P D Gonz Ález

P Érez

BIJECTIVENESS OF THE NASH MAP FOR QUASI-ORDINARY HYPERSURFACE SINGULARITIES

Keywords: 2000 Mathematics Subject Classification. Primary 14J17; Secondary 32S10, 14M25 Nash problem, arc space, quasi-ordinary singularities, singularities

In this paper we give a positive answer to a question of Nash, concerning the arc space of a singularity, for the class of quasi-ordinary hypersurface singularities extending to this case previous results and techniques of Shihoko Ishii.

Introduction

In a 1968 preprint, published later as [N], Nash introduced a map, nowadays called the Nash map (resp. local Nash map), from the set of families of arcs with origin at the singular locus of a variety X (resp. at a fixed singular point x ∈ X), to the set of essential divisors over the singular locus of X (resp. over the point x). These families of arcs are called Nash components (resp. local Nash components). Obviously, these maps coincide if the singular locus of X is reduced to an isolated singular point x. Nash showed that these maps are injective and asked if they were surjective. Ishii and Kollár have shown an affine four dimensional variety with singular locus reduced to a point, for which the Nash map is not bijective (see [I-K]). The answer of Nash question for surface and threefolds singularities is not known in general. Plenat has given sufficient conditions for the surjectivity of the Nash map for isolated surface singularities in [START_REF] Plénat | À propos du problème des arcs de Nash[END_REF]. In the case of surface singularities the Nash map it is known to be bijective in the following cases: for minimal singularities by Reguera [START_REF] Reguera-Lopez | Families of arcs on rational surface singularities[END_REF], for sandwiched singularities by the work of Lejeune and Reguera [LJ-R] and [START_REF] Reguera-Lopez | Image of the Nash map in terms of wedges[END_REF], for rational double points of type A n , already studied by Nash [N], and of type D n by Plenat [START_REF] Plénat | Résolution du problème des arcs de Nash pour les points doubles rationnels Dn[END_REF] (the result for rational double points in general is announced in [START_REF] Plénat | Résolution du probleme des arcs de Nash pour les points doubles rationnels[END_REF]). Plenat and Popescu-Pampu have shown a class varieties of dimension two and higher for which the Nash map is bijective in [Pl-PP1] and [Pl-PP2]; a similar result in the surface case is announced by Morales [Mo].

Ishii and Kollár have shown that Nash question has a positive answer in the case of normal toric varieties, see [I-K]. Ishii has generalized this result for the class pretoric algebraic varieties, which contains in particular the class of toric varieties (non necessarily normal). Petrov formulated Nash question for pairs (X, B), consisting of a variety X and a proper closed subvariety B containing the singular locus of X, and exhibited a positive answer in the case of pairs, (X, B), formed by a normal toric variety X and an invariant set B. He applied this result to prove the bijectiveness of the Nash map for the class of stable toric varieties, a class of reduced but non necessarily irreducible varieties introduced by Alexeev [Al], which generalize normal toric varieties (see [Pe]). Ishii has shown recently that the local Nash map is bijective for analytically pretoric singularities, a class of singularities containing toric and analytically irreducible quasi-ordinary hypersurface singularities (see [I3]). The class of quasi-ordinary singularities appears classically in Jung's strategy to obtain resolution of surface singularities from the embedded resolution of plane curves (see [J], [A] and [L1]).

The purpose of this Note is to show that the Nash map is bijective for any reduced germ of quasiordinary hypersuperface singularity, Theorem 4. We show that the results and approach for the class of pretoric singularities, analyzed by Ishii in [I2], can be extended to the case of an analytically irreducible quasi-ordinary hypersurface germ (X, x) by showing a property, Proposition 1, deduced from Lipman's result on the structure of the singular locus (see [L2], §7). Moreover, if B is a proper closed subgerm of (X, x) containing the singular locus of X, we prove that Nash question for the pair (X, B) has a positive answer under a natural technical condition (see Hypothesis 1 and Theorem 2).

In the case of a quasi-ordinary hypersurface germ (X, x) with several irreducible components X i , i = 1, . . . , t, the analysis of the pre-image of the singular locus of X by the normalization map (see Lemma 5) is essential to deduce the main result Theorem 4, the bijectiveness of the Nash map for X, from the bijectiveness of the Nash map for suitable pairs (X i , B i ) for i = 1, . . . , t.

Finally, we compare the notions of pretoric singularity and analytically pretoric singularity, introduced by Ishii in [I2] and [I3] respectively, with the notion of toric quasi-ordinary singularity introduced [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF] and we show that our main result holds in a slightly larger category, which we call strongly analytically pretoric (see Corollary 5).

The explicit description of the essential divisors over the singular locus of a quasi-ordinary hypersurface singularity, given in this paper, is applied by Hernando and the author in [GP-H] to prove that the essential divisors of an irreducible germ (X, x) of quasi-ordinary hypersurface determine, through a suitable notion of Poincaré series, the characteristic monomials, an analytical invariant which in the analytic case encodes the embedded topological type of the germ characterized by the work of Gau and Lipman (see [L2] and [Gau]). It should be noticed that the Poincaré series associated only to the essential divisors over the point x, which correspond by Ishii's result [I3] to the local Nash components, does not contain enough information to recover the characteristic monomials of the quasi-ordinary hypersurface (X, x) in general, see [GP-H]. This property reflects the fact that quasi-ordinary singularities are rarely isolated: X does have an isolated singularity at x if and only if the singular germ (X, x) is of dimension one or normal of dimension two. The use of all essential divisors, over the different components of the singular locus of X, and also over x, is crucial to recover the characteristic monomials of the germ (X, x), as main local invariants.

Basic definitions on the arc space and relative Nash problem

In this section we give basic definitions and results on the relative Nash problem, also called Nash problem for a pair. These notions are natural extensions of the corresponding ones on the classical Nash problem (see [Pe]). In this paper the scheme X is a pure dimensional reduced algebraic (resp. algebroid germ of) variety, defined over a field k, algebraically closed of characteristic zero. Let B ⊂ X be a reduced proper k-subscheme containing the singular locus Sing(X) of X.

A resolution of singularities of (X, B) is a proper modification φ :

Y → X such that φ |Y -φ -1 (B) : Y -φ -1 (B) → X -B is an isomorphism. The resolution φ is divisorial is φ -1 (B) is a divisor. An exceptional divisor E over X relative to B is an exceptional divisor such that the center of E over X is contained in B. An exceptional divisor E over X relative to B is essential if for every resolution φ : Y → X of the pair (X, B) the center of E on Y is an irreducible component of φ -1 (B). This center is called an essential component on Y . If k ⊂ K is a field extension an arc over X is a morphism α : SpecK[[t]] → X.
We denote respectively by 0 and η the closed point and generic point of SpecK

[[t]]. If m ≥ 0 is an integer, an m-jet over X is a morphism α : SpecK[[t]]/(t m+1 ) → X.
The set X m of m-jets can be given the structure of scheme of finite type over k. We have canonical morphisms

X m+k → X m , corresponding to SpecK[[t]]/(t m+1 ) → SpecK[[t]]/(t m+k+1
), for all m, k ≥ 0. The arc space X ∞ := lim ← -X m has the structure of scheme over k, not of finite type. A point z ∈ X ∞ corresponds to an arc α z : SpecK [[t]] → X such that K is the residue field at z. We have that K-valued points of X ∞ correspond to arcs of the form SpecK [[t]] → X bijectively. We often denote the point z and the corresponding arc α z by the same symbol. We have a canonical projection π X : X ∞ -→ X, defined by z → α z (0). A morphism of varieties (resp. of algebroid germs) ψ : Y → X, corresponds to a morphism at the level of arcs:

ψ ∞ : Y ∞ → X ∞ . See [D-L] for the general construction of arcs spaces. A Nash component of X relative to B is an irreducible component of π -1 X (B) which is not con- tained in B ∞ ⊂ X ∞ .
If the field k is of characteristic zero, as it is assumed in this paper, then the proof of Lemma 2.12 [I-K] imply that the Nash components of X relative to B coincide with the irreducible components of π -1 X (B). Denote by i C i the union of Nash components of X relative to

B. Let φ : Y → X be a divisorial resolution of (X, B), i..e, φ -1 (B) is a divisor with irreducible com- ponents E 1 , . . . , E l . Then the restriction φ ∞ to l j=1 π -1 Y (E j ) -→ i C i .
is dominant and bijective outside B ∞ . For all i there exists a unique

j i such that π -1 Y (E j ) → C i is dominant.
The analogous statement of Nash Theorem in this relative situation is that E j i is an essential divisor of X relative to B and that the relative Nash map C i → E j i is an injection, from the Nash components and to the set of essential divisors of X relative to B (see Lemma 2.14 [I3] for an analogous proof in this relative situation or Theorem 2.17 [Pe] for a sketch of proof in the algebraic case). See also [I-K] for a modern proof of the classical statement of Nash [N], when B = Sing(X)). The Nash problem for X relative to B, i.e., the Nash problem for the pair (X, B), is to determine if this correspondence is bijective.

Basic notations on normal toric varieties

We give some basic definitions and notations (see [O] or [Fu] for proofs). If N is a lattice we denote by M the dual lattice, by N R the real vector space spanned by N and by , the canonical pairing between the dual lattices N and M (resp. vector spaces N R and M R ). A rational convex polyhedral cone τ in N R , a cone in what follows, is the set τ := pos{a 1 , . . . , a s } of non negative linear combinations of vectors a 1 , . . . , a s ∈ N . The cone τ is strictly convex if τ contains no linear subspace of dimension > 0. We denote by • τ the relative interior of a cone σ. The dual cone τ ∨ (resp. orthogonal cone τ ⊥ ) of τ is the set {w ∈ M R / w, u ≥ 0} (resp. w, u = 0) ∀u ∈ τ }). A fan Σ is a family of strictly convex cones in N R such that any face of such a cone is in the family and the intersection of any two of them is a face of each. If τ is a cone in the fan Σ, the semigroup τ ∨ ∩ M is of finite type, it spans the lattice M and the variety Z τ,N = Spec k[τ ∨ ∩ M ], which we denote by Z τ when the lattice is clear from the context, is normal. The affine varieties Z τ corresponding to cones in a fan Σ glue up to define the toric variety Z Σ . The torus

T N := Z {0} ∼ = (k * ) rk N is embedded in Z Σ
as an open dense subset and there is an action of T N on Z Σ which extends the action of the torus on itself by multiplication. We have a bijection between the relative interiors of the cones of the fan and the orbits of the torus action,

• τ → orb Z Σ (τ ), which inverses inclusions of the closures. We denote the orbit orb Z Σ (τ ) by orb(τ ) when the toric variety Z Σ is clear from the context.

In this paper σ denotes a rational strictly convex cone in N R of maximal dimension. Any non

zero vector v ∈ σ ∩ N defines a valuation val v of the field of fractions of k[σ ∨ ∩ M ] (resp. of the m-adic completion k[[σ ∨ ∩ M ]] of the localization of k[σ ∨ ∩ M ] at the maximal ideal, (σ ∨ ∩ M ) \ {0},
defining the origin o σ of the toric variety Z σ ). This valuation, called monomial or toric valuation, is defined for an element 0

= φ = c u X u ∈ k[[σ ∨ ∩ M ]], by val v (φ) = min cu =0 n, u .
If the ray ρ := vR ≥0 belongs a fan Σ subdividing σ, the closure D ρ of the orbit orb(ρ) is an invariant divisor (we denote it also by D v if the vector v is primitive for the lattice N ). We denote by val Dρ the associated divisorial valuation. If v = qv 0 for q ∈ Z ≥1 and v 0 a primitive vector for the lattice N then we have that val v = qval Dρ . Following Ishii we say that the valuation val v is a toric divisorial valuation (see [I2]). The cone σ induces a partial order on N R , defined by u ≤ σ u ′ iff u ′ ∈ u + σ (similar definition for ≤ σ ∨ on M R holds).

Quasi-ordinary hypersurfaces: singular locus and normalization

A quasi-ordinary hypersurface singularity X is defined by Spec

k[[x]][y]/(f ) where x = (x 1 , . . . , x d ) and f ∈ k[[x]][
y] is a quasi-ordinary polynomial, i.e., a Weierstrass polynomial such that the discriminant ∆ y f with respect to y is of the form x δ ǫ where δ ∈ Z d ≥0 and ǫ ∈ k [[x]] is a unit. By the Jung-Abhyankar Theorem the roots of quasi-ordinary polynomials are fractional power series with particular properties. Namely, if f is irreducible of degree n a root of f is of the form:

ζ = c λ x λ ∈ k[[x 1/n ]], where x 1/n = (x 1/n 1 , . . . , x 1/n d )
, and the terms appearing in this expansion verify certain properties. In particular if f is of degree > 1 in y, certain monomials, determined by comparing the different roots of f and called characteristic or distinguished, appear in the expansion of ζ with non zero coefficient. The corresponding exponents, which are also called characteristic, can be relabelled in the form

λ 1 ≤ σ ∨ λ 2 ≤ σ ∨ • • • ≤ σ ∨ λ g . These exponents determine the following nested sequence of lattices: M 0 ⊂ M 1 ⊂ • • • ⊂ M g =:
M where M 0 := Z d and M j := M j-1 + Zλ j for j = 1, . . . , g with the convention λ g+1 = +∞. We have that the exponents appearing in the expansion of ζ belong to M . See [L2] and also [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF]. We have ring extensions:

(1) k[[σ ∨ ∩ M 0 ]] = k[[x]] -→ O X ∼ = k[[x]][ζ] -→ k[[σ ∨ ∩ M ]]
where σ ∨ denotes the positive quadrant R d ≥0 and M 0 = Z d (we denote by σ, N 0 and N the corresponding dual objects of σ ∨ , M 0 and M respectively). In [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF] it is proved that the ring extension

O X → k[[σ ∨ ∩ M ]
] is the inclusion of O X in its integral closure in its field of fractions. Geometrically, (1) corresponds to a sequence of finite maps:

(2) (X σ,N , o σ ) = ( X, x) ν -→ (X, x) ρ -→ (Z, x) = (X σ,N 0 , o σ ) = (k d , 0).
Since the map ρ • ν is equivariant, it maps the orbit orb X τ to orb Z τ , for each face τ < σ.

We recall Lipman's description of the singular locus of a quasi-ordinary hypersurface, see Theo-rem7.3 [L2], for a precise statement (cf. also the reformulation of this result given in [PP]).

Theorem 1. With the previous notations if (X, x) is analytically irreducible we have that the irreducible components of Sing(X) are of codimension one or two. The codimension one (resp. two) components are intersections of X with x i = 0, (resp. with x i = 0 and x j = 0) for some suitable coordinate sections, in each case, determined by the characteristic monomials.

Proposition 1. The set ν -1 (Sing(X)) is a germ of closed set at the origin of X, which is invariant by the torus action on X.

Proof. By Lipman's theorem, the irreducible components of Sing(X) are the germs ρ -1 (orb Z (τ )) at the point x, where τ runs certain set of a one (resp. two ) dimensional faces of σ. It follows from this and the previous discussion that the irreducible components of ν -1 (Sing(X)) are of codimension one or two. The codimension one (resp. two) components are of the form

(ρ • ν) -1 (orb Z (τ )) = orb X (τ ).
If x i = X u i for i = 1, . . . , d, in (1) then x i = 0 (resp. x i = x j = 0) defines in Z the closure of the orbit orb Z (τ ) where the cone τ is characterized by τ ⊥ ∩σ ∨ = pos k =i (u k ) (resp. by τ ⊥ ∩σ ∨ = pos k =i,j (u k )).

Relative Nash problem, the irreducible case

We follow Ishii's approach in [I2] and [I3]. Let (X, x) be an irreducible germ of quasi-ordinary hypersurface. We study the relative Nash problem for a proper closed set B ⊂ X such that Sing(X) ⊂ B. We introduce the following hypothesis on B.

Hypothesis 1. Any irreducible component of ν -1 (B) is an orbit closure orb X (τ ) corresponding to some face τ of σ.

Notice that Sing(X) verifies this hypothesis by Proposition 1. When the hypothesis above is verified any irreducible component in the closure of the set ν -1 (B)-Sing( X) is an orbit closure corresponding to some regular face τ < σ (with respect to the lattice N ). We denote by τ 1 , . . . , τ r the regular faces determined in this way and by e i ∈ N , the barycenter of τ i (i.e., the sum of the primitive integral vectors, for the lattice N , in the edges of τ i ), for i = 1, . . . , r.

Let {v j } s j=1 the set of minimal elements, with respect of the partial order ≤ σ in the set:

S := τ <σ, τ singular • τ ∩N.
By [I-K] the toric divisors {D v j } j=1,...,s are the essential divisors of toric variety X, and also the essential divisors of the germ of X at the closed orbit, by Lemma 4.9 [I3]. This characterization of essential divisors generalizes a result of Bouvier [Bo], see also [Bo-GS].

Lemma 2. Each e i , for i = 1, . . . , r is minimal among {e i , v j } j=1,...,s i=1,...,r for the order ≤ σ . Proof. See Lemma 5.7 [I2]. Lemma 3. Let {e i , v j } j=1,...,w i=1,...,r , (w ≤ s) be the set of minimal elements of {e i , v j } j=1,...,s i=1,...,r . Then there is an inclusion {essential divisors over X relative to B} ⊂ {D e i , D v j } j=1,...,w i=1,...,r . Proof. The statement can be translated in purely combinatorial terms in terms of the existence of resolutions of singularities of X, which are obtained by composing the normalization map with toric modifications. The precise arguments are the content of the proof of Lemma 5.7 [I2].

Following Ishii,[START_REF] Ishii | Arcs, valuations and the Nash map[END_REF], we associate to a non zero vector v ∈ σ∩N a subset T X ∞ (v) of X ∞ , containing only arcs which lift to arcs with generic point in the torus T N of X:

T X ∞ (v) := {α ∈ X ∞ | α(η) ∈ ν(T N ), ord t α * (x u ) = v, u , for u ∈ M }. The sets T X ∞ (v)
, defined similarly, are orbits of a natural action of (T N ) ∞ on the arc space of the normal toric variety X (see [I1]). Lemma 4. Let {e i , v j } j=1,...,w i=1,...,r , (w ≤ s) be the set of minimal elements of {e i , v j } j=1,...,s i=1,...,r . Then, the following closures are distinct Nash components of X ∞ :

T X ∞ (e i ), i = 1, . . . , r and T X ∞ (v j ), j = 1, . . . , w. If v ∈ {e i , v j } j=1,...,s i=1,...,r the image of the component T X ∞ (v)
by the Nash map is the divisor D v . Proof. The proof is analogous to the proofs of Lemma 4.6 and 4.7 in [I3]. Notice that with our hypothesis the proof holds not only for vectors v ∈ • σ ∩N but also for vectors v ∈ σ ∩ N . See also the proof of Lemma 5.10 and Theorem 5.11 [I2].

Theorem 2. Let (X, x) be a irreducible germ of quasi-ordinary hypersurface singularity. Let B a closed subscheme verifying Hypothesis 1. Then the Nash map between the set of Nash components of π -1 X (B) and the set of essential divisors of X relative to B is bijective.

Proof. Let u ∈ {e i , v j } j=1,...,w i=1,...,r . Then the sequence u → T X ∞ (u) → D u defines an injection from the set {e i , v j } j=1,...,w i=1,...,r to the set of essential divisors over X, by Lemma 4 and by the injectivity of the Nash map. By Lemma 3 the set of essential divisors is of cardinality less or equal than r + w, hence it follows that this set is of cardinality equal to r + w and the injection above is a bijection. This implies that the Nash map is bijective.

Corollary 3. If (X, x) is an analytically irreducible quasi-ordinary hypersurface the Nash map is bijective.

Proof. By Proposition 1 the singular locus B = Sing(X) verifies the hypothesis 1 of Theorem 2.

Nash problem for a quasi-ordinary hypersurface

Now we suppose that (X, x) is a germ of reduced quasi-ordinary hypersurface. We denote by f a quasi-ordinary polynomial defining (X, x). The factors f i of the factorization of f = f 1 . . . f t as product of irreducible terms corresponds to the irreducible components of (X, x). The factors f i are quasi-ordinary polynomials, for i = 1, . . . , t.

We denote by B i the intersection:

B i = X i ∩ Sing(X) = Sing(X i ) ∪ j =i j=1,...,t X i ∩ X j .
We denote by ν i : Xi → X the normalization of X i , which is a toric singularity by the previous discussion.

Lemma 5. We have that ν -1 i (B i ) is a germ of invariantly closed set, at the close orbit of the toric singularity Xi .

Proof. We have already shown the statement for ν -1 i (Sing(X i )) by Proposition 1. If j = i then ν -1 (X i ∩ X j ) is defined by f j (ζ (i) ) = 0 where ζ (i) is a fixed root of f i , used to define the ring extension (2) corresponding to X i . We have that the element f j (ζ (i) ) is equal to the product of a monomial by a unit in the local algebra of the toric singularity Xi (this follows easily from the definition, see [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF] for more details). This implies that f j (ζ (i) ) = 0 defines a germ of invariantly closed set, at the close orbit of the toric singularity Xi , which is equal to ν -1 i (X i ∩ X j ). We obtain then a generalization of Corollary 4.12 in [I3]: Theorem 4. Let (X, x) be a reduced germ of quasi-ordinary hypersurface singularity. Then the Nash map between the Nash components of π -1 X (Sing(X)) and the essential divisors of X is bijective. Proof. We keep the notations given above for the irreducible components of X. Notice that π -1 X (Sing(X)) = t i=1 π -1 X i (B i ) by definition of B i . It follows from this that:

{Nash components of X} ⊂ t i=1
{Nash components of X i relative to B i }.

(See the proof of Lemma 4.11 [I3]). We prove that:

{essential divisors over X} ⊂ t i=1
{essential divisors over X i relative to B i }.

Let φ i : Y i → X i be a resolution of (X i , B i ). Then the composite φ : Y → X defined by:

Y := t i=1 Y i φ i → t i=1 X i → X
is a resolution of the pair (X, Sing(X)) by the definition of B i . Let E be an essential divisor of X.

The center of E in Y is an irreducible component of

φ -1 (Sing(X)) = t i=1 φ -1 i (B i ),
thus an irreducible component of φ -1 i (B i ) for some i. This implies the assertion. The hypothesis 1 is verified by B i with respect to X i for i = 1, . . . , t by Lemma 5. By Theorem 2 applied to the pair (B i , X i ) the Nash map between the set of Nash components of π -1 X i (B i ) and the set of essential divisors of X i relative to B i is bijective.

6. An extension of the results to a larger category In Definition 4.1 [I3] the notion of analytically pretoric singularity is introduced in the algebroid category. A germ (X, x) is called in [I3] analytically pretoric if there exists a sequence of injective local homomorphisms:

k[[σ ∨ ∩ M 0 ]] ρ * → O X,x ν * → k[[σ ∨ ∩ M ]] such that i. ν * • ρ * : k[[σ ∨ ∩ M 0 ]] → k[[σ ∨ ∩ M ]] is the canonical injection corresponding to a finite index lattice inclusion M 0 ⊂ M , ii. the morphism ν : Speck[[σ ∨ ∩ M ]] → X corresponding to ν * is the normalization map, iii. the restriction of ν to the torus Speck[[σ ∨ ∩ M ]][M ] is an isomorphism onto its image.
Notice that Ishii introduced the notion of pretoric variety in the algebraic category in [I2] Definition 5.1. The first two conditions for a variety X to be pretoric are the formulations of axioms i. and ii. above in the algebraic category, while the third condition above is to be replaced by iii'. The closed subset ν -1 (Sing(X)) is invariant for the torus action of X. We say that a germ of algebroid singularity is strongly analytically pretoric if it verifies conditions i., ii., and iii'. Notice that in the algebroid case condition iii'. implies condition iii in Ishii's definition of analytically pretoric singularity.

Corollary 5. If (X, x) is a germ of strongly analytically pretoric singularity then the associated Nash map is bijective.

Proof. The analysis and results done in the sections 4 and 5 extends formally for any algebroid singularity (X, x) verifying conditions i., ii., and iii'.

We end this section by formulating some natural questions which come out from the comparison of the notions of (strongly) analytically pretoric with that of toric quasi-ordinary singularity, introduced in [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF] as a suitable generalization of the notion of quasi-ordinary singularity (in [START_REF] Pérez | Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant[END_REF] the this class of singularities was restricted to the case of relative hypersurface germ in Z σ × C). A germ of complex analytic variety (X, x) of pure dimension d is a toric quasi-ordinary singularity if there exists:

a. An affine normal toric variety X σ,N 0 = Spec k[σ ∨ ∩ M 0 ], defined by a d dimensional rational strictly convex cone σ for the lattice N 0 (dual of σ ∨ and M 0 respectively). b. A finite morphism of germs (X, x) ρ → (X σ,N 0 , o σ ), where o σ is the closed orbit, which is unramified over the torus of X σ,N 0 . Condition b. means that for each representative of the morphism ρ there exists an open neighborhood of o σ such that ρ is unramified over its intersection with the torus. If the cone σ is simplicial then (X, x) is a quasi-ordinary singularity: we reduce to this case by replacing the lattice M 0 by a sublattice M ′ 0 of finite index such that σ ∨ is regular for M ′ 0 and hence Z σ,N ′ 0 = C d , for N ′ 0 the dual lattice of M ′ 0 . Then the normalization ( X, x) is a germ of toric variety X σ,N = Spec C[σ ∨ ∩ M ] for some lattice M ⊇ M 0 , at is closed orbit, and the composite ρ • ν is a germ of toric equivariant map defined by the change of lattices (see Theorem 5.1 in [PP], or see [START_REF] González Pérez | Toric embedded resolutions of quasi-ordinary hypersurface singularities[END_REF] in the hypersurface case). The definition of analytically pretoric singularity can be easily adapted to the complex analytic category. It is immediate that if (X, x) is a germ of analytically pretoric singularity then it is toric quasi-ordinary singularity, since the map ρ is then unramified over the torus by axiom i and iii. It follows from the discussion above that if the cone σ is simplicial the notions of toric quasi-ordinary singularity and that of analytically pretoric singularity coincide (at least in the complex analytic category). It seems quite reasonable that both notions coincide also in the algebroid category and without any assumption on simpliciality on the cone σ appearing in both Definitions. Is to be conjectured that for a equidimensional germ of algebroid singularity conditions i., ii., iii. and i., ii., iii'. are equivalent.

Acknowledgement. The author is grateful to Profesor Shihoko Ishii for stimulating conversations during the Hayashibara Forum on Singularities 2006 at IH ÉS. During the preparation of this work the author has benefited from the hospitality of Institut de Mathématiques de Jussieu in Paris.

The authors' research was supported in part by Programa Ramón y Cajal and MTM2004-08080-C02-01 grants of Ministerio de Educación y Ciencia, Spain.

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Pedro Daniel González Pérez

Departamento de Algebra. Facultad de Ciencias Matemáticas. Universidad Complutense de Madrid. Plaza de las Ciencias 3. 28040. Madrid. Spain.

E-mail address: pgonzalez@mat.ucm.es