in Section 2. Let Π be its polar curve and Π ˚its strict transform in the resolution π : r X Ñ X. A polar wedge about Π is a neighborhood of Π saturated by the π-images of neighbouring curvettes of Π ˚. A ∆-wedge is the ℓ-image of a polar wedge. (For details see [START_REF] Birbrair | The thick-thin decomposition and the bilipschitz classification of normal surface singularities[END_REF]).

According to [START_REF] Birbrair | The thick-thin decomposition and the bilipschitz classification of normal surface singularities[END_REF]Proposition 3.4], ℓ is a Lipschitz map for the inner metric outside a polar wedge A about Π. Moreover, A and the ∆-wedge ℓpAq are union of B-pieces with trivial topology, i.e., the fibers are 2-disks (we call D-pieces such trivial Bpieces). More precisely, if A 0 Ă A is a Dpsq-piece, then ℓpA 0 q is also a Dpsq-piece (with same rate s).

As a consequence of this, each piece of the carrousel decomposition of pC 2 , 0q just constructed lifts to a union of A and B-pieces of the same type in pX, 0q. After absorption of the polar-wedges and of the Dpqq-pieces which do not contain components of the polar curve (see [6, section 13]), one obtains the geometric decomposition of pX, 0q previously described. In the case of minimal singularities, no absorption will be needed so we omit the details of absorption here.

This correspondence between the geometric decomposition of pX, 0q and the carrousel decomposition of pC 2 , 0q can be read through the correspondence between the resolutions ρ 1 and π given by the Hirzebruch-Jung resolution process. This will be a key argument in the proof of Theorem 1.2 in Section 6. A full example is given in Example 5.7.

Characterization of Lipschitz normally embedded surface singularities

Definition 4.1. Let δ be a singular plane curve. We say rate of δ for any rational number which is either a characteristic Puiseux exponent of a branch of δ or the contact exponent between two branches of δ. If δ is smooth its set of rates is empty.

The following easy remark will be useful in the induction of Section 6.

Remark 4.2. Let e be the blow-up of the origin of C 2 and let pδ, 0q Ă pC 2 , 0q be a singular plane curve germ. If δ is irreducible and its set of rates A consists of numbers ě 2, then the set of rates of pδ ˚, pq is A ´1, where ˚denotes strict transform by e. If δ 1 and δ 2 are two components of ∆ whose strict transforms by e meet e ´1p0q at a single point p, and if q is the contact exponent between δ 1 and δ 2 , then the contact exponent between δ 1 and δ 2 equals q ´1.

Let pX, 0q be the germ of a normal complex surface and let ℓ : X Ñ C 2 be a generic plane projection. Let Π be the polar curve of ℓ and ∆ " ℓpΠq be its discriminant curve. Let ρ : Y Ñ C 2 be the resolution of ∆ introduced in Section 3 and let R be the dual graph of ρ. Definition 4.3. A test curve is an irreducible curve pγ, 0q Ă pC 2 , 0q which is the projection by ρ of a curvette γ ˚of an exceptional curve in ρ ´1p0q represented by a node of R such that γ ˚and the strict transforms of p∆, 0q by ρ do not intersect. Definition 4.4. Let pγ, 0q Ă pC 2 , 0q be a test curve. Let C be the component of ρ ´1p0q which intersects the strict transform γ ˚. We define the inner rate q γ of γ as the contact exponent between the ρ-images of two generic curvettes of C.

Let r ℓ : pC n , 0q Ñ pC 2 , 0q be a plane projection which is generic for the curve ℓ ´1pγq. We define the outer rates of γ as the rates of the plane curve r ℓpℓ ´1pγqq. Notice that the outer rates of γ do not depend on the choice of r ℓ.

Theorem 4.5. pX, 0q is Lipschitz normally embedded if and only if for any test curve pγ, 0q, the outer rates of pγ, 0q satisfy the following conditions: p˚1q Any outer rate of γ which is a characteristic exponent equals q γ . p˚2q For any pair δ 1 , δ 2 of components of ℓ ´1pγq, let π 1 : X 1 Ñ X be π composed with a sequence of successive blow-ups of points which resolves the curve δ 1 Y δ 2 . Let E 1 and E 2 be the components of π 1´1 p0q which intersect the strict transforms of δ 1 and δ 2 , and let q 0 be the maximum among minimum of inner rates of vertices along paths joining the vertices v 1 and v 2 in the resolution graph of π 1 . Then the contact exponent between the generic projections r ℓpδ 1 q and r ℓpδ 2 q equals q 0 . Remark 4.6. In fact, an outer rate of γ which is a characteristic exponent is always ě q γ . Indeed, let C ν be the exceptional curve of ρ ´1p0q such that γ ˚is a curvette of C ν . Then B " ρpN pC ν qq is a Bpq γ q-piece in the sense of [20, Section 2] (see also [START_REF] Birbrair | The thick-thin decomposition and the bilipschitz classification of normal surface singularities[END_REF]Section 11]). Let l : pX, 0q Ñ pC 2 , 0q be another generic projection. Note that lpℓ ´1pBqq is also a Bpq γ q-piece, so any characteristic exponent of a complex curve inside it is ě q γ . Similar arguments show that the contact exponent between r ℓpδ 1 q and r ℓpδ 2 q, as in Condition (˚2), is always ě q 0 . Remark 4.7. Notice that Condition p˚2q for γ does not depend on the choice of the resolution π 1 . In fact, one could take for π 1 the resolution π ρ : X Ñ X of pX, 0q obtained by taking the pull-back of ρ by ℓ, then normalizing and then resolving the remaining quasi-ordinary singularities. It is easy to prove that π ρ is a good resolution for the lifting by ℓ of any test curve of ρ.

Proof of Theorem 4.5. The proof will use some arguments already presented in [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF], in particular in the proof of [20, 20.1].

Assume that there exists a test curve pγ, 0q Ă pC 2 , 0q which does not satisfy Condition p˚1q. Let δ be a component of the lifting ℓ ´1pγq such that r ℓpδq admits a characteristic exponent q ą q γ . Let pxpwq, ypwqq " pw n , ř iěn b i w i q be a parametrization of γ. Fix w 0 P C ˚and consider the algebraic arc p : r0, 1q Ñ γ defined by pptq " pxptw 0 q, yptw 0 qq. Then p lifts to two semi-algebraic arcs γ 1 , γ 2 : r0, 1q Ñ δ such that for all t, ℓpγ 1 ptqq " ℓpγ 2 ptqq " pptq and d out pγ 1 ptq, γ 2 ptqq " Opt nq q.

Since γ 1 ptq and γ 2 ptq belong to different sheets of the cover ℓ, then for any path σ between them, the loop ℓpσq will have to travel through the Bpq γ q-piece containing the point pptq, so lengthpℓpσqq ě Opt nqγ q. Since ℓ is Lipschitz for the inner metric (outside polar-wedges which can be avoided by multiplying the length of ℓpσq by a factor of π), we also have that the length of σ is ě Opt nqγ q. Therefore, we get

d inn pγ 1 ptq, γ 2 ptqq ě Opt nqγ q. So lim tÑ0 d out pγ 1 ptq, γ 2 ptqq d inn pγ 1 ptq, γ 2 ptqq " 0,
which implies that pX, 0q is not Lipschitz normally embedded. Assume now that pγ, 0q Ă pC 2 , 0q does not satisfy Condition p˚2q of Theorem 4.5, i.e., there exists two components δ 1 and δ 2 of the lifting ℓ ´1pγq such that q ą q 0 , where q 0 is defined as in Condition (˚2) and q equals the contact exponent q r ℓpδ1q, r ℓpδ2q . Now lift the arc p defined before to two semi-algebraic arcs γ 1 : r0, 1q Ñ δ 1 and γ 2 : r0, 1q Ñ δ 2 . We then have d out pγ 1 ptq, γ 2 ptqq " Opt nq q. On the other hand, any path σ in X from γ 1 ptq to γ 2 ptq corresponds to a path σ in the resolution graph Γ joining the vertices v 1 and v 2 . By the same argument as before, the length of a minimal σ will be Opt nq0 q where q 0 is the minimal rate of the vertices on σ. Therefore d inn pγ 1 ptq, γ 2 ptqq " Opt nq0 q and we conclude as before that pX, 0q is not Lipschitz normally embedded.

Therefore, if pX, 0q is Lipschitz normally embedded, then any test curve pγ, 0q Ă pC 2 , 0q satisfies conditions p˚1q and p˚2q.

We now want to prove that, conversely, if any test curve satisfies conditions p˚1q and p˚2q, then pX, 0q is Lipschitz normally embedded.

Let ℓ 1 , ℓ 2 and ℓ 3 : pX, 0q Ñ pC 2 , 0q be three distinct generic projections for pX, 0q and for i " 1, 2, 3, let A i be a polar wedge for ℓ i such that A 1 , A 2 and A 3 are pairwise disjoint outside the origin. Then for any pair of points p, q P X t0u sufficiently close to 0, p and q are both outside A i for at least one of i P t1, 2, 3u. Choose such an i and set ℓ pp,qq " ℓ i .

By a straightforward adaptation of the argument in the last page of [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF], one shows that there exist a neighborhood V of 0 in X and a constant L ě 1 such that for any pair p, q P V , 1 L ˆdinn pp, q 1 q `dout pq 1 , qq ˙ď d out pp, qq where q 1 is the extremity of the lifting by ℓ " ℓ pp,qq of the segment rℓppq, ℓpqqs with origin p. In particular, ℓpq 1 q " ℓpqq.

Thus the result follows from the following Lemma 4.8, which is proved later.

Lemma 4.8. Let ℓ : pX, 0q Ñ pC 2 , 0q be a generic projection for pX, 0q and let pA, 0q be a polar wedge for ℓ. Assume that any test curve for ℓ satisfies conditions p˚1q and p˚2q. Then there exists ǫ ą 0 and a constant K ě 1 such that for any p P B 4 ǫ t0u and any pair of distinct points p 1 , p 2 P X A such that ℓpp 1 q " ℓpp 2 q, we have d inn pp 1 , p 2 q ď Kd out pp 1 , p 2 q.

In order to prove Lemma 4.8, we need two preliminary lemmas 4.9 and 4.10. Lemma 4.9. Let γ be a test curve. Then Condition p˚1q implies that the restriction of ℓ to any component δ of ℓ ´1pγq is an isomorphism.

Proof. Let C ν be the irreducible component of ρ ´1p0q such that γ ˚is a generic curvette of C ν . Then B " ρpN pC ν qq is a Bpq γ q-piece. Let r B be the component of ℓ ´1pBq which contains δ (by a component of a semi-algebraic germ pZ, 0q, we mean the closure of a connected component of Z t0u).

Let ℓ 1 : pX, 0q Ñ pC 2 , 0q be another generic projection and let ∆ 1 be its discriminant curve. Then ℓ 1 p r Bq is up to higher order a Bpq γ q-piece of a carrousel decomposition of ∆ 1 . Choose a generic test curve γ 1 in it and let δ 1 be the component of the lifting ℓ 1´1 pγ 1 q inside r B. Let y " ř qiě1 a i x qi be a Puiseux expansion of γ. Since the curve ℓpδ 1 q is inside B and since B is a q γ -neighbourhood of γ, then the Puiseux expansion of ℓpδ 1 q coincides with that of γ up to exponent q i " q γ , with a distinct non zero coefficient for x qγ .

Assume ℓ | δ : δ Ñ γ is not an isomorphism, then multpδq " k multpγq where k is an integer ě 2. Since the curves δ and δ 1 are isomorphic, then multpδq " multpδ 1 q. Since ℓ is generic for δ 1 , then multpℓpδ 1 qq " multpδ 1 q. So we get multpℓpδ 1 qq " k multpγq. Since q γ is the greatest characteristic Puiseux exponent of γ and since the Puiseux expansions of γ and ℓpδ 1 q coincide up to exponent q γ , we then obtain that ℓpδ 1 q admits a characteristic exponent q ą q γ . Since δ and δ 1 are isomorphic, q is an outer rate of γ and Condition p˚1q is not satisfied.

Let π : r X Ñ X be a resolution of pX, 0q described after Definition 3.2. It factors through the Nash modification ν : q X Ñ X and through the blow-up of the origin and no two nodes of its resolution graph Γ are adjacent. Let σ : r X Ñ Gp2, nq be the map induced by the projection p 2 : q X Ă X ˆGp2, nq Ñ Gp2, nq. The map σ is well defined on E " π ´1p0q and its restriction to E is constant on any connected component of the complement of P-curves in E ([10, Section 2], [22, Part III, Theorem 1.2]). The connected subgraphs of Γ obtained by removing all P-nodes and adjacent edges are called P-Tjurina components. Lemma 4.10. Let γ be a test curve and δ 1 and δ 2 two components of ℓ ´1pγq whose strict transforms meet E 1 and E 2 at smooth points p 1 and p 2 of E.

Assume E 1 and E 2 are not P-curves and that there exists a path in the graph of π joining v 1 to v 2 through vertices with rates ě q γ . Then Condition p˚2q implies σpp 1 q ‰ σpp 2 q.

Proof. Consider a semi-algebraic arc p : r0, 1q Ñ γ such that ||pptq|| " Optq and two distinct liftings γ 1 : r0, 1q Ñ δ 1 and γ 2 : r0, 1q Ñ δ 2 of p by ℓ. Then d out pγ 1 ptq, γ 2 ptqq " Opt q q where q is an outer rate of γ.

For q 1 ą q γ sufficiently close to q γ , there exists an Apq γ , q 1 q-piece A such that γ is in the outer boundary of A and A X ∆ " H. Let A 1 and A 2 be the liftings of A containing respectively δ 1 and δ 2 . A 1 and A 2 are inside N pΓ 1 q and N pΓ 2 q where Γ 1 and Γ 2 are the P-Tjurina components containing v 1 and v 2 . Assume σpp 1 q " σpp 2 q. Adapting the arguments of the proof of Lemma 11.7 in [START_REF] Neumann | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity[END_REF] inside the pieces A 1 and A 2 , one obtains d inn pγ 1 ptq, γ 2 ptqq " Opt qγ q and lim tÑ0 d inn pγ 1 ptq, γ 2 ptqq d out pγ 1 ptq, γ 2 ptqq " 8.

This contradict Condition p˚2q (here q 0 " q γ ).

Remark 4.11. A consequence of Lemma 4.10 is that for any P-Tjurina component Γ 1 , πpN pΓ 1 qq contains at most one component of ℓ ´1pγq.

Proof of Lemma 4.8. Consider a Bpqq-piece B of the carrousel decomposition of C 2 with respect to the discriminant curve ∆ of ℓ. Let N 1 and N 2 be two components of ℓ ´1pBq, so N 1 and N 2 are Bpqq-pieces, possibly with N 1 " N 2 . Let q 0 be the maximum among minimum of inner rates of pieces along paths joining N 1 t0u to N 2 t0u in X t0u, so q 0 ď q and q 0 " q if and only if N 1 " N 2 or N 1 or N 2 can be joined by a path through pieces with higher rates.

Claim 1. There exists K 1 , K 2 ą 0 and ǫ ą 0 such that for all p P pB 4 ǫ t0uq X B and for all any pair of distinct points p 1 , p 2 such that p 1 P pN 1 Aq X ℓ ´1pB 4 ǫ q, p 2 P pN 2 Aq X ℓ ´1pB 4 ǫ q and ℓpp 1 q " ℓpp 2 q " p, we have

d inn pp 1 , p 2 q ď K 1 ||p|| q0 ,
and

K 2 ||p|| q0 ď d out pp 1 , p 2 q.
A straightforward consequence of Claim 1 is that for all p P pB 4 ǫ t0uq X B and p 1 , p 2 as before,

d inn pp 1 , p 2 q d out pp 1 , p 2 q ď K 1 K 2 .
Proof of Claim 1. Choose coordinates px, y, z, . . .q in C n such that ℓ " px, yq and the piece B is foliated by test-curves γ α with Puiseux expansions y "

k ÿ i"1 a i x p k `αx q
where p 1 ă p 2 ă . . . ă p k ă q and where α is in a compact set W Ă C.

For any q 1 ě 1, if a projection ℓ 1 is generic for a curve δ then it is generic for any curve in a q 1 -neighbourhood of δ. Since W is compact, one can choose a finite number of projections ℓ 1 , . . . , ℓ s and a decomposition W " W 1 Y . . . Y W s into compact sets such that for any α P W j , ℓ j is generic for ℓ ´1pγ α q. We will assume s " 1 since the proof is similar for s ě 2 taking minimums of bounds. So we choose ℓ 1 such that for each α P W , the projection ℓ 1 is generic for ℓ ´1pγ α q and we choose the coordinates px, y, z, . . .q in C n so that ℓ 1 " px, zq.

Let α P W and let p P γ α X pB 4 ǫ t0uq. By Lemma 4.9, two distinct points p 1 and p 2 such that ℓpp 1 q " ℓpp 2 q " p belong to distinct connected components of the lifting ℓ ´1pγ α q.

Let δ p1q α Ă N 1 and δ p2q α P N 2 be two distinct irreducible components of the lifting

ℓ ´1pγ α q such that δ p1q α X A " δ p2q α X A " t0u and δ p1q α ‰ δ p2q α if N 1 " N 2 .
Since ℓ 1 is generic for δ Let p " px, yq P γ α and p 1 P δ p1q α and p 2 P δ p2q α such that ℓpp 1 q " ℓpp 2 q " p. Since ||p|| " Op|x|q, there exists L ě 1 such that:

1 L |b p1q q0 pαq ´bp2q q0 pαq| ||p|| q0 ď ||ℓ 1 pp 1 q ´ℓ1 pp 2 q|| ď d out pp 1 , p 2 q. Since W is compact, there exists K 1 2 ą 0 such that for all α P W , K 1 2 ď 1 L |b p1q q0 pαq ´bp2q q0 pαq|. Taking the minimum K 2 of K 1 2 among all pairs δ p1q α , δ p2q
α as before, there exists ǫ ą 0 such that for all p P pB 4 ǫ t0uq X B and for p 1 and p 2 as before, K 2 ||p|| q0 ď d out pp 1 , p 2 q Let us now bound d inn pp 1 , p 2 q We consider the carrousel decomposition of C 2 with respect the the discriminant curve ∆ of ℓ and we decompose X into pieces consisting of components of inverse images by ℓ of pieces of the decomposition of C 2 (see also end of Section 3).

Since p 1 and p 2 are on different sheets of the cover ℓ, a path from p 1 to p 2 with minimal length will have to travel through a Bpq 0 q-piece N of pX, 0q and it only travels through pieces with rates ě q 0 . Now ℓpN q is a Bpq 0 q-piece in pC 2 , 0q and there exists a ą 0 such for ǫ ą 0 sufficiently small, if px, y 1 q and px, y 2 q are in ℓpN q X B 4 ǫ then |y 1 ´y2 | ď a|x| q0 . Let m " mpX, 0q (so m is at most the order of the cover ℓ| N : N Ñ ℓpN q), and let κ be the local bilipschitz constant of ℓ outside the polar wedge A. Then d inn pp 1 , p 2 q is less than or equal to mκ times the diameter of ℓpN q providing one avoids the ∆-wedge ℓpAq. So we obtain

d inn pp 1 , p 2 q ď mpπaqκ||p|| q0 ,
where we use the factor πa instead of a to allow replacing a segment by a path avoiding ℓpAq. Setting K 1 " mpπaqκ completes the proof of Claim 1.

Let us now consider an Apq, q 1 q-piece A 0 of the carrousel decomposition of C 2 with respect to ∆ with q ă q 1 , so A 0 X ∆ " H. According to Lemma 4.9 and Remark 4.11, if A 1 is a component of ℓ ´1pA 0 q, the restriction ℓ| A 1 : pA 1 , 0q Ñ pA 0 , 0q is a homeomorphism and two distinct components of ℓ ´1pA 0 q have distinct corresponding P-Tjurina components.

Let A 1 and A 2 be two distinct components of ℓ ´1pA 0 q, so A 1 and A 2 are Apq, q 1 qpieces. Let q 0 be the maximum among minimum of inner rates of pieces along paths joining A 1 t0u to A 2 t0u in X t0u. (So q 0 ď q and q 0 " q if and only if A 1 or A 2 can be joined by a path through pieces with rates ě q 0 .) Claim 2. There exists K ą 0 and ǫ ą 0 such that for all p P pB 4 ǫ t0uq X A 0 and for all p 1 P A 1 X ℓ ´1pB 4 ǫ q and p 2 P A 2 X ℓ ´1pB 4 ǫ q such that ℓpp 1 q " ℓpp 2 q " p, we have d inn pp 1 , p 2 q d out pp 1 , p 2 q ď K.

Proof of Claim 2. Let Γ 1 and Γ 2 be the P-Tjurina components such that A i Ă πpN pΓ i qq, i " 1, 2. Let P 1 (resp. P 2 ) be the values of σ on

Ť vj PΓ1 E j (resp. Ť vj PΓ2 E j ).
In suitable coordinates the semi-algebraic set A 0 is defined by inequalities

α|x| q 1 ď |y ´k ÿ i"1 a i x pi | ď α|x| q ,
where α ą 0 and 1 ď p 1 ă p 2 ă . . . ă p k " q.

Let n " lcmpdenompq 1 q, denompp i q, i " 1, . . . , kq. Then A 0 is the union of the images the maps φ ξ1,ξ2 : r0, 1s ˆr0, 1q Ñ pC 2 , 0q parametrized by pξ 1 , ξ 2 q P S 1 ˆS1 and defined by @ps, tq P r0, 1s ˆr0, 1q, φ ξ1,ξ2 ps, tq " pxptq, yps, tqq with xptq " ξ 1 t n and yps, tq " k ÿ i"1 a i xptq pi `αξ 2 ˆsxptq q 1 `p1 ´sqxptq q Ṅotice that ||φ ξ1,ξ2 ps, tq|| " Opt n q. Case 1. Assume first that A 1 or A 2 can be joined by a path through pieces with rates ě q 1 . Fix ξ 1 , ξ 2 P S 1 ˆS1 and consider two liftings φ 1 : r0, 1s ˆr0, 1q Ñ A 1 and φ 2 : r0, 1sˆr0, 1q Ñ A 2 of φ by ℓ, i.e., φ 1 " pℓ| A1 q ´1 ˝φξ1,ξ2 and φ 2 " pℓ| A2 q ´1 ˝φξ1,ξ2 . Let `cs,t ˘be a continuous family of paths in X t0u parametrized by ps, tq P r0, 1s ˆp0, 1q such that, ' c s,t has origin φ 1 ps, tq and extremity φ 2 ps, tq ' c s,t consists of the path φ 1 prs, 1s ˆttuq followed by a path c 1 t from φ 1 p1, tq to φ 2 p1, tq (independent of s) through pieces with rates ą q 1 followed by the reversed φ 2 prs, 1s ˆttuq.

Fix s P r0, 1q. As t tends to 0, the ratio

lengthpc 1 t q
lengthpφiprs,1sˆttuqq tends to zero for i " 1, 2 and the path c s,t tends to the union of two segments whose angles with the kernel of ℓ depends only on P 1 and P 2 (see Figure 1). Since P 1 ‰ P 2 (Lemma 4.10) and since the projection ℓ is generic, we obtain that for all s P r0, 1q lim tÑ0 lengthpc s,t q d out pφ 1 ps, tq, φ 2 ps, tqq " a, where a ą 0 just depends on P 1 and P 2 (so it is independent of ξ 1 , ξ 2 and s). Since d inn pφ 1 ps, tq, φ 2 ps, tqq ď lengthpc s,t q, this proves Claim 2 in that case. Case 2. Assume now that any path from A 1 or A 2 goes through pieces with rates ď q. Let B (resp. B 1 ) be the Bpqq-piece (resp. Bpq 1 q-piece) attached to the outer (resp. inner) boundary of A 0 and let ǫ, K 1 and K 2 (resp. K 1 1 and K 1 2 ) be constants associated to B (resp. B 1 ) as in Claim 1.

Fix ξ 1 , ξ 2 P S 1 ˆS1 and consider again φ 1 and φ 2 as defined in Case 1. We have for all ps, tq P r0, 1s ˆp0, 1q, d inn pφ 1 ps, tq, φ 2 ps, tqq ď d inn pφ 1 p0, tq, φ 2 p0, tqql engthpφ 1 pr0, ss ˆttuqq `lengthpφ 2 pr0, ss ˆttuqq Let ǫ ą 0 be sufficiently small and let κ ą 0 be a bound for the local bilipschitz constant of the restriction ℓ : X X ℓ ´1pB ǫ q A Ñ B ǫ , where A is a polar wedge around the polar curve of ℓ. We then have for t sufficiently small, i.e., such that φ ξ1,ξ2 ps, tq P B ǫ , d inn pφ 1 ps, tq, φ 2 ps, tqq ď d inn pφ 1 p0, tq, φ 2 p0, tqq `2καspt nq ´tnq 1 q ď K 2 1 t nq0 , where K 2 1 " K 1 `2κα. Notice that K 2 1 is independent of ξ 1 , ξ 2 and s. Since ||φ ξ1,ξ2 ps, tq|| " Opt n q, we then have proved that there exist a constant C ą 0 and ǫ ą 0 such that for any p P B ǫ X A 0 ,

d inn pp 1 , p 2 q ď C||p|| q0
where p 1 and p 2 are the liftings of p to A 1 and A 2 respectively.

Let us now deal with the outer distance. As t tends to 0, the two arcs φ 1 pr0, 1s ttuq and φ 2 pr0, 1s ˆttuq tend to the union of two coplanar segments which are opposite sides of a trapezoid (Figure 2). Then for t ą 0 sufficiently small and for any s P r0, 1s, d out pφ 1 ps, tq, φ 2 ps, tqq ě p1 ´ηq min ˆdout pφ 1 p0, tq, φ 2 p0, tqq, d out pφ 1 p1, tq, φ 2 p1, tqq ẇith η small (the constant 1 ´η is for the case when P 1 " P 2 in the previous notation, i.e., the trapezoid is a rectangle).

Applying Claim 1 to the pieces B and B 1 adjacent to A 0 , there exist K 2 ą 0 and K 1 2 ą 0 such that for all t ą 0 sufficiently small, K 2 ||φ ξ1,ξ2 p0, tq|| q0 ď d out pφ 1 p0, tq, φ 2 p0, tqq, and K 1 2 ||φ ξ1,ξ2 p1, tq|| q0 ď d out pφ 1 p1, tq, φ 2 p1, tqq.

Since for all s, ||φ ξ1,ξ2 ps, tq|| " Opt n q, we then obtain that there exists C 1 ą 0 and ǫ ą 0 such that for any p P B ǫ X A 0 ,

C 1 ||p|| q0 ď d out pp 1 , p 2 q.
where p 1 and p 2 are the liftings of p to A 1 and A 2 respectively. This proves Claim 2 in Case 2.

The geometric decomposition of a minimal singularity

The aim of this section is to describe the geometric decomposition with rates of a minimal surface singularity germ pX, 0q and its correspondence with the carrousel decomposition of pC 2 , 0q with respect to the discriminant curve ∆ of a generic projection of pX, 0q and the resolution ρ of ∆ (see Section 3).

Let us first recall the definition of the minimal (also called fundamental) cycle Z min of a normal surface singularity pX, 0q. We refer to [START_REF] Némethi | Five lectures on normal surface singularities[END_REF] for details. Let π : p r X, Eq Ñ pX, 0q be a resolution and let E 1 , . . . , E r be the irreducible components of the exceptional divisor E " π ´1p0q. The minimal cycle Z min the minimal element of the set of divisors Z " ř r i"1 m i E i whose coefficients m i are strictly positive integers and such that @j " 1, . . . , r, Z ¨Ej ď 0. A reduced minimal cycle means that Z min " ř r i E i , i.e., m i " 1 for all i " 1, . . . , r. If f : pX, 0q Ñ pC, 0q is an analytic function, then its total transform pf q " pf ˝πq ´1p0q decomposes into pf q " Zpf q `f ˚where f ˚is the strict transform and Zpf q a positive divisor with support on E. For each j " 1, . . . , r, one has pf q¨E j " 0. Hence Zpf q ¨Ej ď 0 for all j " 1, . . . , r. If h : pX, 0q Ñ pC, 0q is a generic linear form, then Zphq is the minimal element among divisors Zpf q, and Z min ď Zphq. For any rational singularity, a fortiori for minimal, the minimal resolution resolves the basepoints of the family of generic linear forms and Zphq " Z min (see [START_REF] Michael Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Némethi | Five lectures on normal surface singularities[END_REF]). So, for a rational singularity, the L-nodes in a resolution graph, and then, the thick-thin decomposition, are topologically determined.

We now restrict to minimal singularities. In order to describe the geometric decomposition of pX, 0q, we will use the description by Spivakovsky ([22]) of the minimal resolution π of the pencil of polar curves of generic plane projections pX, 0q Ñ pC 2 , 0q and the description by Bondil ([2,[START_REF] Bondil | Fine polar invariants of minimal singularities of surfaces[END_REF]) of the resolution ρ of the family of projected polars ℓpΠ D q, where ℓ : pX, 0q Ñ pC 2 , 0q is a generic plane projection.

In [START_REF] Spivakovsky | Sandwiched singularities and desingularization of surfaces by normalized Nash transformations[END_REF], Spivakovsky gives the following combinatorial characterization of the dual resolution graphs of minimal singularities which immediately furnishes the Lnodes. Let π 1 : X 1 Ñ X be the the minimal good resolution of pX, 0q and let Γ 1 be its dual graph. Denote by V pΓ 1 q the set of vertices of Γ 1 . If v P V pΓ 1 q, we denote by E v the corresponding irreducible component of the exceptional divisor pπ 1 q ´1p0q, we set wpvq " E 2 v and we denote by νpvq the valence of v, i.e., the number of edges adjacent to v. Proposition 5.1. [START_REF] Spivakovsky | Sandwiched singularities and desingularization of surfaces by normalized Nash transformations[END_REF] A surface singularity is minimal if and only if Γ 1 is a tree of rational curves and for all vertices v P V pΓ 1 q, ´wpvq ě νpvq.

Remark 5.2. Since the minimal cycle is reduced, a vertex of Γ 1 is an L-node if and only if ´wpvq ą νpvq.

Spivakovsky introduced the function s : V pΓ 1 q Ñ N defined as follows: spvq is the number of vertices on the shortest path from v to an L-node. So spvq " 1 if and only if v is an L-node.

Since minimal singularities are rational they can be resolved by only blowing up points, as Tjurina showed in [START_REF] Nikolaevna | Absolute isolatedness of rational singularities and triple rational points[END_REF], and spvq is the number of blow-ups it takes before E v appears in the successive exceptional divisors.

We now state Theorem 5.4 in Chapter III of [START_REF] Spivakovsky | Sandwiched singularities and desingularization of surfaces by normalized Nash transformations[END_REF] in a formulation inspired by Bondil in [START_REF] Bondil | Discriminant of a generic projection of a minimal normal surface singularity[END_REF].

Theorem 5.3. [22, Chap. III, Theorem 5.4] Let pX, 0q be a minimal surface singularity. Let ℓ : pX, 0q Ñ pC 2 , 0q be a generic linear projection and let Π be its polar curve. Let π 1 : pX 1 , 0q Ñ pX, 0q be the minimal resolution of pX, 0q. Consider the cycle S :" ř spvqE v , where the E v are the irreducible components of pπ 1 q ´1p0q. Then the strict transform Π ˚of Π by π 1 is smooth. It consists of exactly ´pS Èv q¨E v ´2 curvettes of each E v and one component through each intersection point E v X E w for which spvq " spwq. Moreover, the latter intersection points are the only basepoints of the family of generic polars Π D , and they are simple, i.e., they are resolved by one blow-up.

Following the terminology of [START_REF] Spivakovsky | Sandwiched singularities and desingularization of surfaces by normalized Nash transformations[END_REF], an edge of Γ 1 between two vertices v and w is central if spvq " spwq, and a vertex v is central if there are at least two neighboring vertices w, w 1 such that spvq´1 " spwq " spw 1 q. Using this, the above theorem says that for each central edge there is one component of Π ˚through the intersection point of the corresponding curves and that for each central vertex v, there is at least one component of Π ˚which is a curvette of E v . Any other components of Π go through L-nodes. In [START_REF] Bondil | Discriminant of a generic projection of a minimal normal surface singularity[END_REF], Bondil shows that the minimal resolution of pX, 0q obtained by only blowing up points is also the minimal resolution of Π just described. Then, blowing up the points corresponding to central edges, we get the resolution π 0 : p r X 0 , Eq Ñ pX, 0q introduced in Section 3, i.e., the minimal resolution which factors through the blow-up of the origin and through Nash blow-up.

We then know the resolution graph Γ 0 of π 0 together with its nodes. So we topologically know the geometric decomposition of pX, 0q from the resolution graph. We now need to determine the rate q of each node. In order to do this, we will use a more precise description of the polar curve and of the discriminant curve presented by Bondil in [START_REF] Bondil | Discriminant of a generic projection of a minimal normal surface singularity[END_REF].

An A n -curve is a germ of an analytic curve isomorphic to the plane curve y 2 xn`1 " 0. If n is odd, then A n consists of a pair of smooth curves with contact exponent n`1 2 while if n is even, A n is an irreducible curve Theorem 5.4 ([2, 3]). Let pX, 0q be a minimal singularity, and let Π be the polar of a generic linear projection. Then

(1) Π decomposes as a union of

A ni -curves Π " Ť i C i and each C i meet a single irreducible component E vi of the exceptional divisor of π 0 (2) If E 2
vi " ´1 (i.e., E vi comes from blowing up a central edge), then C i is an (irreducible) A 2spviq´2 -curve. Otherwise C i consists of two smooth curves forming an A 2spviq´1 -curve (3) The contact exponent between C i and C j equals the minimal value of spvq on the shortest path in Γ 0 between the vertices v i and v j .

Using the fact that each branch of Π is isomorphic to a plane curve and that the restriction ℓ| Π : Π Ñ ∆ is generic, Bondil deduces from Theorem 5.4 the following description of the discriminant curve: Proposition 5.5 ([2, 3]). The discriminant curve ∆ of a generic projection ℓ of pX, 0q is a union of A n -curves in one-to-one correspondence with the curves C i of Proposition 5.4, and their pairwise contact exponents equal that of the corresponding C i 's. Moreover, the minimal resolution of ∆ is the resolution ρ : Y Ñ C 2 which resolves the base points of the family of generic polar curves pℓpΠ D qq DPΩ .

We deduce from this the rates of the pieces Bpqq of the geometric decomposition of pX, 0q: Corollary 5.6. Let pX, 0q be a minimal surface singularity and let Γ 0 be the dual resolution graph of the resolution π 0 described above. The rate q v of a node v of Γ is given by:

q v " # spvq if E 2 v ă ´1 spvq ´1{2 if E 2 v " ´1,
Proof. The rate q v is the contact exponent between the π-images of two generic curvettes of E v . When v is a node such that E 2 v ă ´1, the images of two generic curvettes of E v form a A 2spvq´1 -curve so their contact exponent equals spvq. When E 2

v " ´1 the images of two generic curvettes of E v are A 2spvq´2 -curves whose contact exponent equals spvq ´1 2 .

Example 5.7. Let pX, 0q be a minimal singularity with the following resolution graph:

´4 ´2 ´3 ´3 ´2 ´2 ´2 ´2 ´2
The negative weights are the self-intersections of the exceptional curves. The L-nodes are the vertices v such that ´wpvq ą νpvq (Remark 5.2). They are in black in the graph.

The following graph on the left shows two different things. First, the arrows represent the components of the polar curve of a generic plane projection. The gray node represents a curve obtained by blowing up the minimal resolution at the intersection point of two exceptional curves corresponding to a central edge. There are four pairs C 1 , . . . , C 4 of smooth components, and one component C 5 with multiplicity 2. Secondly, it shows the geometric decomposition of pX, 0q. The rational numbers in italics are the rates of the nodes.

The graph on the right is the resolution graph of the discriminant curve ∆ " ℓpΠq. The arrows represent the components of ∆ " Ť 5 i"1 where ∆ i " ℓpΠ i q, i " 1, . . . , 5. The root-vertex is the black vertex and each vertex is weighted by the corresponding rate.

C 1 C 2 C 5 C 4 C 3 1 5 {2 2 1 2 1 1 3 ∆ 1 ∆ 2 ∆ 3 ∆ 4 ∆ 5 3 5 {2 3 2 1 2

Minimal implies Lipschitz normally embedded

The aim of this section is to prove one direction of Theorem 1.2: any minimal surface singularity is Lipschitz normally embedded. We first state and prove the key Proposition 6.1.

Let pX, 0q be a normal surface germ and let ℓ : pX, 0q Ñ pC 2 , 0q be a generic projection. Let U be an open neighborhood of 0 in C 2 and let e : U 1 Ñ U be the blow-up of the origin. Let p X be the pull-back of ℓ and e and let p ℓ : p X Ñ U 1 and p e : p X Ñ X be the two projections. Let n : X 1 Ñ p X be the normalization of p X. By [4, Prop. 2.15], e 1 " p e ˝n is the normalized blowup of the maximal ideal of pX, 0q. We then have a commutative diagram:

X 1 e 1
❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ℓ 1 ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ p X p e / / p ℓ X ℓ U 1 e / / U
When pX, 0q is rational, a fortiori when minimal, e 1 is the blowup of the maximal ideal; no normalization is needed ( [START_REF] Nikolaevna | Absolute isolatedness of rational singularities and triple rational points[END_REF]). Proposition 6.1. Let pX, 0q be a minimal surface singularity. Let Π be the polar curve of a generic projection ℓ : pX, 0q Ñ pC 2 , 0q and let E 1 " e 1´1 p0q. Choose p P E 1 such that if it is a smooth point of X 1 , then it is not on the strict transform Π ˚of Π. Set q :" ℓ 1 ppq. Then (i) the germ pX 1 , pq is a minimal singularity with multiplicity the number of irreducible components of E 1 through p; (ii) the map germ ℓ 1 : pX 1 , pq Ñ pU 1 , qq is a generic projection for pX 1 , pq. Remark 6.2. If pX 1 , pq is smooth and p P Π ˚, then, according to [START_REF] Bondil | Discriminant of a generic projection of a minimal normal surface singularity[END_REF] either pΠ ˚, pq is the strict transform of a moving polar (i.e., p is not a basepoint of the family of generic polars) or pX, 0q is the singularity A 2 : x 2 `y2 `z3 " 0. In both cases, it is easy to see that the degree of ℓ 1 at pX 1 , pq equals 2 while the multiplicity of pX, p 1 q is 1. So ℓ 1 : pX 1 , pq Ñ pU 1 , qq is not generic. Remark 6.3. The fact the pX 1 , pq is minimal is [START_REF] Bondil | Résolution des singularités de surfaces par éclatements normalisés[END_REF]Théorème 5.9]. The authors prove it there without using the existence of a resolution of pX, 0q. We give here a short proof using this fact.

Proof of Proposition 6.1. Let π : Y Ñ X be the minimal resolution of pX, 0q and let Γ be its resolution graph. Since pX, 0q is rational, then π factors through the blow-up of the maximal ideal ( [START_REF] Michael Artin | On isolated rational singularities of surfaces[END_REF]). Assume pX 1 , pq is not smooth. Then pX 1 , pq has minimal resolution graph one of the connected components Γ 1 of Γ minus the L-nodes. So Γ 1 is a rational graph and pX 1 , pq is rational. Moreover, since the minimal cycle of pX, pq is reduced, the minimal cycle of pX 1 , pq is also reduced and the multiplicity m 1 of pX 1 , pq equals the number of L-nodes adjacent to Γ 1 , i.e., the number of irreducible components of E 1 containing p.

Assume now that pX 1 , pq is smooth. Then by hypothesis, p R Π ˚. Since there is a branch of Π ˚through any singular point of E 1 , it implies that p is a smooth point of E 1 . So the number of branches of E 1 through p equals 1. This proves (i).

In order to prove (ii), we have to check that ℓ 1 : pX 1 , pq Ñ pU 1 , qq satisfies Conditions (1) and (2) of Definition 2.2.

The map ℓ 1 : X 1 Ñ U 1 is a branched cover with degree the multiplicity m " mpX, 0q of pX, 0q. Its discriminant locus is included in the strict transform ∆ of ∆ by e union the exceptional curve C " e ´1p0q. This divisorial discriminant is computed in [START_REF] Bondil | Résolution des singularités de surfaces par éclatements normalisés[END_REF]Proposition 6.1] for any normal surface germ pX, 0q: it equals ∆ ˚`pm ´rqC where r the number of branches of the generic hyperplane section of pX, 0q. In our case, r " m since pX, 0q is minimal ([4, Lemma 5.4 and Theorem 5.8 ]). So, the discriminant of ℓ 1 is just the reduced curve ∆ ˚and the branching locus of ℓ 1 is Π ˚. In particular the polar curve of ℓ 1 : pX 1 , pq Ñ pU 1 , qq is the germ pΠ ˚, pq.

Assume pX 1 , pq is smooth and p R Π ˚. Then ℓ 1 : pX 1 , pq Ñ pU 1 , qq has empty polar curve, so it is an isomorphism and ℓ 1 : pX 1 , pq Ñ pU 1 , qq is a generic projection. This proves (ii) in that case.

Assume now pX 1 , pq is not smooth. Then p P Π ˚since pX, 0q is resolved by a sequence of blowing-ups of points on the successive strict transforms of Π ([2, Lemma 3.1]. Since the projection ℓ is generic for its polar curve (Condition (1) of Definition 2.2), then the following Lemma 6.4 (which will be used again later) implies that ℓ 1 : pX 1 , pq Ñ pU 1 , qq is a generic projection for its polar curve pΠ ˚, pq, i.e., it satisfies Condition (1) of Definition 2.2. Lemma 6.4. Denote by e 0 : U 0 Ñ C N the blow-up of the origin of C N and let pγ, 0q Ă pC N , 0q be a curve germ whose strict transform γ ˚by e 0 intersects e ´1 0 p0q at a unique point p. Let ℓ : C N Ñ C 2 be a linear projection which is generic for the curve germ pγ, 0q.

We may choose coordinates so that ℓ is the map pz 1 , . . . , z N q Þ Ñ pz 1 , z 2 q. Denote by P n´3 the subset of e ´1 0 p0q given by z 1 " z 2 " 0. Let e : U 1 Ñ U be the blowup of the origin of C 2 , where U is a neighborhood of 0. Then there is a map ℓ 0 : pU 0 P n´3 q Ñ U 1 with e ˝ℓ0 " ℓ ˝e0 . By genericity of ℓ, the point p is in U 0 P n´3 .

Set q " ℓ 0 ppq. Then the map germ ℓ 0 : pU 0 P n´3 , pq Ñ pU 1 , qq is generic for the curve germ pγ ˚, pq.

Proof. We will use the criterion of genericity introduced in the proof of Theorem 5.1 in [START_REF] Neumann | Lipschitz geometry of complex curves[END_REF]. Let us first assume that pγ, 0q is irreducible and that the coordinates are chosen so that pγ, 0q admits a Puiseux parametrization of the form ω Þ Ñ pz 1 pωq, . . . , z N pωqq " pω n , ÿ jěn a 2j ω j , . . . , ÿ jěn a N j ω j q Set A :" tj : Di, a ij ‰ 0u and call an exponent j P A tnu an essential integer exponent if gcdti P tnu Y A : i ď ju ă gcdti P tnu Y A : i ă ju. Denote by B the set of essential integer exponents of pγ, 0q.

Genericity criterion ([19, Section 5]) The projection ℓ is generic for the curve germ pγ, 0q if and only if B Ă tj, a 2j ‰ 0u.

We can assume our coordinates are chosen so that pγ, 0q is tangent to the z 1axis and then A Ă tj : j ą nu We consider for e 0 and e the chart over z 1 ‰ 0 so that writing pz 1 , v 2 , v 3 , . . . , v N q the corresponding local coordinates of U 0 and pz 1 , v 2 q that of U 1 , we have: e 0 pz 1 , v 2 , v 3 , . . . , v N q " pz 1 , z 1 v 2 , z 1 v 3 , . . . , z 1 v N q and epz 1 , v 2 q " pz 1 , z 1 v 2 q. Then q is the origin of the local coordinates of U 0 and the strict transform γ ˚of γ by e 0 has the following Puiseux parametrization in the coordinates pz 1 , v 2 , v 3 , . . . , v N q: ω Þ Ñ pω n , ÿ jěn a 2j ω j´n , . . . , ÿ jěn a N j ω j´n q Since B Ă ta 2,j ‰ 0u, the set of essential integer exponents of γ ˚is tj ´n; j P Bu. Since ℓ 0 is given by ℓ 0 pz 1 , v 2 , v 3 , . . . , v N q " pz 1 , v 2 q, then, according to the above genericity criterion, ℓ 0 is generic for pγ ˚, qq.

The proof when pγ, 0q is reducible is essentially the same using the extension of the genericity criterion in [START_REF] Neumann | Lipschitz geometry of complex curves[END_REF]Section 5] taking account of the contact exponents between branches.

Let us now prove that Condition (2) of Definition 2.2 is satisfied. Let π : r X Ñ X be the resolution introduced in Section 3. By definition, it factors through the blow-up e 1 . Consider the map π 1 : r X Ñ X 1 defined by π " e 1 ˝π1 . According to Theorem 5.3, its restriction over pX 1 , pq is a resolution of pX 1 , pq which factors through the normalized Nash transform of pX 1 , pq and the P-curves of pX, 0q and pX 1 , pq over p coincide. Now, take any D Ă Ω, where pℓ D : pX, 0q Ñ pC 2 , 0qq DPΩ denotes the family of generic projections of pX, 0q. Let ℓ 1 D : pX 1 , pq Ñ pC 2 , 0q be the projection defined by e ˝ℓ1 D " ℓ D ˝e1 . We know that the polar curve Π 1 D of ℓ 1 D equals the germ pΠ D , p 1 q where ˚means strict transform by e 1 . Therefore the family of polars pΠ 1 D q DPΩ of projections ℓ 1 D coincide with the family of germs pΠ D , p 1 q DPΩ , which is equisingular in terms of strong simultaneous resolution. This shows that Condition (2) of Definition 2.2 is satisfied for the family pℓ 1 D q DPΩ .

We now prove the "if" direction of Theorem 1.2.

Let pX, 0q be a minimal surface singularity with generic projection ℓ : pX, 0q Ñ pC 2 , 0q. Let ρ : Y Ñ C 2 be the sequence of blow-ups which resolves the base points of the family of curves pℓpΠ D qq DPΩ and let R be its resolution graph. We have to check Conditions p˚1q and p˚2q of Theorem 4.5 for any test curve pγ, 0q associated to a node of R. In fact, we will check these conditions for the ρ-image of a curvette of any irreducible curve in ρ ´1p0q (so any vertex of R, not only nodes). In the proof we say test curve for such a curve even if it correspond to a non node vertex. By Proposition 5.5, the discriminant curve ∆ of ℓ is a union of A n -curves, and ρ : Y Ñ C 2 is the minimal resolution of ∆. We consider the following two cases: Case 1. γ is the ρ-image of a curvette of ρ ´1p0q whose inner rate q γ is an integer n (in particular, γ is smooth); Case 2. γ is the ρ-image of a curvette of ρ ´1p0q such that q γ " n `1{2 with n ě 1 an integer;

Case 1. We will proceed by induction on q γ , so assume first q γ " 1, i.e., γ is a generic line through the origin of C 2 , so pℓ ´1pγq, 0q is a generic hyperplane section of pX, 0q. Since pX, 0q is minimal, the generic hyperplane section pℓ ´1pγq, 0q also a minimal singularity ([12, Lemma 3.4.3]) so it is a union of mpX, 0q smooth transversal curves, where mpX, 0q denotes the multiplicity of pX, 0q. Therefore γ has a single outer rate which equals 1 and Conditions p˚1q and p˚2q are satisfied.

Let n be an integer ě 2. Assume that for any minimal singularity, any test curve with inner rate n ´1 satisfies Conditions p˚1q and p˚2q. Let γ be the ρ-image of a curvette with inner rate q γ " n. We use again the notations ℓ, e, ℓ 1 and e 1 introduced for Proposition 6.1 and we set C " e ´1p0q.

Consider the point q " γ ˚X C, where ˚means strict transform by e. Since n ě 2, the strict transform ∆ ˚contains q. Since ℓ is generic for its polar curve and since e 1 is the blow-up of the origin, then the fiber ℓ 1´1 pqq contains a unique point p which belongs to the strict transform Π ˚of Π by e 1 . Claim 1. γ satisfies condition p˚1q.

Proof. Let σ be a component of ℓ ´1pγq. We have to prove that pσ, 0q is smooth. Assume first that the strict transform σ ˚of σ by e 1 meets E 1 " e 1´1 p0q at a point p 1 P ℓ 1´1 pqq distinct from p. Then p 1 does not belong to the strict transform Π ˚of Π by e 1 . Therefore pX 1 , p 1 q is smooth, p 1 is a smooth point of E 1 " e 1´1 p0q by (i) of Proposition 6.1 and the map germ ℓ 1 : pX 1 , p 1 q Ñ pU 1 , qq is an isomorphism by (ii) of Proposition 6.1. Since γ ˚is a smooth curve transverse to C at q, then pℓ 1´1 pγ ˚q, p 1 q " pσ ˚, p 1 q is a curvette of E 1 . Since pX, 0q is minimal, the multiplicity of a generic linear form on pX, 0q has multiplicity 1 along E 1 . By [11, 1.1], this implies that σ is a smooth curve of pX, 0q.

Assume now that σ ˚X E " p. According to Proposition 6.1, the map germ ℓ 1 : pX 1 , pq Ñ pU 1 , qq is a generic projection of pX 1 , pq. Moreover, its discriminant and polar curves are respectively the strict transform p∆ ˚, qq of p∆, 0q by e and the strict transform pΠ ˚, pq of pΠ, 0q by e 1 . Since γ has rate n ě 2, then its strict transform pγ ˚, qq by the blow-up of 0 is a test curve with inner rate n ´1 (Remark 4.2). Taking pγ ˚, qq as test curve for pX 1 , pq, we apply the induction assumption: pγ ˚, qq satisfies Condition p˚1q, i.e., pγ ˚, qq is a smooth curve on pX 1 , p 1 q. Let π 0 : pX 0 , Eq Ñ pX, 0q be the minimal resolution of pX, 0q. It factors through e 1 . Let π 1 : X 0 Ñ X 1 be the resolution of X 1 such that π 0 " e 1 ˝π1 . Since σ ˚is a smooth curve on pX 1 , p 1 q, then by [11, 1.1], its strict transform σ 2 by π 1 is a curvette at a smooth point of π 1´1 pp 1 q (along which the multiplicity of the maximal cycle is 1). The curve σ 2 is also the strict transform of σ by π 0 , and since the maximal cycle of pX, 0q is reduced, then again by [11, 1.1] (we use here the converse statement) then pσ, 0q is a smooth curve on pX, 0q. Claim 2. γ satisfies Condition p˚2q.

Proof. Let us write ℓ ´1pγq as the union ℓ ´1pγq " η 1 Y η 2 where η 2 is the union of components of ℓ ´1pγq whose strict transforms by e 1 contain p.

Let σ be a component of η 1 and let p 1 " σ ˚X E 1 . The strict transform by e 1 of another component σ 1 of ℓ ´1pγq meets E 1 at a point in ℓ 1´1 pqq different from p 1 . Therefore σ and σ 1 have distinct tangent lines so their contact exponent equals 1. This proves that any component of η 1 has contact exponent 1 with any other component of ℓ ´1pγq.

It remains to prove Condition p˚2q for two components δ 1 and δ 2 of η 2 . Let q 0 the rate associated to δ 1 and δ 2 as in Condition p˚2q. The strict transforms δ 1 and δ 2 of δ 1 and δ 2 by e 1 are two components of the liftings ℓ 1´1 pγ ˚q of the test curve pγ ˚, qq for the surface germ pX 1 , p 1 q, and the rate associated to δ 1 and δ 2 as in Condition p˚2q for pX 1 , p 1 q equals q 0 ´1. We now use the induction assumption: since γ ˚satisfies Condition p˚2q as a test curve of pX 1 , p 1 q with rate n ´1, then q 0 ´1 " n ´1, so q 0 " n. This proves Claim 2. Case 2. We now assume γ is a curvette of ρ ´1p0q with inner rate q γ " n `1{2 where n is an integer ě 1. Then, in suitable coordinates x and y, γ is a curve with a Puiseux expansion of the form y " ax n`1 2 and there is a unique component ∆ 1 of ∆ with same type y " a 1 x n`1 2 `higher order. Let q " γ ˚X C " ∆ 1˚X C as in Case 1, and let Π 1 Ă Π such that ℓpΠ 1 q " ∆ 1 and p " ℓ 1´1 pqq X Π 1˚.

We will proceed again by induction on n, using similar arguments as in Case 1. Assume first n " 1, i.e., q γ " 3{2. Then ∆ 1 and γ are 3{2-cusps, i.e., equisingular to u 2 ´v3 " 0, with contact exponent 3{2. The strict transforms ∆ 1˚a nd γ ˚by e are smooth curves meeting e ´1p0q at the same point q.

Let p 1 P ℓ 1´1 pqq be distinct from p. Then the map germ ℓ 1 : pX 1 , p 1 q Ñ pU 1 , qq is an isomorphism, so the germ pℓ 1´1 pγ ˚q, p 1 q is a smooth curve tangent to E 1 . Therefore, there is a unique component σ of ℓ ´1pγq whose strict transform by e 1 contains p 1 , and it has multiplicity 2. So σ is a plane curve, and since it is smooth after one blow-up, it is a 3{2-cusp. Moreover, a similar argument as in the proof of Claim 1 shows that σ has contact exponent 1 with any other component of ℓ ´1pγq.

Let us now deal with pℓ 1´1 pγ ˚q, pq. According to Theorem 5.4, p is a smooth point of an exceptional curve obtained by blowing-up the intersection point between two exceptional curves of ℓ 1´1 pCq corresponding to a central edge in the resolution graph, and pℓ 1´1 pγ ˚q, pq consists of the strict transform of a component of ℓ ´1pγq which is equisingular to Π 1 . So this component is a cusp, i.e., its unique rate is 3{2. This implies that γ satisfies Conditions p˚1q and p˚2q.

The rest of the induction uses the same arguments as in Case 1. This completes the proof of the "if" direction of Theorem 1.2.

Explicit example of lifting of test curves

The aim of this section is to give in an explicit example a flavor of Conditions p˚1q and p˚2q of Theorem 4.5 in the case of a minimal singularity. We return back to Example 5.7, and we give for some examples of test curves pγ, 0q, the resolution graph of the lifting ℓ ´1pγq on the left and the resolution of its generic projection ℓ 1 pℓ ´1pγqq on the right. Figure 3 is for the test curve given by a generic line.

Rational and Lipschitz normally embedded implies minimal

In this section, we prove the other direction of Theorem 1.2: any rational surface singularity which is Lipschitz normally embedded is minimal. Remark 8.1. A Lipschitz normally embedded surface singularity is not necessarily minimal. A counter-example is given by the (non rational) hypersurface in C 3 with equation xypx `yq `z4 " 0. It is a superisolated singularity. The graph of its minimal resolution factorizing through Nash has four vertices. It consists of a central vertex and three bamboos of length one, these three leaves being the L-nodes, and the central vertex the single P-node. Proof. Let p r X, Eq be the minimal resolution of pX, 0q, Z the minimal cycle and E " Ť E i . The multiplicity of Z at any L-curve is 1, since pX, 0q is Lipschitz normally embedded. Consider Laufer's algorithm for finding Z ([13, Proposition 4.1]), and let E j Ă E be the last curve one adds in the algorithm before one obtains Z. Assume that E j is not an L-curve, so Z ¨Ej " 0. Let Z 1 be the penultimate cycle obtained by Laufer's algorithm. Then Z 1 " Z ´Ej and Z 1 ¨Ej " ´E2 j ą 1 which contradicts pX, 0q being rational by Laufer's criterion [START_REF] Henry | On rational singularities[END_REF]Theorem 4.2]. So the last curve added by Laufer's algorithm is always an L-curve.

One can always run Laufer's algorithm such that each curve is added once, before any curve is added a second time. So unless Z " ř E i there would be an L-curve with multiplicity ą 1, which is a contradiction. Thus pX, 0q is minimal.

p1q α Y δ p2q αb 1 b

 p2q1 , Condition p˚2q implies that the curves ℓ 1 pδ p1q α q and ℓ 1 pδ p2q α q have Puiseux expansions respectively: j pαqx rj `bp1q q0 pαqx q0 `h.o, j pαqx rj `bp2q q0 pαqx q0 `h.o,where ' r 1 ă r 2 ă . . . ă r m ă q 0 , ' b j pαq, j " 1 . . . ,m, b p1q q0 pαq and b p2q q0 pαq depend continuously on α, ' for all α P W, b p1q q0 pαq ‰ b p2q q0 pαq, and where "`h.o." means plus higher order terms.

  Figure 1.

φ 2

 2 p0, tq φ 1 p0, tq φ 2 ps, tq φ 1 ps, tq φ 2 p1, tq φ 1 p1, tq

Figure 2 .

 2 Figure 2.

Figures 4 and 5

 5 are for two test curves which are ρ-images of curvettes of the exceptional curves corresponding respectively to the vertices v 1 and v 2 .

1 Figure 4 .

 14 Figure 3.

v 2 Figure 5 .

 25 Figure 5.