Olivia Zurita-Miranda

Coralie Fourcade-Dutin

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Hussein Mourtada

Camille Plénat

Jet schemes and minimal toric embedded resolutions of rational double point singularities

Keywords: 2010 Mathematics Subject Classification. 14E15, 14E18, 14M25 Embedded Nash problem, Resolution of singularities, Toric Geometry

Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every "minimal embedded toric resolution". We prove that this correspondence is bijective except for the E 8 singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.

Introduction

In this article, we construct embedded resolutions of surfaces having rational double point singularities; also called simple singularities. The word simple refers to the fact that they have no moduli (i.e., a hypersurface singularity with the same topological type of a simple singularity is analytically isomorphic to the simple singularity [LeT]). But at the same time, they are simple from a resolution of singularities point of view, i.e., easy to resolve. The traditional approach to resolve singularities is to iterate blowing ups at smooth centers in order to make an invariant drop. This invariant takes values in a discrete ordered set with a smallest element (which detects smoothness). It should not only detect smoothness, but also be easy to compute so that its behavior can be followed when iterating the blowing ups. In this article, we adopt a different strategy to resolve a simple singularity X ⊂ C 3 : we construct an embedded toric resolution of X ⊂ C 3 . This construction is based on a deep invariant, the set EC (essential components) of irreducible components of the jet schemes satisfying some natural properties. Recall that for m 0, the m-th jet scheme of a variety X parametrizes morphisms Spec C[t]/(t m+1 ) -→ X (see section 2 for details). These are finite dimensional approximations of the space of arcs which parametrize germs of curves drawn on X. Actually the geometry of the m-th jet scheme is intimately related with the geometry of the set of arcs in a smooth ambient space containing X and which have "contact" with X larger than m. This explains in parts why while the arc space of X detect information about abstract resolution of singularities (see [N]), jet schemes detect information about embedded resolution of singularities (see [START_REF] Mourtada | Jet schemes and minimal generating sequences of divisorial valuations in dimension 2[END_REF], [LMR], [ELM]).

The other subject which this article considers is the minimality of embedded resolutions of singularities. In contrast with the abstract resolution case, there is no "universal" minimal embedded resolution for surface singularities. Therefore we need to make this notion more precise. Actually, the embedded resolution of X ⊂ C 3 that we construct is toric and is obtained from a particular regular subdivision of the Newton dual fan Γ associated with X (see section 3 for the definition of Γ). Since simple singularities are Newton non-degenerate (see section 3), regular subdivisions of Γ give toric embedded resolutions of X ⊂ C 3 (see [Ho], [Va], [LJ]). This is equivalent to say that an abstract resolution of singularities of the toric variety Z Γ defined by the fan Γ gives an embedded toric resolution of X ⊂ C 3 . But the toric variety Z Γ is of dimension 3 and hence thanks to a theorem by Bouvier and Gonzalez-Sprinberg [BGS], we know that there exists a toric resolution of singularities of Z Γ where all the irreducible divisors of the exceptional locus are essential, i.e., the centers of the divisorial valuations associated with these divisors give irreducible components of the exceptional locus of any other toric resolution of singularities of Z Γ . We call such a resolution of singularities a minimal toric resolution; in [BGS], it is called G-desingularization; we call a toric embedded resolution of X ⊂ C 3 minimal if it corresponds to a minimal toric resolution of Z Γ . Note that in general, a minimal toric embedded resolution of singularities is not unique, but the divisorial valuations associated with the irreducible divisors of its exceptional locus are the same for all minimal toric embedded resolution of singularities. We say that these last divisorial valuations are embedded essential. We will prove that the toric embedded resolution of X ⊂ C 3 which we construct from jet schemes is minimal for all simple singularities except for E 8 (Note that the E 8 singularity behaves exceptionally also from many other points of view [START_REF] Lê | Combinatorics of rational singularities[END_REF]). In particular we describe a bijection between the set of essential components of the jet schemes of simple singularities (except for E 8 ) and the set of embedded essential divisorial valuations. This can be thought as a solution of an embedded version of the Nash problem.

The choice of this class of singularities is related to the following facts:

First, for m big enough the number of irreducible components of their m-th jet scheme is constant; this simplifies the classification of the irreducible components we are interested in, when m varies. This is not the case in general (see [START_REF] Mourtada | Jet schemes of complex branches and equisingularity[END_REF], [START_REF] Mourtada | Jet schemes of toric surfaces[END_REF]).

Second, these singularities are Newton non-degenerate with respect to their Newton polyhedron, and therefore they have a toric embedded resolution; see [AGS], [O1], [O2]. Defining the class of irreducible components of jet schemes, mentioned above, is more subtle for Newton degenerate singularities [LMR]. This is related to a conjecture of Teissier on embedding any singularity in such way that it can be resolved by a toric morphism (see [T1], [T2], [START_REF] Mourtada | Jet schemes and minimal generating sequences of divisorial valuations in dimension 2[END_REF]).

We would like to thank Shihoko Ishii, Monique Lejeune-Jalabert, Patrick Popescu-Pampu and Bernard Teissier for several discussions about this article.

Jet schemes

In this section, we begin by giving some preliminaries on jet schemes; we then recall from [START_REF] Mourtada | Jet schemes of rational double point singularities, Valuation Theory in Interaction[END_REF] the structure of jet schemes of rational double point singularities and we extract from this stucture some information about particular irreducible components of these jet schemes. These information will be used in the next section to obtain the canonical resolution of these singularities. We will all the cases of simple singularities (see their defining equations below); the case of E 6 (see below the defining equation) has been treated in [START_REF] Mourtada | Jet schemes of rational double point singularities, Valuation Theory in Interaction[END_REF], but we consider it briefly here for the convenience of the reader.

Let k be an algebraically closed field of arbitrary characteristic and X be a k-algebraic variety. For m ∈ N, the functor

F m : k -Schemes -→ Sets Spec(A) -→ Hom k (SpecA[t]/(t m+1 ), X)
where A is a k-algebra, is representable by a k-scheme X m [Is]. X m is the m-th jet scheme of X, and F m is isomorphic to its functor of points. In particular the closed points of X m are in bijection with the k[t]/(t m+1 ) points of X. For m, p ∈ N, m > p, the truncation homomorphism A[t]/(t m+1 ) -→ A[t]/(t p+1 ) induces a canonical projection π m,p : X m -→ X p . These morphisms verify π m,p •π q,m = π q,p for p < m < q, and they are affine morphisms, they define a projective system whose limit is a scheme that we denote by X ∞ . This is the arc space of X. Note that X 0 = X. We denote the canonical projection π m,0 : X m -→ X 0 by π m , and denote by Ψ m the canonical morphisms X ∞ -→ X m .

We now assume that X ⊂ C 3 is defined by one of the following equations:

A n , n ∈ N : xy -z n+1 = 0. D n , n ∈ N, n 4 : z 2 -x(y 2 + x n-2 ) = 0. E 6 : z 2 + y 3 + x 4 = 0 . E 7 : x 2 + y 3 + yz 3 = 0. E 8 : z 2 + y 3 + x 5 = 0 .
We denote by X 0 m := π -1 m,0 (O), where O is the origin of C 3 ; it is the singular locus of X. We will now associate a divisorial valuation over C 3 with the irreducible component of C m ⊂ X 0 m satisfying the property ( ) that for every irreducible component

C m+1 ⊂ X 0 m+1 , if π m+1,m (C m+1 ) ⊂ C m then codim(C m+1 ) > codim(C m ). ( ) For m ∈ N, let ψ a m : C 3 ∞ -→ C 3
m be the canonical morphism, here the exponent a stands for "ambient". For p ∈ N, we consider the following cylinder in the arc space

Cont p (f ) = {γ ∈ C 3 ∞ ; ord t f • γ = p}.
Since ψ a m is a trivial fibration, for every irreducible component C m ⊂ X 0 m , we have that

ψ a m -1 (C m ) ∩ Cont m+1 (f )
is an irreducible component of Cont m+1 (f ), whenever this last intersection is non-empty. But from the property ( ) we have codim(C m+1 ) > codim(C m ), this implies

ψ a m -1 (C m ) ∩ Cont m+1 (f ) = ∅.
Let γ be the generic point of ψ a m -1 (C m ) ∩ Cont m+1 (f ), then for every h ∈ C[x, y, z], we define the map ν Cm : C[x, y, z] -→ N, as follows ν Cm (h) = ord t h • γ.

(1)

It follows from corollary 2.6 in [ELM], that C m is a divisorial valuation (see also [dFEI], [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF]prop. 3.7 (vii)

applied to ψ a m -1 (C m )).
Given m 1, with an irreducible component C m of X 0 m , we associate the following vector that we call the weight vector:

v(C m ) = (ν Cm (x), ν Cm (y), ν Cm (z)) ∈ N 3 .
In the following definition, we consider the irreducible components of X 0 m that will be meaningful for the problem we are considering.

Definition 2.1. The following set of particular irreducible components of X 0 m :

EC(X) := {C m ⊂ X 0 m , m 1 is an irreducible component satisfying ( ) and v(C m ) = v(C m-1 ) for any component C m-1 verifying π m,m-1 (C m ) ⊂ C m-1 }, (2)
is called the set of essential components of X.

We also consider the following set of associated valuations :

EE(X) := {ν Cm , C m ∈ EC(X)},
Where EE stands for Embedded-Essential, to say that these valuations are supposed to appear in every embedded toric resolution.

Note that the definitions of the sets EC(X) and EE(X) are ad-hoc to simple singularities. Indeed, the fact that these singularities are non-degenerate with respect to their Newton polyhedron implies that the meaningful valuations are monomial; this motivates the definition above of the set EC(X). From another point of view, if we check carefully the equations, we will figure out that the components which does not belong to the set EC(X), are at their generic points trivial fibrations above components which belong to EC(X); that is why they are not useful and this phenomenon reflects the fact that the singularities we consider are Newton non-degenerate.

Note also that on one hand we are interested in the embeddings of the singularities in C 3 that are defined by the equations we gave at the beginning of this section; this includes that we consider the variables x, y, z as given and hence the embedding of the torus in C 3 is also given. On the other hand, since we are interested in toric resolutions, the exceptional locus of such a resolution will be defined as the inverse image of the complement of the torus, i.e., of the coordinate hyperplanes. So the fact that the x axis (defined by y = z = 0 in C 3 ) and the y axis (defined by x = z = 0 in C 3 ) belong to the singularities A n , the y axis belongs to the singularities D n and the z axis belongs to the singularity E 7 , suggests that if we want to find a toric resolution, we need also to consider the jets which go through respectively these coordinate axis. But the family of m-th jets whose center is a generic point on one of these axis is irreducible because our surfaces have isolated singularities at the origin. So the components which come from those families are easy to determine; again since our singularities are Newton non-degenerate, we are only interested in the vectors associated with these components, so we only consider such components for small m, whenever the associated vector changes. For m large enough, the associated vector stabilizes.

A careful reading of [START_REF] Mourtada | Jet schemes of normal toric surfaces[END_REF] (and of [START_REF] Mourtada | Jet schemes of complex branches and equisingularity[END_REF]) produces a determination of the set EC for a rational double point (X, 0). We will denote it by EC(X). Let f (x, y, z) ∈ C[x, y, z] be the defining equation of a rational double point singularity X. We write

f ( m i=0 x i t i , m i=0 y i t i , m i=0 z i t i ) = i=m i=0 F i t i mod t m+1 , ( ) then the m-th jet scheme X m of X is naturally embedded in C 3(m+1) = (C 3 ) m = SpecC[x i , y i , z i , i = 0, . . . , m]
and is defined by the ideal I m = (F 0 , F 1 , ..., F m ). From Section 3 in [START_REF] Mourtada | Jet schemes of normal toric surfaces[END_REF] and section 3.2 in [START_REF] Mourtada | Jet schemes of rational double point singularities, Valuation Theory in Interaction[END_REF], we are able to determine the set EC(X). This is the subject of the following lemma.

We do not treat the E 8 singularity (which will be in some sense an exception to part of our formulation) because it takes too much space; we only give the weight vectors associated with the components in EC(E 8 )) in the corollary that follows the lemma.

Lemma 2.2. For an A n singularity, EC(A n ) is given by the components centered at the origin

{V (x 0 , . . . , x l-1 , y 0 , . . . , y m-l , z 0 ), l = 1, . . . , m and m = 1, . . . , n},
and the components centered at the x respectively y axis are {V (y 0 , . . . , y l , z 0 ), l = 0, . . . , n}, respectively {V (x 0 , . . . , x l , z 0 ), l = 0, . . . , n}.

For a D 2n singularity, EC(D 2n ) is given by the components centered at the origin

{X 0 1 = V (x 0 , y 0 , z 1 ), X 0 2 = V (x 0 , y 0 , z 0 , z 1 ), H 2k = V (x 0 , y 0 , z 0 , z 1 , y 1 , y 2 , ..., y k-1 , z 2 , z 3 , ..., z k ), k = 2, . . . , n -1, H 2k+1 = V (x 0 , y 0 , z 0 , z 1 , y 1 , y 2 , ..., y k , z 2 , z 3 , ..., z k ), k = 1, . . . , n -2, L 2k+1 = V (x 0 , x 1 , y 0 , z 0 , z 1 , y 1 , y 2 , ..., y k-1 , z 2 , z 3 , ..., z k ), k = 1, . . . , 2n -1},
and the components centered at the y-axis are V (x 0 , z 0 ), and V (x 0 , x 1 , z 0 ).

For an E 6 singularity we have

EC(E 6 ) = {V (x 0 , y 0 , z 0 ), V (x 0 , y 0 , z 0 , z 1 ), V (x 0 , y 0 , y 1 , z 0 , z 1 ), V (x 0 , x 1 , y 0 , y 1 , z 0 , z 1 , z 2 ), V (x 0 , x 1 , y 0 , y 1 , y 2 , z 0 , z 1 , z 2 , z 3 ), V (x 0 , x 1 , x 2 , y 0 , y 1 , y 2 , y 3 , z 0 , z 1 , z 2 , z 3 , z 4 , z 5 )}.
For an E 7 singularity, EC(E 7 ) is given by the components centered at the origin x 1 ,x 2 ,y 0 ,y 1 ,y 2 ,z 0 ),V (x 0 ,x 1 ,x 2 ,y 0 ,y 1 ,z 0 ,z 1 ),V (x 0 ,x 1 ,x 2 ,x 3 ,y 0 ,y 1 ,y 2 z 0 ,z 1 ),V (x 0 ,x 1 ,x 2 ,x 3 ,x 4 ,y 0 ,y 1 ,y 2 z 0 ,z 1 ),V (x 0 ,x 1 ,x 2 ,x 3 ,x 4 ,y 0 ,y 1 ,y 2 ,y 3 z 0 ,z 1 ),V (x 0 ,x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,y 0 ,y 1 ,y 2 ,y 3 z 0 ,z 1 ,z 2 ),V (x 0 ,x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ,y 0 ,y 1 ,y 2 ,y 3 ,y 4 z 0 ,z 1 ,z 2 ), V (x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 6 , x 8 , y 0 , y 1 , y 2 , y 3 , y 4 z 0 , z 1 , z 2 , z 3 )}, and those centered above the z axis {V (x 0 , y 0 ), V (x 0 , y 0 , y 1 )}.

{V (x 0 , y 0 , z 0 ), V (x 0 , x 1 , y 0 , z 0 ), V (x 0 , x 1 , y 0 , y 1 , z 0 ), V (x 0 , x 1 , x 2 , y 0 , y 1 , z 0 ), V (x 0 ,
From lemma 2.2, we deduce directly the set EE. First recall that a monomial valuation, defined on the ring C[x, y, z] and associated with a vector a = (a 1 , a 2 , a 3 ) ∈ N 3 , is defined by: for

h = i=(i 1 ,i 2 ,i 3 )∈N 3 c i x i 1 y i 2 z i 3 ∈ C[x, y, z] then ν a (h) = min i∈N 3 ;c i =0 a 1 i 1 + a 2 i 2 + a 3 i 3 .
(3)

From lemma 2.2 we deduce:

Proposition 2.3. The valuations that belong to EE(A n ) are monomial. They are associated with the vectors v(C m ) where C m ∈ EC(A n ). These vectors are (l, m -l + 1, 1), l = 1, . . . , m and m = 1, . . . , n (l, 0, 1), l = 1, . . . , n + 1, (0, l, 1), l = 1, . . . , n + 1.

The valuations that belong to EE(D 2n ) are monomial. They are associated with the vectors v(C m ) where C m ∈ EC(D 2n ). These vectors are

U 1 = (1, 1, 1), U 2 = (1, 2, 2), . . . , U n-1 = (1, n -1, n -1), W 1 = (1, 1, 2), W 2 = (1, 2, 3), . . . , W n-1 = (1, n -1, n), V 1 = (2, 1, 2), V 2 = (2, 2, 3), . . . V 2n-2 = (2, 2n -2, 2n -1), W 0 = (1, 0, 1), V 0 = (2, 0, 1).
The valuations that belong to EE(E 6 ) are monomial. They are associated with the vectors v(C m ) where C m ∈ EC(E 6 ) :

(1, 1, 1), (1, 1, 2), (1, 2, 2), (2, 2, 3), (2, 3, 4), (3,4,6) The valuations that belong to EE(E 7 ) are monomial. They are associated with the vectors v(C m ) where C m ∈ EC(E 7 ) :

(1, 1, 0), (1, 2, 0), (1, 1, 1), (2, 1, 1), (2, 2, 1), (3, 2, 1), (3, 3, 1), (3, 2, 2), (4, 3, 2), (5, 3, 2), (5, 4, 2), (6, 4, 3), (7, 5, 3), (9, 6, 4) The valuations that belong to EE(E 8 ) are monomial. They are associated with the vectors v(C m ) where C m ∈ EC(E 8 ) :

(1, 1, 1), (1, 1, 2), (1, 2, 2), (1, 2, 3), (2, 2, 3), (2, 3, 4), (2, 3, 5), (2, 4, 5), (3,4,6) ,(3,5,7),(3,5,8),4,6,8),(4,6,9),(4,7,10),(5,7,11),(5,8,11)(5,8,12),(5,9,13) (5,9,14),(6,10,14),(6,10,15) Proof. Let a = (a 1 , a 2 , a 3 ) ∈ N 3 . Assume that

C m = V (x 0 , . . . , x a 1 -1 , y 0 , . . . , y a 2 -1 , z 0 , . . . , z a 3 -1 ) ⊂ C 3 m is an irreducible component of X m , which belongs to EC(X). Then, a generic point of Ψ a m -1 (C m )∩ Cont m+1 (f ) is of the shape (x a 1 t a 1 +• • • , y a 2 t a 2 +• • • , z a 3 t a 3 +• • • ) where (x a 1 , y a 2 , z a 3 ) is generic. For h ∈ C[x, y, z],
it follows from the definition of ν Cm (h) (see equation ( 1)) and from the definition of ν a (h) (see equation ( 3)) that ν Cm (h) = ν a (h). The proposition follows.

Toric minimal embedded resolutions

In this section, we define minimal toric embedded resolutions, and we give the motivations for this definition.

Let N Z 3 be a lattice and let σ ⊂ N R 3 = N Z R be the strictly convex cone generated by the vectors v 1 , ..., v r , i.e.

σ =< v 1 , . . . , v r >= { r i=1 λ i v i , λ i ∈ R + }. Let f = i=(i 1 ,i 2 ,i 3 )∈N 3 c i x i 1 y i 2 z i 3 be a polynomial in C[x, y,
z] with f (0) = 0. We denote by N P + (f ) its Newton polyhedron at the origin: this is the convex hull in R 3 + of the set

{(i 1 , i 2 , i 3 ) + R 3 + , c i = 0},
where we have supposed that f is given by its Taylor expansion at the origin and let N P (f ) be the union of the compact boundaries of N P + (f ), called Newton boundary. This polyhedron and its boundary depend on the choice of coordinates. Let Γ(f ) be the dual fan of N P (f ); we will call covectors vectors which lie in Γ(f ). A covector defines a linear map on R 3 . For a positive covector L, i.e. with positive coefficients, we define the distance d(L, f ) as the minimal value of the linear map L at x

∈ N P + (f ); let ∆(L, f ) := {x ∈ N P + (f ) : L(x) = d(L, f
)} be the dual face to L. We can then define f L (x) as the restriction of f to the dual face of L, ∆(L, f ); that is

f L (x) = i=(i 1 ,i 2 ,i 3 )∈∆(L,f ) c i x i 1 y i 2 z i 3 .
We can find more details about Newton polyhedra, their dual fans and Newton nondegeneracy in [O2](see also [AGS]).

Definition 3.1. Let f ∈ k[x, y, z] be such that f (0) = 0. We say that the hypersurface {f = 0} is non-degenerate with respect to a covector L if the variety

F * (L) = {u ∈ k * 3 /f L (u) = 0}
is a reduced smooth hypersurface in the torus k * 3 . We say that f is non degenerate at the origin 0 with respect to its Newton Polyhedron N P (f ) if it is non degenerate for each covector.

Recall that a fan Σ is said to be regular if every cone σ ∈ Σ is regular, i.e. the vectors generating σ are part of a basis of Z 3 as a Z-module, or equivalently (for σ a cone maximal dimension), if σ =< v 1 , v 2 , v 3 > then the absolute value | det(v 1 , v 2 , v 3 ) |= 1. A very useful result about toric embedded resolutions of this type of singularities goes as follows.

Theorem 3.2. ( [Va], [O1], [AGS]) We consider a pair X ⊂ C 3 where X = {f = 0} is a Newton non-degenerate singularity, then the folllowing properties are equivalent:

1) A subdivison Σ of Γ(f ) is regular, 2) The proper birational morphism µ Σ : Z Σ -→ C 3 = Spec k[x, y, z]
induced by Σ is an embedded resolution of singularities of the pair. Here Z Σ is the toric variety associated with the fan Σ.

Note that the morphism µ Σ is explicit in term of Σ; more precisely, a regular cone σ =< v 1 , v 2 , v 3 >⊂ Σ of maximal dimension determines an affine chart of Z Σ which is isomorphic to the affine space C 3 = Spec k[q, r, s] and the rectriction of µ Σ to this chart is given by

x = q v 1,1 r v 2,1 s v 3,1 y = q v 1,2 r v 2,2 s v 3,2 z = q v 1,3 r v 2,3 s v 3,3 , where v i = (v i,1 , v i,2 , v i,3
). Moreover, an edge of Σ (a cone of dimension 1) determines an orbit of Z Σ whose Zariski closure defines a divisor on Z Σ which is an irreducible component of the exceptional divisor of µ Σ ( [O]).

And the strict transform of X = {f = 0} is defined as follows:

Definition 3.3. Let Σ be a regular subdivision of Γ(f ). The strict transform of {f = 0} by µ Σ is the Zariski closure of (µ Σ ) -1 (C * 3 ∩ {f = 0}).

We now recall the notion of "essential divisor", which is one of the motivations of this work. For a singular variety, we can find infinitely many resolutions of singularities; the notion of essential divisor searches for intrinsic data in all resolutions of singularities of a given singular variety.

Definition 3.4. (Essential divisors) Let X be a singular variety and π : ( X, E) → (X, sing(X)) be a resolution of singularities of (X, Sing(X). Let E i ∈ E be a divisor which is an irreducible component of the exceptional locus E. We say that E i is an essential divisor if for any other resolution of singularities π : (X , E ) → (X, Sing X), the center on X of the divisorial valuation determined by

E i is an irreducible component of E . A divisor E i is said inessential if it is not essential.
It should be noted that the notion of essential divisor given above is associated with abstract resolutions of singularity and not with embedded resolutions of singularities. For surface singularities, a divisor is essential if it appears on the minimal resolution of singularities. Note that a minimal resolution does not exist in general for a singular variety of dimension larger than 2.

For normal toric singularities, it is possible to determine essential divisors. For that we need the following definition of a minimal system of generators of a cone. Let N ∼ = Z n be a lattice and let N R be the real vector space N ⊗ Z R. Definition 3.5. ( [BGS], see also [AM]) Let σ ⊂ N R be a strongly convex rational polyhedral cone. The minimal system of generators of σ is the set:

G σ = {x ∈ σ ∩ N \0 | ∀n 1 , n 2 ∈ σ ∩ N, x = n 1 + n 2 ⇒ n 1 = 0 or n 2 = 0}
An element in G σ is also called irreducible. It is primitive by definition.

It follows from propostion 1.3 in [BGS], that the elements of G σ appear on every regular fan which subdivides σ as an extremal vector, i.e. a primitive generator of a 1-dimensional cone of the fan. Note that such a subdivision determines a (toric) resolution of singularities of the toric variety defined by σ, and dimension 1 cones determine the irreducible components of the exceptional divisors. This gives a feeling for the following theorem which characterizes essential divisors on toric varieties.

Theorem 3.6. ( [BGS], [IK]) Let σ ⊂ N R be a strongly convex rational polyhedral cone. The minimal system of generators G σ is in bijection with the set of essential divisors of the toric variety V σ . Therefore a primitive vector is in G σ if and only if it is an extremal vector of any regular subdivision of the cone σ.

Actually, the fact that the divisors which correpond to elements of G σ are toric essential (i.e. essential for toric resolutions of singularities) follows from [BGS] and the fact that they are essential for all resolutions of singularities follows from [IK]. It also follows from [BGS] that for a three dimensional toric variety defined by a cone σ there exists a toric resolution of singularities which is associated with a subdivision of σ all of whose 1-dimensional cones are rays associated to elements of G σ . Such a resolution is called a G σ -resolution. We think of such a resolution as a minimal toric resolution. Note that such a resolution does not exist in general for normal toric varieties of dimension larger than 3. Together with theorems 3.2 and 3.6, this discussion motivates the following two definitions: Definition 3.7. (Toric embedded-essential divisors) Let X ⊂ C 3 where X = {f = 0} is a Newton non-degenrate singularity. A toric embedded essential divisor E is a divisor which appears on every toric resolution of singularities of X(such a resolution exists thanks to theorem 3.2) as an irreducible component of the exceptional divisor. This is equivalent to say that E corresponds to an element of G σ for some cone σ ∈ Γ(f ) of maximal dimension. Definition 3.8. (Minimal toric embedded resolution) Let X = {f = 0} ⊂ C 3 be a Newton nondegenerate singularity at the origin. A toric embedded resolution of X ⊂ C 3 which is associated with a subdivision Σ of Γ(f ) is minimal if the only 1-dimensional cones which appear in Σ are determined by elements of G σ for some cone σ ∈ Γ(f ) of maximal dimension and the 1-dimensional cones of Γ(f ). This is equivalent to say that all the irreducible components of the exceptional divisor of a minimal toric embedded resolution are toric embedded essential divisors.

Note that these definitions make sense also for non-isolated singularities as in [ACT] for instance.

Toric minimal embedded resolutions for simple singularities

In this section, we construct toric embedded resolutions from the data of the jet schemes of simple singularities that we gave in section 2. We also prove that for all singularities of this type, except for the E 8 singularity, the essential components of the jet schemes (introduced in section 2) correspond to toric embedded-essential divisors. We will treat each type of singularity separately: first we construct a toric resolution of singularities using the weight vectors associated in section 2 with essential components, then we construct a regular subdivision of the dual fan of the singularity embedded in C 3 , and finally by using a theorem from [BGS] (see also [AM]) which characterizes toric essential divisor for toric varieties, we will prove for simple singularities (except for the singularity of type E 8 ) the minimality of our toric embedded resolution of singularities.

Remark 4.1. Instead of drawing the dual fan of each polyhedron in R 3 (with coordinates x, y, z) we use the trace of the dual fan on the hyperplane defined by the equation x + y + z = 1. Vectors are then represented by points. We call this trace the Newton face or simply, with some abuse of notation dual fan.

The A n singularities

Theorem 4.2. The weight vectors of A n give a toric minimal embedded resolution of the singularity. In particular the set of essential components EC(A n ) is in bijection with the divisors which appear on every toric minimal embedded resolution of these singularities.

The Newton polyhedron and its dual fan are the following:

(1,1,O) x y z (1,0,0) (0,1,0) (n+1,0,1) (0,n+1,1) (0,0,1) ((0,0,n+1)
The dual fan is not simplicial; a resolution of this singularity is given by a regular and simplicial subdivision of the Newton dual fan. The set EE(A n ) of valuations, found after a careful reading of [START_REF] Mourtada | Jet schemes of rational double point singularities, Valuation Theory in Interaction[END_REF], is (l, m -l + 1, 1), l = 1, . . . , m and m = 1, . . . , n (l, 0, 1), l = 1, . . . , n + 1, (0, l, 1), l = 1, . . . , n + 1.

(see lemma 2.2 and proposition 2.3). We reorder this set in the following way in order to study them more easily (see the remarks below):

4. Homogeneous vectors (α, β, γ) such that α + β + γ = k, for 3 k n + 2 are also on the same line.

All these remarks show us a way to obtain an almost symmetric resolution from the vectors. Many algorithms work, we propose one of them to the reader.

ALGORITHM

First place (1, 1, 1) and make the triangle ((0, 0, 1), (n + 1, 0, 1), (0, n + 1, 1)) simplicial. Then place successively the vectors (k, 0, 1), . . . (1, 0, 1) , joining each new vector to (1, 1, 1), to make the resolution simplicial. Do the same symmetrically with (0, k, 1), . . . , (0, 1, 1). One has

(k, 0, 1) = k n + 1 (n + 1, 0, 1) + n + 1 -k n + 1 (0, 0, 1)
Thus these vectors are primitive vectors.

For A 2n : Place the vectors (n, 1, 1) and (1, n, 1) and the corresponding edges. One can remark that the vectors (k, 1, 1) lie on the line (n, 1, 1), (1, 1, 1)). It is at this point that the resolution is not symmetrical: if one chooses (n, 1, 1), then one makes the trapezium simplicial by adding the edges ((n, 1, 1), (0, 1, 0)) and ((n, 1, 1), (1, 0, 0)). So in the toric resolution, one will have the edges joining the vectors on ((n + 1, 0, 1), (0, n + 1, 1)) to (0, 1, 0). Conversely, if one chooses (1, n, 1), one will have the edges joining the vectors on ((n + 1, 0, 1), (0, n + 1, 1)) to (1, 0, 0). For A 2n+1 , one had [( n+1 2 , n+1 2 , 1)] first, the trapezium thus becomes divided into three parts and the resolution become symmetrical.

Then add alternatively (2, 2, 1), (n -1, 2, 1), (2, n -1, 1), (3, 3, 1) . . . ( n+1 2 , n+1 2 , 1),and if n is even ( n+1 2 + 1, n+1 2 , 1),( n+1 2 , n+1 2 + 1, 1) . Then add the remaining vectors in the natural order, with the corresponding edges.

(3,1,1)

(5,0,1) (0,5,1)

(1,1,1) (4,0,1)

(3,0,1)

(2,0,1)

(1,0,1) (0,1,1) (0,2,1) Proof.

(0,3,1) (0,4,1) (1,0,0) !0,1,0) (0,0,1) ((4,1,1) (1,4,1) (3,2,1) (2,3?1) (2,2,1) (1,2,1) (2,1,1) (1,3,1)
• Let us look at triangles [(1, 1, 1), (k, 0, 1), (k + 1, 0, 1)] and [(1, 1, 1), (0, k, 1), (0, k + 1, 1)] for 1 k n, we have:

1 0 0 1 k k + 1 1 1 1 = -1 and 1 k k + 1 1 0 0 1 1 1 = 1
• We also have the triangles [(1, 0, 1), (1, 1, 1), (0, 0, 1)] et (0, 1, 1), (1, 1, 1), (0, 0, 1)] of determinant 1.

• Now, we look at

[(k, n -k, 1), (k -1, n -(k -1), 1), (0, 1, 0)]. We have k k -1 0 n -k n -(k -1) 1 1 1 0 = 1 If n is even, we have the triangles t [( n+1 2 -1, n+1 2 , 1), ( n+1 2 , n+1 2
+1, 1), (0, 1, 0)] and if n is odd, we have the triangles [(

n+1 2 + 1, n+1 2 , 1), ( n+1 2 , n+1 2 , 1), (0, 1, 0)] et [( n+1 2 , n+1 2 - 1, 1), ( n+1 2 , n+1
2 , 1), (0, 1, 0)] with determinant 1.

• The cones [e 1 , (n + 1, 0, 1), (n, 1, 1)] and [e 1 , e 2 , (n, 1, 1)] are non singular.

• Finally we have to look at the triangles

[(k + l, k, 1), (k + l -1, k, 1), (k + 1, k + 1, 1)] and [(k + l, k, 1), (k + l -1, k, 1), (n + 2 -k, k -1, 1)] for all 1 k n+1 2
and k l n -1(and to the symmetrical one too). One has

k + l k + l -1 k + 1 k k 1 1 1 1 = 1 and k + l k + l -1 n + 2 -k k k k -1 1 1 1 = 1
Now we have to prove that in fact this resolution is minimal or a G-resolution (according to C. Bouvier and G. Gonzalez-Sprinberg). First of all, the weight vectors (l, m -l + 1, 1), l = 1, . . . , m and m = 1, . . . , n are clearly primitive as their third component is 1. Consider the simplicial cone generated by e 3 , (n + 1, 0, 1), (0, n + 1, 1); all the weight vectors belong to this cone. Let us give the same names to the points in R 3 which are the end of the vectors starting from 0 and the vectors themselves.

Lemma 4.3. The points in R 3 {e 3 , (n+1, 0, 1), (0, n+1, 1), (l, m-l+1, 1), l = 1, . . . , m and m = 1, . . . , n} belong to the face (e 3 , (n+1, 0, 1), (0, n+1, 1)) of the thetahedron [0, e 3 , (n+1, 0, 1), (0, n+ 1, 1)]. This implies that the set {e 3 , (n+1, 0, 1), (0, n+1, 1), (l, m-l+1, 1), l = 1, . . . , m and m = 1, . . . , n} is free over Z, i.e. none of the vectors is reducible into other vectors of this set.

Suppose that the lemma is proved. Then the only primitive vectors that could generate the characteristic vectors should be vectors inside the tetrahedron [0, e 3 , (n+1, 0, 1), (0, n+1, 1)]. Suppose that there is such a vector. By th. 1.10 of [BGS], it corresponds to an essential divisor in the toric resolution of the cone [e 3 , (n + 1, 0, 1), (0, n + 1, 1)], i.e., it appears in every regular subdivision as an extremal vector. But the vectors (l, m -l + 1, 1), l = 1, . . . , m and m = 1, . . . , n}, we get such a regular subdivision, so there is no primitive vector in the tetrahedron [0, e 3 , (n + 1, 0, 1), (0, n + 1, 1)].

Let us prove the lemma:

Proof. The equation of the face (e 3 , (n + 1, 0, 1), (0, n + 1, 1)) is z -1 = 0. Clearly all the points {e 3 , (n + 1, 0, 1), (0, n + 1, 1), (k, n -k, 1), (k -1, n -(k -1), 1), (0, 1, 0), (l, m -l + 1, 1), l = 1, . . . , m and m = 1, . . . , n} belongs to it.

The D n singularities

Theorem 4.4. The weight vectors of D n give a "canonical" toric minimal embedded resolution of the singularity. In particular the set of essential components EC(D n ) is in bijection with the divisors which appear on every toric minimal embedded resolution of these singularities.

Proof. Let us do it for D 2n .

Let us recall what are the Newton polyhedron for D 2n and the associated dual cone: First of all, to obtain a toric resolution of singularities of D 2n with equation z 2 + x(y 2 + x 2n-2 ) = 0, as it is not commode, one has to blow up the y axes, i.e. to add the vector (1, 0, 1). Let us now consider the weight vectors and show some of their properties:

(2n 1,0,0) (2,0,1) (2,2n 2,2n 1) (1,0,0) (0,1,0) (0,0,1) (0,0,2) (1,2,0)
• The vectors U i = (1, i, i) (1 i n -1) are primitive and belong to the cone C 2 . More precisely they lie onto the line (e 1 , (0, 1, 1)):

In fact we have the following equalities:

U i = 2n -2i -1 2n -1 e 1 + i 2n -1 e 2 + i 2n -1 (2, 2n -2, 2n -1)
and if we note U 0 = (1, 0, 0) , then for 1 i n -1,

U i = U i-1 + (0, 1, 1)
• The vectors W i = (1, i, i + 1) (1 i n -1) are primitive and belong to the cone C 3 . More precisely they lie onto the line ((1, 0, 1), (0, 1, 1)): In fact we have the following equalities:

W i = n -1 2n -2 e 3 + n -i -1 2n -2 (2, 0, 1) + i 2n -2 (2, 2n -2, 2n -1)
and if we note W 0 = (1, 0, 1) , then for 1 i n -1, W i = W i-1 + (0, 1, 1)

• The vectors V i = (2, i, i + 1) (1 i 2n -2) are primitive and belong to the cones C 1 and C 3 . More precisely they lie onto the line ((2, 0, 1), (0, 1, 1)):

In fact we have the following equalities:

V i = 2n -2 -i 2n -2 (2, 0, 1) + i 2n -2 (2, 2n -2, 2n -1)
and, if we note V 0 = (2, 0, 1) , then for 1 i < 2n -2, V i = V i-1 + (0, 1, 1).

CLAIM: the simplicial decomposition of the dual Newton cone of D 2n obtained by adding, the edge [e 1 , (2, 2n-2, 2n-1], the vector (1, 0, 1), then the vectors U 1 to U n-1 in the natural order, the vectors W 1 to W n-1 in the natural order and the vectors V 1 to V 2n-2 in the natural order too, give a toric minimal embedded resolution of D 2n . The resolved dual cone is of the form:

(2,2n-2,2n-1) (0,1,0)

(2,0,1)

(1,0,1) (0,1,1)

(1,0,0) (0,0,1)

V2 V1 V3 U1 U2 U3 W1 W2 Figure 4 : Resolution of D 2n
Proof of the claim: by the previous observations, the vectors U i , V i , W i above are primitive; thus it remains to prove that the cones obtained are non singular, i.e. they have determinant +1 or -1 and using the same proof as for A n , we prove then that the vectors are irreducible.

We add first the vector (1, 0, 1), which corresponds to the blowing up of the y-axes. Then, the cone C 3 is decomposed into two cones : C 31 = ((1, 0, 1), (2, 0, 1), (2, 2n -2, 2n -1)) and C 32 = ((1, 0, 1), e 3 , (2, 2n -2, 2n -1)) . By the above observation, we have now that each group of characteristic vectors are in different cones.

• Decomposition of the cone C 1 :

Let us note U 0 = e 1 . We add the vectors from U 1 to U n-1 and show by induction that we obtain at the end regular cones. At each step the cone where lie the new A i is decomposed into three cones:

C31

(2,0,1)

(2,2n 2,2n 1)

(1,0,0) (0,1,0) (0,0,1)

(2,1,2)

(1,1,2)

(1,0,1) (0,1,1)

(1,1,1)

C1

Figure 5 : Resolution of D 4 step by step (U i , U i-1 , e 2 ), ((U i , U i-1 , (2, 2n -2, 2n -1)) and (U i , e 2 , (2, 2n -2, 2n -1). That last one contains U i+1 .

1 1 0 i i -1 1 i i -1 0 = 1, 1 1 2 i i -1 2n -2 i i -1 2n -1 = -1, 1 2 0 i 2n -2 1 i 2n -1 0 = n -1 -2i
That last determinant is not equal to +1 or -1, so we add U i+1 and so on. For i = n -1 the last determinant is equal to one.

• Decomposition of the cone C 31 :

Let us note W 0 = (1, 0, 1). We add the vectors from W 1 to W n-1 and show by induction that we obtain at the end regular cones. At each step the cone where lie the new W i is decomposed into three cones: (W i , W i-1 , e 3 ), ((W i , W i-1 , (2, 2n -2, 2n -1)) and (W i , e 3 , (2, 2n -2, 2n -1). That last one contains W i+k for k 1.

We have :

1 1 0 i i -1 0 i + 1 i 1 = -1, 1 1 2 i i -1 2n -2 i + 1 i 2n -1 = 1, 1 2 0 i 2n -2 0 i + 1 2n -1 1 = 2n-2-2i
That last determinant is not equal to +1 or -1, so we add A i+1 and so on. For i = n -1 the last determinant is equal to one.

• Decomposition of the cones C 32 and C 2 :

Let us note V 0 = (2, 0, 1). We add the vectors from V 1 to V 2n-2 and show by induction that we obtain at the end regular cones. At each step the two cones where lie the new C i are decomposed into two cones themselves: (V i , V i-1 , e 1 ), (V i , V i-1 , (1, 0, 1)) and (V i , e 3 , (2, 2n -2, 2n -1), (V i , (1, 0, 1), (2, 2n -2, 2n -1). Those last ones contain V i+1 and are not regular.

2 2 1 i i -1 0 i + 1 i 0 = 1 16 2 2 1 i i -1 0 i + 1 i 1 = 2 2 1 i 2n -2 0 i + 1 2n -1 0 = 2ni -i -2ni + 2i -2n + 2 = i -2n + 2 and 2 2 1 i 2n -2 0 i + 1 2n -1 1 = 2n -2 -i
For i = 2n -3 the two last determinants are equal to +1 or -1.

Moreover we have obtained a G-resolution:

As for A n singularity, one can show that W i (resp.U i and V i ) belongs to the base of the cone (e 3 , (2, 0, 1), (2, 2n-2, 2n-1)) (resp. (e 1 , (2, 0, 1)(2, 2n-2, 2n-1))∩(e 3 , (2, 0, 1), (2, 2n-2, 2n-1)), and (e 1 , e 2 , (2, 2n-2, 2n-1)) , whose equation is y-z +1 = 0 (resp. y-z +1 = 0; x+y-z -1 = 0 and x + y -z -1 = 0). With these vectors we obtain a regular subdivision of each cone, which implies that there are no other primitive vectors inside each cone and that the characteristic vectors are irreducible. So we also obtain a G-resolution for D 2n .

Remark: the fact that each vector "belongs" to the base of the cone is equivalent to saying that the volume of the subdivisions stay the same during the process of the resolution.

4.

3 The E 6 singularity, according to [START_REF] Mourtada | Jet schemes of normal toric surfaces[END_REF] Now consider the singularity E 6 given by the equation z 2 + y 3 + x 4 = 0. A simplicial dual Newton polyhedron associated to it is given on figure 6.

Theorem 4.5. The set of weight vectors EE(E 6 ) is (1, 1, 1), (1, 1, 2), (1, 2, 2), (2, 2, 3), (2, 3, 4), (3,4,6) Call them in the lexicographic order. The weight vectors of E 6 give a "canonical" toric minimal embedded resolution of the singularity. In particular the set of essential components EC(E 6 ) is in bijection with the divisors which appear on every toric minimal embedded resolution of these singularities. Proof. The vectors are in different subcones of the dual fan. To make the fan regular, we divide each cone using the vectors. So we have to look first at the positions of each vector. One has :

{(1, 1, 1), (1, 2, 2), (2, 2, 3), (2, 3, 4)} ⊂ C 1 } {(2, 3, 4), (1, 2, 2)} ⊂ C 2 } {(1, 1, 2), (2, 2, 3)} ⊂ C 3 }
As for the previous singularities, one can prove that adding the weight vectors in each cone (in the alphabetic order, see fig. 6), gives a minimal resolution for each of them. See fig. 6 for the picture.

The E 7 singularity

Now consider the singularity E 7 given by the equation x 2 + y 3 + yz 3 = 0. A simplicial dual Newton polyhedron associated to it is given on figure 7.

Theorem 4.6. The set of weight vectors is (1, 1, 0), (1, 2, 0) = O 2 , (1, 1, 1), (2, 1, 1), (2, 2, 1), (3, 2, 1), (3, 3, 1), (3, 2, 2), (4, 3, 2), (5, 3, 2), (5, 4, 2), (6, 4, 3), (7, 5, 3), (9, 6, 4)

= O 1
Call them in the lexicographic order (except O 1 and O 2 ).

The weight vectors of E 7 give a "canonical" toric minimal embedded resolution of the singularity. In particular the set of essential components EC(E 7 ) is in bijection with the divisors which appear on every toric minimal embedded resolution of these singularities. 7 for denomination of the cones). To make the fan regular, we divide each cone thanks to the vectors. So first we have to look at the positions of each vector. One has :

{(1, 1, 1), (2, 2, 1), (3, 2, 2), (4, 3, 2), (6, 4, 3)} ⊂ C 1 } {(2, 1, 1), (3, 2, 2), (5, 3, 2), (6, 4, 3)} ⊂ C 2 } {(1, 1, 0), (3, 2, 1), (3, 3, 1), (5, 3, 2), (5, 4, 2), (7, 5, 3)} ⊂ C 3 } {(3, 3, 1), (5, 4, 2), (7, 5, 3)} ⊂ C 4 } As for the previous singularities, one can prove that adding the previous vectors in each cone (in the lexicographic order), gives a minimal resolution for each of them. See fig. 7 for the picture.

The E 8 singularity

Now consider the singularity E 8 given by the equation z 2 + y 3 + x 5 = 0. A simplicial dual Newton polyhedron associated to it is given on figure 8.

Proposition 4.7. The weight vector gives an embedded resolution of E 8 but this resolution is not minimal.

Proof. Recall that the set of weight vectors are in this case :

(1, 1, 1), (1, 1, 2), (1, 2, 2), (1, 2, 3), (2, 2, 3), (2, 3, 4), (2, 3, 5), (2, 4, 5), (3, 4, 6), (3, 5, 7), (3, 5, 8), 4, 6, 8), (4, 6, 9), (4, 7, 10), (5, 7, 11), (5, 8, 11)(5, 8, 12), (5, 9, 13) (5, 9, 14), (6, 10, 14), (6, 10, 15) One can show that we get a regular subdivision of each cone thanks to the following characteristic vectors

(1, 1, 1), (1, 1, 2), (1, 2, 2), (1, 2, 3), (2, 2, 3),

(2, 3, 4), (2, 3, 5), (2, 4, 5), (3, 4, 6), (3, 5, 7), (3, 5, 8), (4, 6, 9), (4, 7, 10), (5, 8, 12), (6, 10, 15) = O (We call them in the lexicographical order on fig. 8) Moreover they are irreducible (one can show it in the same way as above), so they give subdivision which induces a minimal embedded toric resolution for E 8 . In fact we have

• (4, 6, 8) = (2, 3, 4) + (2, 3, 4)

• (5, 7, 11) = (4, 6, 9) + (1, 1, 2)

• (5, 8, 11) = (4, 7, 10) + (1, 1, 1)

• (5, 9, 13) = (3, 5, 8) + (2, 4, 5)

• (6, 9, 13) = (3, 4, 6) + (3, 5, 7)

• (5, 9, 14) = (3, 4, 6) + (3, 5, 8)

• (6, 10, 14) = (3, 5, 7) + (3, 5, 7)

4.6 A remark on abstract resolution of singularities of simple singularities

For a simple singularity, the restriction of a minimal embedded resolution to the strict transform of the singularitiy gives its minimal abstract resolution; note that not all the divisors that appear on the minimal embedded resolutions meet the strict transform, only those which corresponds to weight vectors which belong to the two dimensional cones of the Newton dual fan. Moreover, these last divisors may meet the strict tansform along more than one irreducible component.

Figure 2 :

 2 Figure 2 : Resolution of A 4 step by step

Figure 3 :

 3 Figure 3 : Newton polyhedron of D 2n and its dual coneFirst we make the dual polyhedron simplicial by adding the edge [e 1 , (2, 2n-2, 2n-1]. Let us call C 1 be the cone [e 1 , (2, 0, 1), (2, 2n -2, 2n -1)] , C 2 be the cone [e 1 , e 2 , (2, 2n -2, 2n -1)], C 3 be the cone [e 3 , (2, 0, 1), (2, 2n -2, 2n -1)] , C 4 be the cone [e 2 , e 3 , (2, 2n -2, 2n -1)] .

Figure 6 :

 6 Figure 6 : Newton dual polyhedron of E 6 and its resolution

Figure 7 :

 7 Figure 7 : Newton dual polyhedron of E 7 and its resolution

Figure 8 :

 8 Figure 8 : Newton dual polyhedron of E 8 and its resolution

This research was partially supported by the ANR-12-JS01-0002-01 SUSI. The second author is supported by Labex Archimède (ANR-11-LABX-0033) and the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government programme managed by the French National Research Agency (ANR)", and by the CLVA.

EE(A n ) := {(n, 0, 1), . . . (1, 0, 1), (0, n, 1), . . . , (0, 1, 1), (1, 1, 1), (2, 1, 1), . . . , (n, 1, 1), (1, 2, 1) . . . (1, n, 1), (2, 2, 1), (3, 2, 1) . . . (n -1, 2, 1),

The weight vectors are positioned on the following figure :

(1,2,1)

A 4 A_5

(5,0,1) (4,0,1)

(3,0,1)

(2,0,1)

(1,0,1) (0,1,1) (0,2,1)

(2,2,1) CLAIM: these vectors are irreducible, i.e. they represent essential components in the resolution. Before proving this claim, we study the position of the vectors in the Newton face of dual fan, and show that they provide a toric embedded resolution.

Some remarks on the vectors of EE(A n ):

(with abuse of notation, the vectors are considered as points on the Newton face of the cone [(e 3 , (n + 1, 0, 1), (0, n + 1, 1)]).

1. We first prove that the vectors are rational sums of e 1 , e 2 , e 3 , v 1 = (n + 1, 0, 1), v 2 = (0, n + 1, 1). By the symmetry of the singularity, we only have to look at half of the vectors: we have :

for all 1 k n and 1 l n -k + 1.

Moreover (k, 0, 1) = k n+1 v 1 + n n+1 e 3 . This implies that they all belong to the cone (e 3 , (n + 1, 0, 1), (0, n + 1, 1)).

2. One can remark the symmetry we obtain. It will follow that we have to do only half of the computations;

3. The vectors (1, 1, 1) . . . (n, 1, 1) are on the line ((1, 1, 1), (1, 0, 0)) as (k, 1, 1) = (1, 1, 1) + (k -1)(1, 0, 0). The same holds for (k, k, 1) . . . (n + 1 -k, k, 1) for k n+1 2

;


A Labourt-Ibarre

C Voisin

G Cassabois

C Delalande

C Flytzanis

P Roussignol

P Beauvillain

Ultrafast electron thermalization in a magnetic layered Au/Co/Au film

Keywords:

The ultrafast electron dynamics subsequent to a hundred femtosecond photo-excitation is monitored in a few nanometer thick Au/Co/Au film by means of time-resolved transient spectroscopy. We show that the insertion of the thin Co layer in the Au film leads to a drastic modification of the scattering rates of the different thermalization regimes. The electron-electron interactions are dominated by the mobile sp-electrons scattering on the heavy d-electrons, which concomitantly enhances the electron-phonon interactions by up to a factor of 5. In the framework of ultrafast magnetization dynamics, this study provides important data on the electronic processes underlying the spin dynamics in a sample typically designed for magneto-optical applications.

I. INTRODUCTION

In recent years a number of studies in multilayered nanostructured magnetic devices have paved the way to a wealth of applications. As an outstanding example in the context of information storage technology, most of today's hard drive reading heads are based on the Giant Magneto-Resistance effect (GMR) which was initially observed in a sample made of an alternation of metallic and magnetic nanoscale layers [1]. From a fundamental viewpoint these structures have also allowed important findings on the working of magnetism in a regime comprized between atomic and macroscopic scales.

In connection to information storage technology and to the reading and writing speed limiting processes, a research field has emerged in the last decade which is devoted to ultrafast spin dynamics, namely the possibility of changing and reading the magnetization state of a sample on a sub-picosecond time scale [2][3][4][5][6]. Such a femtosecond demagnetization is much faster than any other process observed so far and in particular much faster than the usual process involving the application of an external magnetic field to drive the magnetization. Although the ultrafast demagnetization effect has been confirmed by several teams using various techniques, some issues related to the microscopic mechanisms involved are still controversial. However it appears that the key role is played by the intricate interactions between the hot electrons, the electron bath and the phonons at a subpicosecond time scale. Because of their relevance in a number of issues and applications it is important to delineate the underlying processes and assess their impact.

The electron dynamics can be accessed by means of ultrashort light pulse techniques where the system is brought off equilibrium by a sudden photo-excitation and the ensuing evolution of the electron population is monitored by a delayed probe pulse. In the case of noble metals, earlier studies have allowed to map out the athermal regime and the onset of the thermalization below a few hundreds of femtoseconds and beyond respectively [7][8][9][10][11]. An important finding in these single type spelectron systems is the retardation of the intra-electronic thermalization because of the Pauli exclusion principle in a nearly degenerate Fermi liquid and its entanglement with the electron-phonon processes [10][11][12][13][14].

Here we address this issue of the ultrafast electron dynamics in a nanostructured layered sample -such as those of interest for information technology-made of an alternation of Co and Au layers. We show that although the linear optical properties of the sample are almost identical to the ones of pure gold, its electron dynamics is dramatically affected by the presence of the thin Co layer. Both the electron-electron and electron-phonon interactions are enhanced by a factor of five which we interpret in the light of the peculiar electronic properties of transition metals and the specific characteristics of 3d-electrons. We measure the average electron thermalization times as a function of the Co layer thickness and show that no interface effect is observed and that the dynamics can be modeled as if the sp-electrons were completely delocalized over the whole structure and interacting with 3d-electrons acting as scattering centers in many respects similar to external impurities.

II. SAMPLE

The sample consists of a 28 nm thick layer of c.f.c. gold grown in the (111) direction covered by a layer of (0001) h.c. cobalt and capped by 15 atomic monolayers (ML) of gold [15]. The cobalt layer has a step-like structure, each step being 1.6 mm wide and the thickness of the cobalt layer increasing by one atomic monolayer for each step. The thickness of the cobalt layer spreads from 4 up to 12 atomic monolayers. We also use a film of gold on a glass substrate, the thickness of the former being equivalent to the total thickness of the layered sample (31 ±1 nm).

III. EXPERIMENTAL

The pump-probe setup is based on an amplified Ti:Sapphire laser delivering pulses at a central wavelength of 800 nm with a duration of about 250 fs and a repetition rate of 250 kHz. Part of the beam is used as a pump whereas the other part is sent to an homemade Optical Parametric Amplifier which is tunable from 500 nm through 700 nm and serves as a probe. The overall temporal resolution of the setup is about 350 fs (cf. Fig. 2). The focused spots have a diameter of 40 µm for the probe and 50 µm for the pump, ensuring that we probe an almost homogeneously excited region. We record simultaneously the changes of transmission and reflexion of the sample as a function of the time delay using standard synchronous detection methods involving mechanical chopping of the pump beam and lock-in amplification. Part of the probe beam is picked up before the sample and serves as a reference in order to counterbalance the laser intensity fluctuations leading to a highly reduced noise level. This allows us to achieve detection of relative changes of transmission (∆T /T ) and reflexion (∆R/R) as low as 10 -4 .

As can be inferred from the skin depths, 15 nm in gold and 5 nm in cobalt at a wavelength of 800 nm, and the values of ′′ Au and ′′ Co [16,[START_REF]CRC Handbook of Chemistry and Physics[END_REF], the same amount of energy is homogeneously deposited in the Co and Au layers but with appropriately different densities. With prevailing electron mean free paths, 30 nm in gold and 5 nm in cobalt, i.e. much larger than the thickness of the cobalt layer, and Fermi velocities in the range of 10 6 m.s -1 , the excitation can be taken homogeneous over the whole sample thickness after a few tens of femtoseconds with a density of the order of 100 J/cm -3 .

IV. MODELING

For the analysis of the experimental results the usual application of electromagnetic wave propagation in stratified dielectric media cannot be transposed as such in the present case [START_REF] Visnovsky | [END_REF]. There, the dielectric response being local, the optical coefficients of the different strata are immune to the interfacing. In contrast, in the present case of a metallic structure, the electron are highly delocalized. Taking into consideration the electron characteristics in these two metals (same elementary unit cell volume and sp-electron densities), we adopt an embedment approach of the magnetic metal layer into the nonmagnetic ones and assume that all metal layers possess indistinguishable sp-electrons uniformly extending over the whole composite film with an intercalated layer of delectrons with markedly different characteristics and that essentially act as scattering centers. Since the sp-electron delocalization length is much larger than the layer thickness, we assume an effective Drude-like dielectric contribution for the composite medium. In addition to the above free-electron (intraband) Drude-like contribution, there is a weaker bound-electron (interband) contribution. Practically we take the thickness averaged dielectric contributions from the different layers to account for the effective dielectric function of the sample. It almost reduces to that of pure gold but for the scattering rate which is significantly enhanced by the Co d-electron contribution.

FIG. 1: Transmission (open circles), reflection (black squares) and absorption (stars) of a layered structure consisting of a 28 nm thick gold buffer layer, 6 monolayers of cobalt and 15 monolayers of gold on a glass substrate, computed from the effective medium model using dielectric constants from Ref. [16,[START_REF]CRC Handbook of Chemistry and Physics[END_REF]19]. Light scattering has been neglected. Full red line : experimental reflection and transmission coefficients normalized to the values computed at 2.35 eV

We performed a simulation of the optical properties of a composite layered Au/Co/Au film in the framework of the effective medium dielectric response described above, taking as an input tabulated values of the dielectric matrix elements for gold and cobalt [16,[START_REF]CRC Handbook of Chemistry and Physics[END_REF]19]. The reflection and transmission coefficients of a structure corresponding to 6 monolayers of Co embedded in a 29 nm thick layer of Au is displayed in Fig. 1 as a function of the photon energy. For low photon energy the layered structure shows the trends of a perfect metal with high reflection coefficient and low absorption. In the visible range the transmission shows a maximum around 2.4 eV which is known as the threshold of the interband transitions in gold. As expected for a sample containing 97% of gold atoms, the overall linear optical properties of the composite material are very similar to those of pure gold. Experimental data points for the reflection and transmission coefficient are displayed in the same figure. The overall agreement with the model is very good. As will be reported elsewhere, this formalism also provides a good estimate of the magneto-optical rotations (Kerr and Faraday effects), provided that one introduces a nondiagonal dielectric element proportional to the Cobalt one and to the Cobalt layer thickness. This model (Fig. 1) also allows us to compute the changes of transmission and reflection induced by small changes in the dielectric function of either materials and in particular those resulting from photo-induced changes of the electron population. We find that to an excellent approximation, changes in the dielectric function of cobalt lead to negligible effects as compared to the ones due to changes in the dielectric tensor of gold. Roughly, the relative change of transmission induced by a given change of the dielectric function of gold is 100 times larger than the one induced by the same change in the dielectric function of cobalt. The main conclusion is that the optical properties of the structure basically reflect those of gold, that is those involving the delocalized sp-electrons. This corresponds to the fact that the photo-excitation leads to significant changes in the occupation number of the only states lying close to the Fermi level.

V. EXPERIMENTAL RESULTS

Figure 2 displays the transmission and reflection transients of the layered sample together with those of the reference gold film at different probe wavelengths. Compared to the pump-probe cross-correlation, the transmission transients show a finite rise time (related to the internal thermalization through electron-electron collisions) of the order of one hundred femtoseconds and a recovery time of the order of one picosecond (related to the external thermalization through electron-phonon interactions).

At any probe wavelength we observe that both the rise and decay times are smaller in the layered system than in pure gold. This means that both the electron-electron and electron-phonon interactions are significantly enhanced by the presence of the thin cobalt layer.

In contrast to the transmission transients, the changes of reflection show a strong wavelength dependence in the spectral range investigated in this paper. The reflection transients are negative at low probe photon energy and become positive at high photon energy. For the intermediate photon energies 2.34 eV and 2.38 eV, the gold signal shows a bipolar temporal profile. The results obtained for the gold film are very similar to those reported by Sun et al. especially regarding the sign of ∆R/R as a function of the probe wavelength [8]. This behavior has been accounted for with a two-population model describing both the thermalized and off-equilibrium electron populations. It has been shown that in this probe photon energy range, slightly below the interband transition threshold, the thermalized and out-of-equilibrium populations give contributions to ∆R/R with opposite signs, which explains the bipolar profiles observed for probe energies of 2.34 eV and 2.38 eV for instance. At lower energies (2.3 eV and below) and at higher energy (2.43 eV and above) ∆R/R is dominated by the thermalized contribution. Extending this analysis to the case of the layered system gives insight into the electron interactions in this system and confirms the modeling of the effective dielectric response as previously outlined.

VI. INTERNAL THERMALIZATION A. Results

First, the overall spectral evolution of the transients (sign of ∆R/R) is very similar to the one of the gold film (negative change of reflection at low energy, positive at high energy). This confirms our anticipation that the optical response of cobalt is not distinguishable from that of gold to the extent that it stems from the mobile sp-electrons that share the same characteristics. However transients measured in the layered system and in the gold film are qualitatively different in the spectral region where both the thermalized and out-of-equilibrium populations give comparable contributions. Especially the bipolar profiles observed in ∆R/R for the gold film at 2.34 eV and 2.38 eV are absent in the layered system. For instance for a gold film probed at 2.34 eV the positive bump (see Fig. 2) has been shown to be due to the off-equilibrium electronic population whereas the negative one is due to the thermalized population [8]. In the layered system, only the negative one remains, i.e. no nonthermal electronic population is observable within our temporal resolution. In fact, because of the presence of the d-electrons in the intercalated Co-layer the sp -d collisions are dominant, suppressing the nonthermal electronic population within our temporal resolution. Thus we conclude that the internal electronic thermalization is achevied within 300 fs which corresponds to a strong enhancement of the electron-electron scattering with respect to the case of pure gold. Consistently, at 2.43 eV ∆R/R becomes positive, as expected from the gold case and still no evidence of any nonthermal population is to be seen.

In both spectral regions where the thermalized contribution is expected to dominate we find that ∆R/R and ∆T /T have the same sign as in pure gold. Thus we can analyze the transients for these wavelengths in the light of a two-temperature model which in principle allows to extract both electron-electron and electron-phonon relaxation times [20]. We fit the transients to the convolution of the experimental pump-probe cross-correlation with eq.( 1) :

X(t) =H(t){A(1 -e -t τee )e -t τ eph + B(1 -e -t τ eph )} (1)
where H(t) stands for the Heaviside function, τ ee stands for the average sp-d-electron relaxation time, τ eph stands for the average electron/phonon relaxation time and the second term accounts for the long living plateau corresponding to an overall heating of the sample after electrons and phonons have reached thermal equilibrium. The decay time of this plateau is of the order of several nanoseconds and is not observed on the time scale shown here.

Far off the threshold of interband transitions, we find identical time constants for both the transmission and reflection transients as expected from the two-temperature model. In the case of pure gold at a probe photon energy of 2.47 eV we get τ ee = 490 fs and τ eph = 1.1 ps. These values are in agreement with the ones reported so far in the literature for the low perturbation regime [8,10]. We checked carefully that these values do not depend on the pump power over one order of magnitude. In the layered system, the rise time of the signal (internal relaxation) is found to be less than 100 fs, roughly 5 times smaller than in bulk gold. However the pump-probe cross-correlation being of the order of 350 fs, our capability to quantify this rise time is limited and hampers any tentative measurement as a function of the Co layer thickness. On a logarithmic scale (see Fig. 3) it is clear that in the layered system the decay becomes exponential earlier than in pure gold, which is consistent with the above mentioned enhanced internal thermalization.

B. Interpretation

In the electron dynamics, the key role is played by the electron states around the Fermi level and is essentially determined by the broad half-filled sp-band which is roughly the same for the two metals. The main difference between noble metals and magnetic ones is in the position of the d-bands with respect to the Fermi level [21][START_REF] Dugdale | The Electrical Properties of Metals and Alloys[END_REF][START_REF] Rossiter | The Electrical Resistivity of Metals and Alloys[END_REF]. These are much narrower than the spbands and lie well inside them. In noble metals, they are doubly degenerate, filled and located well below the Fermi level, roughly 2.5 eV in the case of Au. In the magnetic case in contrast, the d-bands are only partially filled with the Fermi level cutting through at different positions for spin up and spin down states as the d-bands are Zeeman split by the internal magnetization mean field. The d-bands being much narrower and less dispersive than the sp ones, their density of states is larger by almost a factor five. Thus we may assume the delectrons immobile and not contributing to the dielectric response other than as scattering centers -in several respects similar to static impurities-for the lighter mobile sp-electrons. This analysis is a transposition of the two-electron liquid model introduced by Mott for the static/low frequency conductance in magnetic metals to the high frequency/optical dielectric response of these metals [START_REF] Mott | [END_REF][25][26]. The d-scattering channels are very efficient and are responsible for the much lower conductivity of magnetic metals as compared to the one of noble metals. From the Fermi golden rule, one can qualitatively argue that the scattering rate of an s electron onto s and d electrons is proportional to their density of states. In the case of Co the latter is about 5 times larger than the former. Although located in the narrow Co layer in our case, these d-electrons are efficient scatterers for the spelectrons of the adjacent Au layers as well to the extent that the sp-electron delocalization length in Au is larger than the layer thickness, leading to a sizeable increase of the overall scattering rate in agreement with the experimental data.

VII. ELECTRON-LATTICE THERMALIZATION

Similarly, the electron-phonon scattering rate can be deduced from a fit of the data to eq. ( 1). Since the external relaxation is much slower than the internal one, one can also get an estimate of the electron-phonon scattering rate from a monoexponential fitting of the data for time delays larger than 3 τ ee , after substraction of a long-lasting background due to the final heating of both electrons and lattice (cf. Fig. 3). External relaxation is found to be significantly faster in the layered film than in gold, the relaxation rate depending on the cobalt layer thickness.

Fig. 4 shows the variations of the relaxation rate as a function of the cobalt layer thickness, as well as a linear fit to the data. The extrapolated value of the relaxation rate for a vanishing cobalt layer thickness is very close to the one measured in the reference gold film. This means that interface effects are negligible in the overall relaxation processes in the layered system with respect to bulk effects. This is consistent with our assumption that hot electrons are delocalized over the whole structure.

We can deduce an estimate of the effective electronphonon scattering rate in the cobalt layer from the variations of the relaxation rate as a function of the cobalt layer thickness. We simply assume that the electronphonon relaxation rate of the layered structure is given by the average of the relaxation rates in both materials weighted by the probability of presence in each layer. Since the electrons are almost ballistic, this probability is proportional to the thickness of the layer divided by the Fermi velocity in the related material. We use a Fermi velocity of 0.33x10 6 m.s -1 in cobalt and 1.4x10 6 m.s -1 in gold. It turns out that the electron-phonon scattering rate in cobalt is of the order of 5 ± 1 ps -1 which corresponds to a scattering time of the order of 200 ± 50 fs.

This scattering rate turns out to be about 5 times larger than in noble metals, which is again a direct consequence of the interplay between sp and d electrons : d-electrons being heavy and bound, collisions between sp and d electrons provide an important path to electronphonon coupling as discussed above [START_REF] Ziman | Electrons and Phonons[END_REF]. Data are available in the literature for nickel which is another ferromagnetic transition metal. Although relatively dispersed the values reported so far are in the range of 300 to 800 fs [4,6,[START_REF] Melnikov | [END_REF]29]. Cobalt shows a slightly faster electronphonon scattering time which might be related to its specific band structure around the Fermi level.

VIII. CONCLUSION

In conclusion we have studied the electron dynamics in a layered Au/Co/Au film. We have shown that in the limit of a small cobalt layer thickness, the linear optical properties of the system are almost identical to those of a pure gold film. In contrast, the nonlinear properties are strongly affected by the presence of the transition metal layer. Especially both internal and external scattering rates show a sizeable increase for less than 3 % of cobalt atoms. From a quantitative analysis as a function of the layer thickness, we find that interface processes are negligible. Therefore the accelerated relaxation observed in this structure is due to an intrinsic enhancement of the electronic interactions. This effect is attributed to the large density of d-states available near the Fermi level in transition metals. These highly enhanced electronic interactions and the distinct dynamics of the two electron populations represent a valuable input for forthcoming studies about ultra-fast photo-induced spin dynamics in such nanostructured devices.

This work has been done in the framework of ACI Jeunes Chercheurs Nanomag.

FIG. 2 :

 2 FIG. 2: (a) Transient changes of reflection (left side) and transmission (right side) of a layered Au/Co/Au film (black squares) and of a 30 nm thick reference Au film (open blue circles) for several probe energies. The pump photon energy is kept at 1.55 eV. The dashed line is the pump-probe crosscorrelation.

FIG. 3 :

 3 FIG. 3: Logarithm of the transient change of transmission of a layered Au/Co/Au film (black squares) and a reference gold film (open circles). The thermal background has been subtracted. The solid lines are monoexponential fits to the data.

FIG. 4 :

 4 FIG. 4: External (electron-phonon) scattering rate as a function of the cobalt layer thickness (black squares). Scattering rate in the reference gold film (open circle). The solid line is a linear fit to the data.