The asymptotic Samuel function and invariants of singularities

The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. In this paper, we use this function to introduce the notion of the Samuel slope of a Noetherian local ring, and we study some of its properties. In particular, we focus on the case of a local ring at singular point of a variety, and, among other results, we prove that the Samuel slope of these rings is related to some invariants used in algorithmic resolution of singularities.


Introduction
Let X be an equidimensional algebraic variety of dimension d defined over a perfect field k.If X is not regular, then the set of points of maximum multiplicity, Max mult X , is a closed proper set in X.We will denote by max mult X the maximum value of the multiplicity at points of X.A simplification of the multiplicity of X is a finite sequence of blow ups, (1.0.1) where π i : X i → X i−1 is the blow up at a regular center contained in Max mult X i−1 .
Simplifications of the multiplicity exist if the characteristic of k is zero (see [36]), and resolution of singularities follows from there.Recall that Hironaka's line of approach to resolution makes use of the Hilbert-Samuel function instead of the multiplicity [20].The centers in the sequence (1.0.1) are determined by resolution functions.These are upper semi-continuous functions and their maximum value, max f X i , achieved in a closed regular subset Maxf X i ⊆ Max mult X i , selects the center to blow up.Hence, a simplification of the multiplicity of X, X ← X L , is defined as a sequence of blow ups at regular centers. ( where max f X i denotes the maximum value of f X i for i = 0, 1, . . ., L. Usually, f X is defined at each point as a sequence of rational numbers.The first coordinate of f X is the multiplicity, and the second is what we refer to as Hironaka's order function in dimension d, ord   [16,Theorem 7.6 and §7.11]), thus, we usually say that this is the main invariant in constructive resolution.Therefore, the last set of coordinates can be though as a refinement of the function ord (d) X .As we will see, the function ord X can always be defined if k a perfect field.
Example 1.1.Let k be a perfect field, let S be a smooth k-algebra of dimension d and define R = S[x] as the polynomial ring in one variable with coefficients in S. Suppose X is a hypersurface in Spec(R) of maximum multiplicity m > 1 given by an equation of the form Set β : Spec(R) → Spec(S) and let ζ ∈ X be a point of multiplicity m.Then one can define a Rees algebra, R, on S, which we refer to as elimination algebra, that collects information on the coefficients a i ∈ S, i = 1, . . ., m. Hironaka's order function at the point ζ, ord (d) X (ζ), is defined using R (see Section 6).If the characteristic of the field k does not divide m, then, after a translation on the variable x, we can assume that the equation is on Tschirnhausen form: And, in such case, it can be shown that: where ν β(ζ) denotes the usual order at the regular local ring S m β(ζ) .As it turns out, with the information provided by the elimination algebra R, which is generated by weighted functions on the coefficients of f (x), one has all the information needed to find a simplification of the multiplicity, at least in the characteristic zero case.However, if the characteristic of the field is p, and if p divides m, then, in general, the equality (1.1.1)does not hold (even if, by chance, the polynomial were in Tschirnhausen form).Philosophically speaking, the elimination algebra R collects information about the coefficients of f (x), but somehow falls short in collecting the sufficient amount of information when the characteristic is positive.This problem motivated in part the papers [5] and [6].There, the function H-ord X was introduced by the first author in collaboration with O. Villamayor.In [4], this function played a role in the proof of desingularization of two dimensional varieties.
To give some insight on how H-ord X is defined, suppose, for simplicity, that m = p ℓ for some ℓ ∈ Z ≥1 , f (x) = x p ℓ + a 1 x p ℓ −1 + . . .+ a p ℓ ∈ S[x], and let ζ be a point of multiplicity p ℓ .Then it can be proved that ord But there are examples where and the inequality remains even after considering translations of the form x ′ = x + s, s ∈ S q , where q = m β(ζ) .This pathology is part of the reasons why the resolution strategy (that works in characteristic zero) cannot be extended to the positive characteristic case.
The previous discussion motivates the definition of the slope of f (x) at ζ as: X (ζ) .
Changes of variables of the form x = x ′ + s with s ∈ S q produce changes on the coefficients of the equation: ( which may lead to a different value of the slope.However, it is possible to construct an invariant from these numbers by setting: H-ord Moreover this supremum is a maximum since there is a change of variables as in (1.1.2) for which H-ord X (ζ) .

H-ord (d)
X can be defined for any hypersurface with maximum multiplicity m, even when m is not a p-th power (see Section 7).Observe that the previous discussion takes care of the case in which X is locally a hypersurface at a singular point ζ, since, after considering a suitable étale neighborhood of X at ζ, it can be assumed that the equation defining X can be written as a polynomial in one variable with coefficients in some regular ring S.
When X is an arbitrary algebraic d-dimensional variety defined over a perfect field, H-ord X can also be defined (in étale topology) using [5], [6] and Villamayor's presentations of the multiplicity in [36].In the latter paper it is proven that, locally, in an étale neighborhood of a closed point ξ of maximum multiplicity m > 0, one can find a smooth k-algebra S of dimension d and polynomials in different variables x i with coefficients in S, f i (x i ) ∈ S[x i ], of degrees m 1 , . . ., m e , with the following property: If we consider (1.1.3)f 1 (x 1 ), . . ., f e (x e ) ∈ R = S[x 1 , . . ., x e ], then each f i (x i ) defines a hypersurface of maximum multiplicity m i , H i = {f i = 0}, so that, X ⊂ Spec(R) and (1.1.4)Max mult X = ∩ i Max mult H i .
In fact, the link between X and the hypersurfaces H i is stronger as we will see in Section 4.
As in the hypersurface case, Hironaka's order function, ord X , is defined by constructing an elimination algebra, R on S, again using certain weighted functions on the coefficients of the polynomials f i (x i ) (see Section 6).And, in the same way, we have that This approach will allow us to work in a situation very similar to the hypersurface case.Details and definitions will be given in Sections 7 and 8.The precise statement of Villamayor's result is given in Theorem 8.1, because it will be used in the proof of our results.

Results
From our previous discussion, the value H-ord X requires the use of local (étale) embeddings, the selection of a sufficiently general finite projection to some smooth scheme, and the construction of a local presentation of the multiplicity as in (1.1.4).Neither of these choices is unique.As a consequence, some work has to be done to show that the values of the function do not depend on any of these different choices.
In this paper we show that the value H-ord More precisely, on the one hand, the value ord X (ζ) can be read studying the Nash multiplicity sequences of arcs in X with center ζ (this was studied in [9] by the last two authors in collaboration with B. Pascual-Escudero).
On the other hand, studying the properties of the asymptotic Samuel function, we came up with the notion of the Samuel slope of a local ring O X,ζ , S-Sl(O X,ζ ) (see Definition 3.3).For a singular point, S-Sl(O X,ζ ) ≥ 1, and we will make a distinction depending on whether S-Sl(O X,ζ ) = 1 (non-extremal case) or S-Sl(O X,ζ ) > 1 (extremal case).Actually, the previous distinction can be made after analyzing properties of the cotangent space m ζ /m 2 ζ .A combination of these pieces of information gives us enough input to compute H-ord (d) X .Our results say that H-ord but more precisely, we can say more: Theorem 8.12.Let X be an equidimensional variety of dimension d defined over a perfect field k.Let ζ ∈ X be a point of multiplicity m > 1. Then: We give an idea of the meaning of this result in the following lines.When the characteristic is zero, the description of the maximum multiplicity locus of X in (1.1.4)goes far beyond that equality.In fact, it can be proven that to lower the maximum multiplicity of X it suffices to work with the elimination algebra R (which is defined on a smooth scheme of dimension d).In other words, a simplification of the multiplicity of the d-dimension variety X becomes a problem about finding a resolution of a Rees algebra defined on a smooth d-dimensional scheme (see Sections 4 and 6).If ord then this indicates that, either the multiplicity of X can be lowered with a single blow up at a regular center, or else, a simplification of the multiplicity of X is a problem that can be solved using certain Rees algebra defined in a (d − 1)-dimensional smooth scheme (see §5.1 for details).Thus, our original problem is, in principle, simpler to solve.And the theorem says that the condition ord The second part of the theorem says that the relevant information from the coefficients of the polynomials in (1.1.3),which, in general, only exists in a suitable étale neighborhood of the point, can already be read through the Samuel slope of the original local ring at the singular point and the sequences of Nash multiplicities of arcs with center the given point.

Organization of the paper
Facts on the asymptotic Samuel function are given in Section 2, and in addition, we study the behavior of this function when consider certain finite extension of rings (Proposition 2.10).In section 3 we define the notion of the Samuel slope of a local ring, and we study his behavior under étale extensions (Propositions 3.10 and 3.11).Rees algebras and their use in resolution of singularities are studied in Sections 4, 5, and 6.The function H-ord X is treated in Section 7. The proof of the main result is addressed in Section 8, here our results from Section 3 are needed.

The asymptotic Samuel function
The asymptotic Samuel function was first introduced by Samuel in [29] and studied afterwards by D. Rees in a series of papers ( [25], [26], [27], [28]).Thorough expositions on this topic can be found in [23] and [31], see also [8] for a generalization to arbitrary filtrations.We will use A to denote a commutative ring with 1.In general, for n ∈ N >1 , the inequality ν I (f n ) ≥ nν I (f ) can be strict.This can be seen for instance by considering the following example.Let k be a field, and let A = k[x, y]/ x 2 − y 3 .Set m = x, y .Then ν m (x) = 1, but ν m (x 2 ) = 3.The asymptotic Samuel function is a normalized version of the previous order that gets around this problem: Definition 2.4.Let I ⊂ A be a proper ideal.The asymptotic Samuel function at I, νI : A → R ∪ {∞}, is defined as:

Definition 2.1. A function
It can be shown that the limit (2.4.1) exists in R ≥0 ∪ {∞} for any ideal I ⊂ A (see [23,Lemma 0.11]).Again, if (A, m) is a local regular ring, then ν m is just the usual order function.As indicated before, this is an order function with nice properties: Proposition 2.5.[23, Corollary 0.16, Proposition 0.19] The function νI is an order function.Furthermore, it satisfies the following properties for each f ∈ A and each r ∈ N:

The asymptotic Samuel function on Noetherian rings
When A is Noetherian, the number ν I (f ) measures how deep the element f lies in the integral closure of powers of I.In fact, the following results hold: Proposition 2.6.[31, Corollary 6.9.1] Suppose A is Noetherian.Then for a proper ideal I ⊂ A and every a ∈ N, Corollary 2.7.Let A be a Noetherian ring and The previous characterization of ν I leads to the following result that give a valuative version of the function.
Theorem 2.8.Let A be a Noetherian ring, and let I ⊂ A be a proper ideal not contained in a minimal prime of A. Let v 1 , . . ., v s be a set of Rees valuations of the ideal Remark 2.9.Let A be a Noetherian reduced ring, and let I ⊂ A be a proper ideal not contained in any minimal prime of A. Set X = Spec(A) and let X be the normalized blow up of X at the ideal I.Then, the sheaf of ideals IO X is invertible and, since X is normal, there is a finite number of reduced and irreducible hypersurfaces H 1 , . . ., H ℓ in X, and there exists an open set U ⊂ X, such that X \ U has codimension at least 2 such that: for some integers c 1 , . . ., c ℓ ∈ Z ≥1 .Denote by v i the valuation associated to O X,h i , where h i is the generic point of H i .Then note that a subset of {v 1 , . . ., v ℓ } has to be a Rees valuation set of I.As an application of Remark 2.9 we can prove the following result about the behavior of the ν function on products of elements.This will be used in the proof of Theorem 8.12.
Proposition 2.10.Let A → C be ring homomorphism of Noetherian rings, where A is regular and C is reduced and equidimensional.Let Q(A) be the fraction field of A. Suppose that no non-zero element of A maps to a zero divisor in C, and that the extension Q(A) → Q(A) ⊗ A C is finite.Let q ∈ Spec(C) and n = q ∩ A. Assume that nC is a reduction of q ⊂ C, and that A/n is regular.If a ∈ A and f ∈ C then: νq (a) = νn (a), and νq (af ) = νq (a) + νq (f ).
Proof.Set X = Spec(C) and Z = Spec(A).Let X be the normalized blow up of X at the ideal q and let Z be the blow up of Z at n. Then there is a commutative diagram (see [3,Lemma 4.2]).The exceptional divisor E of the blow up Z → Z defines only a valuation v 0 in A. The exceptional divisor of X → X defines valuations v 1 , . . ., v ℓ as in Remark 2.9.Note that every valuation v i is an extension of v 0 to C. Then if a ∈ A: for all i = 1, . . ., ℓ.
On the other hand, for each i ∈ {1, . . ., ℓ}, And, again, by Remark 2.9 be have the required equality.

The Samuel slope of a local ring
Let (A, m) be a local Noetherian ring.We will focus on some elements in the associated graded ring Gr m (A) which are nilpotent.They will be used to define the Samuel slope of the local ring.Note that m (≥1) and m (>1) are ideals in A. There is a natural morphism of k(m)-vector spaces, whose kernel is the subspace generated by the degree one nilpotents of Gr m (A).
Remark 3.2.If A is a local regular ring, then ν m = ν m is the usual order function and λ m is an isomorphism.If A is not regular, then we have that dim k(m) m/m 2 = d+t, with t > 0 being the excess of the embedding dimension of (A, m).Note that d = dim(A) = dim(Gr m (A)) = dim(Gr m (A)) red .Therefore, if x 1 , . . ., x d+t ∈ m is a minimal set of generators, then there are at least d elements x i 1 , . . ., x i d , such that their classes in Gr m (A) are not nilpotent.This means that ν m (x i j ) = 1, for j = 1, . . ., d.
Assume that ν m (x 1 ) = . . .= ν m (x d ) = 1.The minimum of ν m (x d+1 ), . . ., ν m (x d+t ) defines a slope with respect to the chosen generators.The Samuel slope is the supremum of all these possible coordinate dependent slopes.
, and let m ⊂ A be the maximal ideal.Then ν m (x 1 ) = 1, ν m (x 2 ) = 5/2 and ν m (x 3 ) = 7/2.It can be checked that S-Sl(A) = 5/2.Remark 3.5.Let Γ be the set of all possible minimal ordered sets of generators x of m.For where the supremum is taken over all the λ m -sequences in the local ring (A, m).
Remark 3.8.Suppose that X is an equidimensional variety of dimension d defined over a perfect field k, and

The Samuel slope and étale extensions.
To prove Theorem 8.12 we will have to work in an étale neighborhood of a given point.To be able to use étale extensions in our arguments, we will first prove that the dimension of ker(λ ζ ) is an invariant under such extensions.From here, it follows that if We do not know if the equality holds in general.However we can prove it for some special cases, which will be enough for our purposes.Lemma 3.9.Let ϕ : (A, m) → (A ′ , m ′ ) be an étale homomorphism of Noetherian local rings.Then Proof.Let d be the Krull dimension of A. Suppose that dim k(m) m/m 2 = d + t, with t > 0. By Lemma 3.9, the result is immediate if dim ker(λ m ) < t, and in fact, in this case, the hypothesis This means that there is some n ≫ 0 such that νm (ρ n ) = νm ′ (θ ′ ).From here we can conclude that given a λ m ′ -sequence θ ′ 1 , . . ., θ ′ t ∈ m ′ we can always find θ 1 , . . ., θ t ∈ m such that : The result now follows by the definition of the Samuel slope and Remark 3.7.
The following result will allow us to compare the Samuel slope of a local ring, at a non closed point of a variety, before and after an étale extension (at least under some special assumptions).This will be used in the proof of Theorem 8.12.
Proposition 3.11.Let (A, m) be a formally d-equidimensional local Noetherian ring.Let p ⊂ A be a prime ideal such that the quotient ring A/p is a (d − r)-dimensional regular ring, with r > 0, and mult m (A) = mult pAp (A p ) = m > 1. Suppose that: • the excess of embedding dimension of (A, m) is t and coincides with the excess of embedding dimension of (A p , pA p ); • both (A, m) and (A p , pA p ) are in the extremal case.Let ϕ : (A, m) → (A ′ , m ′ ) be an étale homomorphism of local rings, and p ′ ⊂ A ′ be a prime ideal such that p ′ ∩ A = p.Assume that: Then there is a λ pAp -sequence at A, γ 1 , . . ., γ t , such that: Proof.We divide the proof in three steps: Step 1.We claim that there are elements y 1 , . . ., y r , y r+1 , . . ., y d ∈ A ′ such that To prove the claim observe first that A ′ := A ′ /p ′ is a regular local ring of dimension (d − r).Therefore, we have that m ′ := m ′ /p ′ = y r+1 , . . ., y d for some y r+1 , . . ., y d ∈ A ′ .Thus we should be able to find r elements, y 1 , . . ., y r in p ′ , with ν m ′ (y i ) = 1 and so that, To see that the last containment is an equality it suffices to prove that q := y 1 , . . ., y r + γ ′ 1 , . . ., γ ′ t is prime and that it defines a (d − r)-dimensional closed subscheme at Spec(B ′ ).But this is immediate since where that last inequality follows because m ′ /q is generated by classes of y r+1 , . . ., y d .
Step 2. Consider the surjective morphism of graded k(m ′ )-algebras: where the T i are variables mapping to the class of To prove the claim, consider the ring of polynomials in d variables over k(m ′ ) localized at the origin, (here we are using the notation from step 1).Setting n := x 1 , . . ., x d ⊂ T , the previous morphism induces another morphism of k(m ′ )-algebras between the graded rings, Gr n (T ) and Gr m ′ (A ′ ), (3.11.4) where ) is a finite extension of Gr n (T ) (here we use the fact that the γ ′ i define nilpotents at Gr m ′ (A ′ )).Now set b := x 1 , . . ., x r ⊂ T .Then we have the following commutative diagram of graded rings: By [24, §5, Theorem 5], ker(ψ ′ ) is nilpotent.Observe that φ is an isomorphism and that D ′ is a finite extension of F (here we use the fact that each [γ To check that the containments are equalities it suffices to observe that Step 3. Consider the commutative diagram, paying attention to the sequence for the n-th degree part from D ′ and D: ).Notice that from Step 2 and [24, §5, Theorem 5], it follows that ker(π From here it follows that κ i,2 ∈ p. Iterating this procedure, we find that and κ i,j ∈ p. Taking n ≫ 0, and setting we have that γ i ∈ p for i = 1, . . ., t, that: and that min i=1,...,t In particular, ν pAp (γ i ) > 1 and ν m (γ i ) > 1 for i = 1, . . ., t.By construction, from where it follows that γ 1 , . . ., γ t ∈ m form both a λ m -sequence and λ m ′ -sequence.We also have that m ′ = y 1 , . . ., y d + γ 1 , . . ., γ t , and that y 1 , . . ., y r + γ 1 , . . ., γ t ⊂ p ′ .To show that the last inclusion is an equality we can argue as in Step 1, to check that A ′ / ( y 1 , . . ., y r + γ 1 , . . ., γ t ) is a (d − r)-dimensional regular local ring.Thus {y 1 , . . ., y r , γ 1 , . . ., γ t } form a minimal set of generators for p ′ A ′ p ′ ⊂ A ′ p ′ , hence γ 1 , . . ., γ t ∈ p form a λ p ′ A ′ p ′ -sequence and therefore a λ pApsequence.

Rees algebras and their use in resolution
The stratum defined by the maximum value of the multiplicity function of a variety can be described using equations and weights ( [36]); and the same occurs with the Hilbert-Samuel function ( [21]).As we will see, Rees algebras happen to be a a suitable tool to work in this setting, opening the possibility to using different algebraic techniques.We refer to [35] and [17] for further details.Definition 4.1.Let A be a Noetherian ring.A Rees algebra G over A is a finitely generated graded A-algebra, G = l∈N I l W l ⊂ A[W ], for some ideals I l ∈ A, l ∈ N such that I 0 = A and I l I j ⊂ I l+j , for all l, j ∈ N. Here, W is just a variable to keep track of the degree of the ideals I l .Since G is finitely generated, there exist some f 1 , . . ., f r ∈ A and positive integers (weights) The previous definition extends to Noetherian schemes in the obvious manner.
In the following lines, we assume that G = ⊕ l≥0 I l W l is a Rees algebra defined on a scheme V that is smooth over a perfect field k (whenever the conditions on V are relaxed it will be explicitly indicated).If we assume V to be affine, then we will write V = Spec(R).
The singular locus of G, Sing(G), is the closed set given by all the points Let m > 1 be the maximum multiplicity at the points of X.Then the singular locus of G = R[f W m ] is the set of points of X having maximum multiplicity m.This idea can be generalized as follows.Suppose X is a d-dimensional variety over a perfect field, and let max mult X be the maximum value of the multiplicity at points of X, Mult X .Then as, explained in the Introduction, using the polynomials in (1.1.3)we have that if . The precise statement of this result will be given in Section 8, since it will play a central role in the proof of Theorem 8.12.
In the previous example, the link between the closed set of points of worst singularities of X and the singular loci of the corresponding Rees algebras is much stronger than just an equality of closed sets Sing(G) = Max mult X .In particular, by defining a suitable law of transformations of Rees algebras after a blow up, we can establish the same link between the closed set of points of worst singularities of the strict transform of X, and the singular locus of the transform of the corresponding Rees algebra (at least if the singularities of X have not improved).This motivates the following definitions.
is the blow up of V at a smooth closed subset Y ⊂ V contained in Sing(G) (a permissible center for G).We use G 1 to denote the (weighted) transform of G by π, which is defined as Definition 4.4.Let G be a Rees algebra over a smooth scheme V .A resolution of G is a finite sequence of blow ups (4.4.1) at permissible centers Y i ⊂ Sing(G i ), i = 0, . . ., L − 1, such that Sing(G L ) = ∅, and such that the exceptional divisor of the composition V 0 ←− V L is a union of hypersurfaces with normal crossings.
Remark 4.5.The Rees algebras of Example 4.2 are defined so that a resolution of the corresponding Rees algebra, G (4.4.1), induces a sequence of blow ups on X, that ultimately leads to a simplification of the multiplicity of X as in (1.0.1).Notice that for these sequences Sing(G i ) = Max mult X i , for i = 0, 1, . . ., L.
Resolution of Rees algebras is known to exists when V is a smooth scheme defined over a field of characteristic zero ( [20], [21]).In [32] and [7] different algorithms of resolution of Rees algebras are presented (see also [16], [15]).More details will be given in the next section.
4.6.On the representation of the multiplicity by Rees algebras.In addition to permissible blow ups, there are other morphisms that play a role in resolution.These are involved in the arguments of Hironaka's trick, and they are used to justify that the resolution invariants are well defined ([10, §21]).Some of these invariants will be treated in the following sections.Apart from permissible blow ups, these morphisms are multiplications by an affine line or restrictions to open subsets.A concatenation of any of these three kinds of morphisms is what we call a local sequence.Therefore, for a given Rees algebra G defined on a smooth scheme V , a G-local sequence over V is a sequence of transformations over V , (4.6.1) where each π i is either a permissible blow up for (and G i+1 is the transform of G i in the sense of Definition 4.3), or a multiplication by a line or a restriction to some open subset of V i (and then G i+1 is the pull-back of G i in V i+1 ).If we assume that sequence (4.4.1) is a G-local sequence over V (instead of just a sequence of permissible blow ups), with G as in Example 4.5, then the equality Max mult X i = Sing(G i ) still holds for each i = 1, . . ., L − 1.Because of this fact we say that the pair (V, G) represents the closed set Max mult X , since there is such a strong link between the two closed sets Sing(G i ) and Max mult X i along the sequence.The same can be said about the representation of the Hilbert-Samuel function in [21].See [14] for precise definitions and results on local presentations.It is worth noticing that for a given Rees algebra G = ⊕ l I l W l there is always some integer N such that G is finite over R[I N W N ] (see [17,Remark 1.3]).
(ii) Rees algebras and saturation by differential operators.Let β : V → V ′ be a smooth morphism of smooth schemes defined over a perfect field k with dim V > dim V ′ .Then, for any integer s, the sheaf of relative differential operators of order at most s, Diff s V /V ′ , is locally free over V ([18, (4) § 16.11]).We will say that a sheaf of O V -Rees algebras G = ⊕ l I l W l is a β-differential Rees algebra if there is an affine covering {U i } of V , such that for every homogeneous element Given an arbitrary Rees algebra G over V there is a natural way to construct a β-relative differential algebra with the property of being the smallest containing G, and we will denote it by Diff V /V ′ (G) (see [34,Theorem 2.7]).Relative differential Rees algebras will play a role in the definition of the so called elimination algebras, see Section 6.
We say that G is differentially closed if it is closed by the action of the sheaf of (absolute) differential operators Diff V /k .We use Diff(G) to denote the smallest differential Rees algebra containing G (its differential closure).See [34,Theorem 3.4] for the existence and construction.
It can be shown that Sing(G) = Sing(G) = Sing(Diff(G)), (see [35,Proposition 4.4 (1), (3)]).In addition, it can be checked that if G represents Max mult X as in Example 4.2, then the integral closure of Diff(G) is the largest algebra in V with this property.The previous discussion motivates the following definition: two Rees algebras on V , G and H, are said to be weakly equivalent if: (i) they share the same singular locus; (ii) any G-local sequence is an H-local sequence, and vice versa, and they share the same singular locus after any G-(respectively H-)local sequence.It can be proven that two Rees algebras G and H are weakly equivalent if and only if Diff(G) = Diff(H) (see [11] and [22]), and, in particular, a resolution of one of them induces a resolution of the other and vice versa.

Algorithmic resolution and resolution invariants
In characteristic zero, an algorithmic resolution of Rees algebras requires the definition of resolution invariants.These are used to assign a string of numbers to each point ζ ∈ Max mult X = Sing(G).In this way one can define an upper semi-continuous function g : Sing(G) → (Γ, ≥), where Γ is some well ordered set, and whose maximum value determines the first center to blow up.This function is constructed so that its maximum value drops after each blow up.As a consequence, a resolution of G is achieved after a finite number of steps.
The most important resolution invariant is Hironaka's order function at a point ζ ∈ Sing(G) which we also refer as the order of the Rees algebra G at ζ, and it is defined as ord It can be shown that two Rees algebras that are weakly equivalent share the same resolution invariants and therefore a resolution of one induces a resolution of the other.In particular, this is the case for G, G and Diff(G) ([17, Proposition 3.4, Theorem 4.1, Theorem 7.18], [37]).
5.1.The role of Hironaka's order in resolution and the use of induction in the dimension.Suppose G is defined on a smooth scheme V of dimension n, and assume that ord ξ (G) = 1 for some closed point ξ ∈ Sing(G).Then, there are two possibilities: (i) Either the point ξ is contained in some codimension-one component Y of Sing(G); in such case it can be proven that Y is smooth, and the blow up at Y induces a resolution of G, locally at ξ ([12, Lemma 13.2]); (ii) Otherwise, it can be shown that, locally, in an étale neighborhood of ξ, there is a smooth projection from V to some smooth (n − 1)-dimensional scheme Z, together with a new Rees algebra R on Z such that a resolution of R induces a resolution of G and vice versa, at least if the characteristic is zero.This is what we call an elimination algebra of G and details on its construction will be given in the next section.Case (ii) indicates that resolution of Rees algebras can be addressed by induction on the dimension when the characteristic is zero.
It is worthwhile mentioning that if the maximum order at the points of Sing(G) is larger than one, then one can attach a new Rees algebra H to the closed points of maximum order, Max ord(G), so that Sing(H) = Max ord(G), and so that the equality is preserved by H-local sequences.Thus H is unique up to weak equivalence.This new Rees algebra H is constructed so that its maximum order equal to one, and the arguments in (i) and (ii) can be applied to it.

Elimination algebras
Along this and the following sections, V (n) denotes an n-dimensional smooth scheme over a perfect field k, and G (n) = ⊕ l I l W l a Rees algebra over V (n) .Our purpose is to search for smooth morphisms from V (n) to some (n−e)-dimensional smooth scheme, for some e ≥ 1, so that Sing(G (n) ) is homeomorphic to its image via β, and so that this condition is preserved by permissible blow ups in some sense that will be specified below.One way to find such smooth morphisms is by considering morphisms from V (n) which are somehow transversal to G (n) .Transversality is expressed in terms of the tangent cone of G (n) at a given point of its singular locus (see Definition 6.4 below).
Let ξ ∈ Sing(G (n) ) be a closed point, and let Gr , where k ′ is the residue field at ξ. Observe that Spec(Gr , where d ξ β denotes the differential of β at the point ξ. , is G (n) -admissible locally at ξ if the following conditions hold: (1) The point ξ is not contained in any codimension-e-component of Sing G (n) ; (2) The Rees algebra G (n) is a β-relative differential algebra (see §4.7 (ii)); (3) The morphism β is G (n) -transversal at ξ.
Regarding condition (1), if ξ is contained in a codimension-e-component of Sing G (n) then this component is a permissible center, see §5.1.Under the previous conditions, it is always possible to construct a G (n) -admissible morphism in an (étale) neighborhood of ξ (see [34] and also [12, §8.3]).Definition 6.5.[34,12] Let β : V (n) → V (n−e) be a G (n) -admissible projection in an (étale) neighborhood of the closed point ξ.Then the O V (n−e) -Rees algebra and any other with the same integral closure in O V (n−e) [W ], is an elimination algebra of G (n) in V (n−e) (see [34,Theorem 4.11]).Example 6.6.Let S be a smooth d-dimensional k-algebra of finite type, with d > 0. Let V (d+1) =

Spec(S[x]). Then the natural inclusion
represents the multiplicity function on X locally at ξ.If the characteristic is zero and if we assume that f has the form of Tschirnhausen (there is always a change of coordinates that leads us to this form): (6.6.1) where a i ∈ S for i = 0, . . ., m − 2, then it can be shown that, up to integral closure, is an elimination algebra of G (d+1) .If the characteristic is positive, the elimination algebra is also defined.In either case, it can be shown that it is generated by a finite set of some symmetric (weighted homogeneous) functions evaluated on the coefficients of f (x) (cf.[33], [34, §1, Definition 4.10]).It is worthwhile noticing that the elimination algebra G (d) is invariant under changes of the form x ′ = x + α with α ∈ S [34, §1.5].Finally, we will see that, to understand elimination algebras in a more general setting, it suffices to treat the hypersurface case, at least for the purposes of this paper (see §8.7, specially (8.7.1) and (8.7.2)).
6.7.Properties of elimination algebras.Let β : V (n) → V (n−e) be a G (n) -admissible projection in an (étale) neighborhood of a closed ξ ∈ Sing(G (n) ), and let G (n−e) ⊂ O V (n−e) [W ] be an elimination algebra.Then Sing(G (n) ) maps injectively into Sing(G (n−e) ), in particular β(Sing(G (n) )) ⊂ Sing(G (n−e) ) with equality if the characteristic is zero, or if G (n) is a differential Rees algebra (see [12, §8.4]).Moreover, If G (n) is a differential Rees algebra, then so is G (n−e) (see [34,Corollary 4.14]).And if 6.8.Hironaka's order of an algebraic variety.Let X be an equidimensional variety of dimension d over a perfect field k and let ζ ∈ X be a point of maximum multiplicity m > 1.We can assume that X = Spec(B) is affine.Let ξ ∈ {ζ} be a closed of multiplicity m.Then, as indicated in Example 4.2, there is an étale neighborhood of Spec(B), X ′ = Spec(B ′ ), an embedding in some smooth (d + e)-dimensional scheme V (d+e) , and a differential Rees algebra G (d+e) representing the top multiplicity locus of X ′ .In §8.7 we will see that under these assumptions, τ G,ξ ′ ≥ e, and there is a G (d+e) -admissible projection to some d-dimensional smooth scheme where an elimination algebra G (d) can be defined.Let ζ ′ ∈ X ′ be a point mapping to ζ.Then by §6.7, ord G (d+e) (ζ ′ ).does not depend on the selection of the étale neighborhood, nor on the choice of Rees algebra representing the top multiplicity locus, nor on the admissible projection.We refer to this rational number as Hironaka's order function of X at ζ in dimension d.

The function H-ord
When facing an algorithmic resolution of the variety X in characteristic zero, the number ord (d) X (ζ) is the most important invariant at the point ζ (after the multiplicity), and there is a strong link between the resolutions of G (d+e) and G (d) : in particular, a resolution of the first induces a resolution of the second and vice versa.When the characteristic is positive, this link between G (d+e) and G (d) is weaker, as illustrated in the following example.
, and let ξ be the singular point of X.The inclusion and β(Sing(G (2) ) = Sing(G (1) ).However, after the blow up at ξ, Sing(G   d) , needs to be refined.This leads us to talk about the function H-ord X , introduced and studied in [5] and [6].We will start with the definition for hypersurfaces, and then we will see that the general case reduces to that of hypersurfaces.7.2.The hypersurface setting.Let V (d+1) be (d + 1)-dimensional smooth scheme over a perfect field k, let X ⊂ V (d+1) be a hypersurface of dimension d, and let ξ ∈ X be a closed point of maximum multiplicity m > 1. Choose a local generator f ∈ O V (d+1) ,ξ defining X in an open affine neighborhood U ⊂ V (d+1) of ξ, which we denote by V (d+1) for simplicity.Define the Rees algebra After applying Weierstrass Preparation Theorem, we can assume that in an étale neighborhood of ξ ∈ V (d+1) , which we again denote by V (d+1) , we have the following situation.There is an affine smooth scheme of dimension d, V (d) = Spec(S), such that V (d+1) = Spec(S[z]), where is z is a variable, and X is defined by (7.2.1) It can be checked that the morphism β : We say that f is written in Weierstrass form with respect to the projection β.
Remark 7.3.[6, §2.15]With the same notation as in §7.2, it can be proved that, in a neighborhood of ξ, G (d+1) has the same integral closure as d) , where G (d) is an elimination algebra of G (d+1) , the ∆ i z are the Taylor differential operators, and we use "⊙" to denote the smallest Rees algebra containing the two that are involved in the expression.Recall that {∆ 0 z , . . ., ∆ r z } is a basis of the free module of S-differential operators of S[z] of order r (see [5,Proposition 2.12]; see also Example 6.6).We will say that (7.3.1) is a simplified presentation of G (d+1) at ξ.The presentation depends on the choice of the smooth morphism β, the variable z and the monic generator f W m .We will use P(β, z, f W m ) to denote this simplified presentation.
does not depend on the choice of the transversal morphism β, nor on the choice of the order-oneelement f W m ∈ G (d+1) (f W m can be replaced by any other order-one-element gW m 1 ∈ G (d+1) non necessarily defining the hypersurface X).Moreover, the supremum in (7.5.1) is a maximum for a suitable selection of z ′ .See [5, §5.2 and Theorem 7.2].
Definition 7.6.[6, §5, Definition 5.12] Let ζ ∈ X be a point of a hypersurface X of multiplicity m > 1, and consider an étale neighborhood X ′ → X of a closed point of multiplicity m, ξ ∈ {ζ}, such that the setting of §7.2 holds, and let ζ ′ ∈ X ′ be a point mapping to ζ.Then we define H-ord Remark 7.7.When the characteristic of the base field k is zero, then it can be shown that for all ζ ∈ Sing(G (d+1) ), H-ord [6, §2.13] and Example 6.6).Thus, this invariant provides new information only when the characteristic of k is positive.For example, if X is as in Example 7.1, it can be checked that H-ord 7.8.p-Presentations.Suppose char(k) = p > 0. Continuing with the notation introduced in §7.2, since G (d+1) is a differential algebra, in order to compute the value β-ord(ξ), it is always possible to find an order-one-element of the form hW p ℓ ∈ G (d+1) , where h is a monic polynomial of degree p ℓ for some ℓ ∈ Z ≥1 , and in Weierstrass form with respect to β.This can be done as follows.Assume that g(z)W N ∈ G (d+1) and that where, for j = 1, . . ., p ℓ −1, bj = c j N ′ b j for some integer c j , and bp ℓ = 1 N ′ b p ℓ .Then h(z)W p ℓ ∈ G (d+1) and P(β, z, h(z)W p ℓ ) is a special type of simplified presentation of G (d+1) .Presentations of the form P(β, z, hW p ℓ ) will be called p-presentations ([5, Definition 2.14]).Compared to general simplified presentations, p-presentations have the advantage that the computation of the slope (7.4.1) becomes simpler.
Hence to maximize the value Sl(P)(ζ) for a given p-presentation P(β, z, hW p ℓ ), one can work with presentations in normal form.For simplicity we restrict the notion of normal form to ppresentations, but a similar concept can be defined for any presentation, see [6, §5.7].
Remark 7.13.Given a hypersurface X and G (d+1) as in §7.2, for a point ζ ∈ Sing(G (d+1) ), and a p-presentation P(β, z, hW p ℓ ) in normal form at ζ, it can be shown that

The general case
Given an equidimensional variety X of dimension d over a perfect field k, and a singular point ζ ∈ X, we would like to emulate the previous statements, which were valid for a hypersurface.To this end, we will use the following result, which can be understood as a generalization of Weierstrass preparation theorem.
Theorem 7.14.[6, Theorem 6.5] Let G (n) be a Rees algebra on a smooth scheme V (n) over k and let ξ ∈ Sing(G (n) ) be a closed point with τ G (n) ,ξ ≥ e ≥ 1.Then, at a suitable étale neighborhood of ξ, a G (n) -transversal morphism, β : V (n) → V (n−e) , can be defined so that the following conditions hold: (i) There are global functions z 1 , . . ., z e in O V (n) such that {dz 1 , . . ., dz e } forms a basis of Ω 1 β , the module of β-relative differentials; (ii) There are positive integers m 1 , . . ., m e ; (iii) There are elements f 1 W m 1 , . . ., f e W me ∈ G (n) , such that: (7.14.1) for some global functions a i ∈ O V (n−e) ; (iv) The Rees algebra G (n) has the same integral closure as: where G (n−e) is an elimination algebra of G (n) on V (n−e) , and the set consists of the relative differential operators described in by the Taylor operators.
Remark 7.15.Observe that since β : are defined to be the Taylor differential operators.
Definition 7.16.[6, Definition 6.6]With the setting and the notation of Theorem 7.14, the data, (7.16.1) that fulfills conditions (i)-(iv) in Theorem 7.14 is a simplified presentation of G (n) .Let X i be the hypersurface defined by Then we can also define H-ord

H-ord
(n−e) X i .
Remark 7.17.Now we go back to Example 4.2, where we consider a representation of the multiplicity of a variety X ⊂ V at a closed point ξ ∈ X, given by a Rees Algebra We will see in §8.7 that Diff(G) satisfies conditions (i)-(iv) in Theorem 7.14.This leads us to define H-ord X i (ζ)}, where X i is the hypersurface defined by f i , i = 1, . . ., e, and ζ ∈ Max mult X .

Main results
In this section we will address the proof of Theorem8.12.For a given point ζ ∈ X of maximum multiplicity m > 0, we will want to compute the value H-ord (d) X (ζ) following the constructions given in Section 7. To this end, we will use Villamayor's presentations of the multiplicity in the étale topology, Theorem 8.1 below.Finally, since we want to that H-ord (d) X (ζ) can actually be computed at O X,ζ , without the need of étale topology, and using the Samuel slope of the local ring, we will be using our results from Section 3. Theorem 8.1.[36, Lemma 5.2, §6, Theorem 6.8] (Presentations for the Multiplicity function) Let X = Spec(B) be an affine equidimensional algebraic variety of dimension d defined over a perfect field k, and let ξ ∈ Max Mult X be a closed point of multiplicity m > 1.Then, there is an étale neighborhood B ′ of B, mapping ξ ′ ∈ Spec(B ′ ) to ξ, so that there is a smooth k-algebra S together with a finite morphism α : Spec(B ′ ) → Spec(S) of generic rank m, i.e., if and there is a commutative diagram: (ii) Let V (d+e) = Spec(R), and let I(X ′ ) be the defining ideal of X ′ at V (d+e) .Then f 1 , . . ., f e ⊂ I(X ′ ); (iii) Denoting by m i the maximum order of the hypersurface represents the top multiplicity locus of X, Max Mult X , at ξ in V (d+e) .
8.2.The setting and the notation for the proof of Theorem 8.12.Let ξ ∈ X be a closed point of multiplicity m > 1, and let (B, m, k(ξ)) the local ring at the point.Applying Theorem 8.1 there is an étale extension (B, m, k(ξ)) → (B ′ , m ′ , k ′ ) for which we can find a smooth k ′ -algebra S and a finite inclusion of generic rank m, Thus, statements (i), (ii) and (iii) of Theorem 8.1 hold for S ⊂ B ′ .In particular, we have a commutative diagram like (8.1.1).With this notation, which we fix for the rest of the section, we will be simultaneously using α(ζ ′ ) and β(ζ ′ ) to denote the image in Spec(S) of a point ζ ′ ∈ Spec(B ′ ).We will choose the first notation if we want to use the properties of the finite projection from Spec(B ′ ).The second notation will be convenient to emphasize the fact that ζ ′ is also a point in the smooth scheme Spec(R).Sometimes we will use V (d+e) to refer to Spec(R).This will help us recall the dimension of the smooth ambient space where Spec(B ′ ) is embedded, and the space where the Rees algebra G (d+e) is defined.And for similar reasons we occasionally will write V (d)  for Spec(S), specially if the elimination algebra G (d) of G (d+e) is involved (see Section 6).
Theorem 8.1 provides three pieces of information that will be specially relevant in our arguments: (I) The existence of the étale neighborhood of B, B ′ together with the finite extension S ⊂ B ′ .
To be able to compare the Samuel slope of B and B ′ (in the extremal case) we will need to know that B ′ can be constructed having the same residue field as B. This issue is addressed in §8.3.(II) The Rees algebra G (d+e) representing the top multiplicity locus of X ′ = Spec(B ′ ).We will see in §8.7 below how to use this Rees algebra to compute the function H-ord X ′ using the results from Section 7. (III) An algebraic presentation of B ′ as an algebra over S, S[θ 1 , . . ., θ e ].We will see in §8. 8 below how to find suitable presentations that will help us computing the Samuel slope in the extremal case.After addressing (I), (II), (III), and after establishing some technical results, we will give the proof of Theorem 8.12.We start by stating a giving an idea of the proof of Proposition 8.4 below.This result was sketched in [36, §6.11] and a complete proof can be found in [14,Appendix A].Here we will focus on the three main steps of the argument that require considering étale extensions.Remark 8.5 and Proposition 8.6 below will be relevant to treat the proof of Theorem 8.12 in the extremal case.
Step 1: If k(ξ) is the residue field at ξ, then, after considering the extension B 1 = O X,ξ ⊗ k k(ξ) it can be assumed that the point of interest is rational.Let m 1 be a maximal ideal of B 1 dominating m ξ .Then if k 1 := B 1 /m 1 , we have that k 1 = k(ξ).
Step 2: After a finite extension of the base field k 1 , k 2 , considering the base change To ease the notation set B 2 := (B 2 ) f .
Step 3: Finally, after considering an étale extension S 3 of S 2 (inside the henselization of the local ring (S 2 ) Y 1 ,...,Y d ; the strict henselization is not needed in this step), it can be assumed that the extension S 3 → B 3 is finite of generic rank equal to m.Let n 3 ⊂ S 3 be the maximal ideal dominating Y 1 , . . ., Y d .Notice that the residue field of S 3 at n 3 is again k 2 .
Regarding to (ii), it suffices to observe that that from the way the finite projection S → B ′ is constructed (see step 2), the morphism Gr m α(ξ ′ ) (S) → Gr m ξ ′ (B ′ ) is injective.Note that the elements κ 1 , . . ., κ d are analytically irreducible over k 2 .
Remark 8.5.In the proof of Proposition 8.4 we have a sequence of étale local extensions: leading to the (étale) extensions of graded rings: X ′ .Theorem 8.1 says that the O V (d+e) -Rees algebra G (d+e) in (8.1.2) represents the maximum multiplicity locus of Spec(B ′ ) in V (d+e) (see §4.6).We can assume that the order m i of each ) represents the maximum multiplicity of the hypersurface defined by f i (x i ) in V (d+1) i = Spec(S[x i ]), for i = 1, . . ., e.By identifying G (d+1) i with its pull-back in V (d+e) , we have that: (8.7.1) ).
As indicated in Remark 7.17, G (d+e) has the same integral closure as which in turns is a simplified presentation of G (d+e) (see Theorem 7.14).We will write: (8.7.4) with a (i) j ∈ S, for j = 1, . . ., m i , and i = 1, . . ., e. (A) The slope of a p-presentation at the closed point of Spec(B ′ ).
From (ii) it follows that, after a translation of the form θ i + s i , for some s i ∈ S, we can also assume that θ i ∈ m ξ ′ for i = 1, . . ., e, and that in addition, m ξ ′ = m α(ξ ′ ) B ′ + θ 1 , . . ., θ e .
With the previous notation, the slope of the p-presentation P at ξ ′ (8.7.5) is (8.7.7) From the exposition in §7.11, it follows that a p-presentation P ′ with Sl(P X ′ (ξ ′ ) can be found starting from the presentation P after considering translations of the form θ ′ i := θ i + s i with s i ∈ S, and so that for each translation (8.7.8) Finally, the restriction of G (d+e) to B ′ , G B ′ , is finite over the expansion of G (d) in B ′ , G (d) B ′ (see [36,Theorem 4.11] and [3,Corollary 7.7]).Write G B ′ = ⊕ n J n W n and define Then, by Proposition 2.10, and using the fact that m α(ξ ′ ) B ′ is a reduction of m ′ , it can be checked that (8.7.9) (here it suffices to use arguments similar to those in the proof of [23,Proposition 0.20]).
(B) The slope of a p-presentation at non-closed points of Spec(B ′ ).
(C) The slope of a p-presentation at non-closed points defining regular subschemes of Spec(B ′ ).Now suppose that η ′ is the generic point of a regular closed subscheme at ξ ′ .In such case, it can be shown that p α(η ′ ) also defines a regular closed subscheme at α(ξ ′ ) (cf. [36,Proposition 6.3]).In addition, after translating the elements θ i by elements in S, it can be assumed that B ′ = S[θ 1 , . . ., θ e ] with θ i ∈ p η ′ , and that moreover, p α(η ′ ) B is a reduction of p η ′ (without localizing at p η ′ , see [3,Lemma 3.6]).
With the previous notation, the slope of the p-presentation P at η ′ (8.7.5) equals to: (8.7.11) see [6,Definition 6.7].Going back to the discussion in §7.11, recall that a p-presentations P ′ with Sl(P ′ )(η ′ ) = H-ord X ′ (η ′ ) can be found after considering translations of the form θ ′ i := θ i + s i with s i ∈ S and so that for each translation, (8.7.12) We emphasize here that there is no need to consider translations with s i ∈ S p η ′ .
To conclude, considering G B ′ as before, recall that, ord η On the other, since p α(η ′ ) B ′ is a reduction of p η ′ , and 8.8.(III) Finding suitable algebraic presentations for B ′ (for the extremal case).

X
, where d is the dimension of X.The function ord(d)X is a positive rational number.At a given singular point ζ ∈ X, f X (ζ) would look as follows:(1.0.3) f X (ζ) = (mult m ζ (O X,ζ ), ord(d)X (ζ), . ..) ∈ N × Q r , where mult m ζ (O X,ζ ) denotes the multiplicity of the local ring O X,ζ at the maximal ideal m ζ .The remaining coordinates of f X (ζ) can be shown to depend on ord

X
(ζ) codifies information from the coefficients of the polynomials in (1.1.3)that only depends on the inclusion S ⊂ R. Observe that the definition of the function H-ord (d)

X
(ζ) can be read from the arc space of X combined with the use of information provided by the asymptotic Samuel function at the maximal ideal of the local ring at ζ.In particular, no étale extensions and no local embeddings into smooth schemes are needed: the information is already present in the cotangent space at ζ, m ζ /m 2 ζ , and the space of arcs in X with center at ζ, L(X, ζ).

Example 2 . 3 .
iii) w(0) = ∞ and w(1) = 0. Remark 2.2.[23, Remark 0.3] If w is an order function then w(x) = w(−x) and if w(x) = w(y) then w(x + y) = min{w(x), w(y)}.Let I ⊂ A be a proper ideal.Then the function ν I : A → R ∪ {∞} defined by ν I (f ) := sup{m ∈ N | f ∈ I m } is an order function.If (A, m) is a local regular ring, then ν m is just the usual order function.

2. 11 .
Notation.Along this paper we will interested in computing the function order ν at points ζ in a variety X over a field k.We will be distinguishing between ν ζ and ν p ζ where p ζ is the prime defining ζ in an affine open set of X.In the first case, for an element f ∈ O X,ζ , ν ζ (f ) is computed using the function ν for the local ring O X,ζ at the maximal ideal m ζ = p ζ O X,ζ .In the second case, for an element f ∈ B, where Spec(B) ⊂ X is an affine open containing ζ, ν p ζ is computed using the function ν for the ring B at the prime ideal p ζ .Note that ν ζ (f ) ≥ ν p ζ (f ).If the local ring O X,ζ is regular then we will use the standard notation ν ζ for the usual order function, and then ν ζ = ν ζ .

Definition 3 . 3 .
Let (A, m) is a Noetherian local ring of dimension d and embedding dimension d + t, with t > 0. Let x = {x 1 , . . ., x d+t } ⊂ m be a minimal set of generators of m.We define the slope with respect to x as Sl x (A) := min{ν m (x d+1 ), . . ., ν m (x d+t )}.The Samuel slope of the local ring A is S-Sl(A) := sup x Sl x (A) = sup x {min {ν m (x d+1 ), . . ., νm (x d+t )}} , where the supremum is taken over all possible minimal set of generators x of m.Example 3.4 be the embedding dimension at ζ, where k(ζ) denotes the residue field of O X,ζ .The Samuel slope of X at ζ is the Samuel slope of the local ring O X,ζ , and a λ ζ -sequence will be a λ m ζ -sequence.

4. 7 .
Uniqueness of the representations of the multiplicity.The Rees algebra of Example 4.2 is not the unique representing Max mult X .To see this, we consider two operations:(i) Rees algebras and integral closure.Two Rees algebras over a (not necessarily regular) Noetherian ring R are integrally equivalent if their integral closure in Quot(R)[W ] coincide.We use G for the integral closure of G, which can be shown to also be a Rees algebra over R ([11, §1.1]).
∅. Thus, when the characteristic is positive, what we consider the first relevant invariant in characteristic zero, ord

Proposition 8 . 4 .
[36,  §6.11], [14, Appendix A] Let X be an equidimensional variety defined over a perfect field k and let ξ ∈ X be a closed point of multiplicity m > 1.Let (B, m, k(ξ)) be the local ring at the point.Then there is a local étale extension (B, m, k(ξ)) → (B ′ , m ′ , k ′ ) such that:(i) There is a smooth k ′ -algebra S and a finite morphism S → B ′ of generic rank equal to m;(ii) If α : Spec(B ′ ) → Spec(S), then the morphism Gr m α(ξ ′ ) (S) → Gr m ξ ′ (B ′ ) is injective, and if, in addition, B is in the extremal case, then
Now we are ready to address the proof of our main theorem: Theorem 8.12.Let X be an equidimensional variety of dimension d defined over a perfect field k.Let ζ ∈ X be a point of multiplicity m > 1. Then:• If S-Sl(O X,ζ ) = 1, then 1 = S-Sl(O X,ζ ) = H-ord (d) X (ζ) ≤ ord (d)X (ζ).In addition, if ζ is a closed point then also ord
s .
is in the non-extremal case.If dim ker(λ m ) = t, then we say that a sequence of elements γ 1 , . . ., γ t ∈ m is a λ m -sequence if their classes γi ∈ m/m 2 form a basis of ker(λ m ).In other words, γ 1 , . . ., γ t ∈ m is a λ m -sequence if their classes in Gr m (A) are nilpotent and γ 1 , . . ., γ t are part of a minimal set of generators of m.
there is a maximal ideal m 2 ⊂ B 2 , dominating m 1 , such that m 2 contains a reduction generated by d elements, κ 1 , . . ., κ d .To achieve this step, a graded version of Noether's Normalization Lemma is used at the graded ring Gr m 2 (B 2 ).Letting k 2 = B 2 /m 2 we get a k 2 -morphism from a polynomial ring in d variables with coefficients in k 2 to some localization of B 2 : ). Proposition 8.6 below guarantees that the field extension in Step 2 of the proof is not needed if (B, m) is in the extremal case.Under this assumption all the graded rings in (8.5.1) are isomorphic.Proposition 8.6.Let X be an equidimensional algebraic variety of dimension d defined over a perfect field k, and let ξ ∈ X be a closed point of multiplicity m > 1 with local ring (O X,ξ , m ξ , k(ξ)).Assume that the embedding dimension at ξ is (d + t) for some t ≥ 1.If ξ is in the extremal case, then m ξ has a reduction a ⊂ m ξ generated by d-elements.Proof.To prove the statement it is enough to show that there are d-elements κ 1 , ...,κ d ∈ m ξ \ m ξ 2 such that if κ 1 , ..., κ d denote their images in m ξ /m 2 ξ , then Gr m ξ (O X,ξ )/ κ 1 , ...,κ d is a graded ring of dimension 0 (see [19, Theorem 10.14]).Since dim k(ξ) m ξ /m 2 ξ = d + t and by hypothesis dim k(ξ) ker(λ ξ ) = t, we can find generators of m ξ , (8.6.1)κ 1 , . .., κ d , δ 1 , . .., δ t such that δ 1 , . .., δ t form a basis of ker(λ ξ ).Notice that the elements δ 1 , . .., δ e are nilpotent in Gr m ξ (O X,ξ )/ κ 1 , . .., κ d (see §3.1).Since the graded ring Gr m ξ (O X,ξ ) is generated in degree one by {κ 1 , . .., κ d , δ 1 , . .., δ t } it follows that the quotient Gr m ξ (O X,ξ )/ κ 1 , . .., κ e is a graded ring of dimension zero and hence κ 1 , . .., κ d is a reduction of m ξ .Observe that the previous proposition holds for any local Noetherian ring in the extremal case.