Differential operators on smooth schemes and embedded singularities

Orlando Villamayor U.

En recuerdo de Angel Larrotonda

Abstract
Differential operators on smooth schemes have played a central role in the study of embedded desingularization.

J. Giraud provides an alternative approach to the form of induction used by Hironaka in his Desingularization Theorem (over fields of characteristic zero). In doing so, Giraud introduces technics based on differential operators. This result was important for the development of algorithms of desingularization in the late 80's (i.e. for constructive proofs of Hironaka's theorem).

More recently, differential operators appear in the work of J. Wlodarczyk ([35]), and also on the notes of J. Kollár ([25]).

The form of induction used in Hironaka's Desingularization Theorem, which is a form of elimination of one variable, is called maximal contact. Unfortunately it can only be formulated over fields of characteristic zero.

In this paper we report on an alternative approach to elimination of one variable, which makes use of higher differential operators. These results open the way to new invariants for singularities over fields of positive characteristic ([34]).

Key words and phrases. Resolution of singularities. Desingularization
2000 Mathematics subject classification. 14E15.

Contents

Part 1.  Introduction.
 1.  Monoidal transformations and Hironaka's topology.
 2.  Integral closure of Rees algebras and a notion of equivalence.
 3.  On differential structures and Kollár's tuned ideals.
 4.  On differential structures and monoidal transformations.
 5.  Idealistic exponents versus basic objects.
 6.  Projection of differential structures and elimination of one variable.
References

Part 1. Introduction.

Let be a smooth scheme over a field of characteristic zero, and let be a singular subscheme. Hironaka proves embedded desingularization of , considering as invariants the Hilbert-Samuel functions at the points of . His proof is based on the reduction of Hilbert Samuel functions by monoidal transformations ([22]).

There is second theorem of Hironaka, used in his proof of reduction of Hilbert Samuel functions, which is called Log-resolution of ideals in smooth schemes. For this second theorem, which we discuss below, the invariant considered is the order of the ideal at the points of the smooth scheme.

In [17], both theorems are linked in a different way. In fact, if is defined by a sheaf of ideals , then desingularization is proved by considering the order of the ideal at points in , and hence avoiding the use of Hilbert Samuel functions.

Let be a smooth scheme over a field , and let be a non-zero sheaf of ideals. Define a function

where denotes the order of at the local regular ring . Let denote the biggest value achieved by this function (the biggest order of ). The pair is the object of interest in Log principalization of ideals. There is a closed set attached to this pair in , namely the set of points where has order ; and there is also a notion of transformation of such pairs by blowing up suitable regular centers.

We will attach to a graded subring of (sheaf of polynomial rings), namely a graded algebra (Rees algebra) of the form

defined uniquely in terms of and .

Actually the Rees algebras that we will consider are closely related to Kollárs notion of tuned ideals.

We will show that there is a closed set in naturally attached to such Rees algebra, and also a notion of transformation. Of course the interest here is on the case of smooth schemes over fields of positive characteristic, where a weak form of elimination of one variable is discussed.

For any non-negative integer the sheaf of -linear differential operators, say , is coherent and locally free over .

There is a natural identification, say , and for each there is a natural inclusions .

If is an affine open set in , each is a differential operator: . We define an extension of a sheaf of ideals , say , so that over the affine open set , is the extension of defined by adding all elements , for all and .

So , and as sheaves of ideals in . Let be the closed set defined by . So

It is simple to check that the order of the ideal at the local regular ring is if and only if .

The previous observations say that is an upper-semi-continuous function, and that the highest order of (at points ) is , if and . Let

denote the blow up of at a smooth irreducible sub-scheme , and is the exceptional hypersurface. If we say that is -permissible. In such case

where is the sheaf of functions vanishing along the exceptional hypersurface .

If is -permissible, has at most order at points of (i.e. that . If, in addition, has no point of order , then we say that defines a -simplification of .

If , let denote the monoidal transformation with center . We say that is -permissible, and set

It turns out that has at most points of order . If it does, define a -permissible transformation at some smooth irreducible center .

For and as before, we define, by iteration, a -permissible sequence

and a factorization

Let denote the strict transform of exceptional hypersurface . Note that:

1) are the irreducible components of the exceptional locus of .

2) The total transform of relates to by an expression of the form

We say that this -permissible sequence defines a -simplication of if has normal crossings, and (i.e. has order at most at ).

When is a field of characteristic zero, and is the highest order of a sheaf of ideals , Hironaka proves that there is a -simplification. Furthermore, taking this as starting point, he indicates how to achieve resolution of singularities.

Hironaka's theorem of resolution of singularities is existential, precisely because his proof of -simplification is existential.

The achievement of constructive resolution of singularities was to provide an algorithm. So given and as before, as input, the algorithm defines a -simplification.

An advantage of a constructive proof of resolution of singularities, over the original existential proof, is that constructive resolutions are equivariant, they provide resolution en étale topology, they are compatible with change of base field etc. (see [32]).

Another advantage of the algorithm of -simplification, already mentioned above, is that it simplifies the proof of desingularization ([17]).

The key point for -simplification, already used in Hironaka's proof, is a form of induction. In fact, Hironaka proves -simplification, by induction on the dimension of the ambient space . To simplify matters, assume that is locally principal, and let denote the highest order of along points in , which is now smooth over a field of characteristic zero. Let

denote the closed set (or say ).

Fix a closed point , and a regular system of parameters at . For any , set , and

If is locally generated by , then has order at , and

The key point is that, the order of at is one. This holds when is a field of characteristic zero.

Recall that locally at . One way to check that has order one at , is to check this at the completion , say . We may choose the system of parameters so that, for a suitable unit :

, and .

As is a field of characteristic zero, , where , and

Then:

A) (in fact ). In particular the ideal has order one at , and the closed set is locally included in a smooth scheme of dimension .

B)(Elimination.) can be described as

C) (Stability of elimination.) Both A), and the description in B), are preserved by any -permissible sequence of transformations.

We will not go into details of A), B) and C). But let us point out the elimination of one variable in (B). In fact the closed set defined in terms of , is also described as where now the involve one variable less.

As indicated above, A),B), and C), together, conform the essential reason and argument in resolution of singularities in characteristic zero. They rely entirely on the hypothesis of characteristic zero. For instance A) does not hold over fields of positive characteristic; so there is no way to formulate this form of induction over arbitrary fields.

The objective of these notes is to report on an entirely different approach to induction, which can at least be formulated over arbitrary fields.

Suppose, for simplicity, that is affine, that is global in , and that is the highest order of . We reformulate the study -sequences of transformations over . In doing so we replace by a graded ring subring of . In this case we consider the subring

In general, if is affine, we define a Rees algebra as a subring of generated by a finite set, say

These subrings can also be expressed as , , and each is an ideal. We say that has differential structure, say Diff-structure, if for , and .

Diff-structures appear in [23] and [24](see 4.2), and they are closely related to the notion of tuned ideals introduced by J Kollár.

It is easy to show that any Rees algebra spans a smallest Diff-structure containing it. Diff-structures are known to have important geometric properties, which make them objects of particular interest. In this paper we report on a characteristic free form of eliminationdefined for Diff-structures (see (B) above).

We also study here a natural compatibility of monoidal transforms and Diff-structures. This is done via Taylor development in positive characteristic (see also [33]). So it makes sense to formulate stability of elimination (see (C) above) over arbitrary fields. Here results are stronger over fields of characteristic zero, where they provide an alternative approach to induction in desingularization theorems.

New invariants for singularities arise, in positive characteristic, when studying this form of elimination in the setting of Diff-structures.

1. Monoidal transformations and Hironaka's topology.

Fix a smooth scheme over a field , an ideal , and a positive integer . Hironaka attaches to these data, say , a closed set, say

where denote the order of at the local regular ring .

Given and , then

where , and . Set formally .

There is also a notion of permissible transformation on these data . Let be a smooth subscheme in , included in the closed , and let

(1.0.1)

be the blow up of at a smooth sub-scheme . Note that

where is the sheaf of functions vanishing along the exceptional hypersurface .

We call the transform of by the permissible monoidal transformation.

If is permissible for both and , then it is permissible for . Moreover, if , , and denote the transforms, then .

We now define a Rees algebra over to be a graded noetherian subring of , say:

where and each is a sheaf of ideals. And we assume that at any affine open set , there is a finite set

and , so that the restriction of to is

To a Rees algebra we attach a closed set:

where denotes the order of the ideal at the local regular ring .

Remark 1.1. Rees algebras are related to Rees rings. A Rees algebra is a Rees ring if, given any affine open set , and as above, all degrees are one.

In general Rees algebras are integral closures of Rees rings in a suitable sense. In fact, if is a positive integer divisible by all , it is easy to check that

is integral over the Rees sub-ring .

Proposition 1.2. Given an affine open , and as above,

Proof. It is clear that for , . So

On the other hand, for any index , is generated by elements of the form , where is weighted homogeneous of degree , provided each has weight . The reverse inclusion is now clear. □

A monoidal transformation (1.0.1) is said to be permissible for if . In such case, for each index , there is a sheaf of ideals, say , so that

One can easily check that

is a Rees algebra over , which we call the transform of .

Let be a Rees algebra on , an affine open set, and let be such that the restriction of to is

Proposition 1.3. Let be a permissible transformation of . There is an open covering of by affine sets , so that:

1) for suitable .

2) The restriction of to is

Proof. 1) follows from Prop 1.2. For 2) argue as in the proof of Prop 1.2, by using the fact that each ideal is generated by weighted homogeneous polynomials on the element of .

Given two Rees algebras over , say and , set in , and define:

as the subalgebra of generated by .

One can check that:

1) . In particular, if in (1.0.1) is permissible for , it is also permissible for and for .

2) Set as in 1), and let , , and denote the transforms at . Then:

2. Integral closure of Rees algebras and a notion of equivalence.

We say that two Rees algebras over , say and , are equivalent, if both have the same integral closure in .

If and are equivalent, then:

1) . In particular, in (1.0.1) is permissible for if and only if it is so for .

2) Set as in 1), and let and denote the transforms at . Then and are equivalent over .

This shows that equivalent Rees algebras define the same closed sets, and the same holds after any sequence of permissible transformations.

Given a smooth scheme , and as in 1, we consider the Rees algebra generated over by (as graded subring of ).

Proposition 2.1. If and are the Rees algebras corresponding to Hironaka's pairs and , then is equivalent to the Rees algebra assigned to .

Proof. Fix an affine open set in , generators of , and generators of . Then:

i) The restriction of to is

ii) The restriction of is

iii) The restriction of to is

iv) The restriction of the Rees algebra assigned to is generated by

One can finally check that both algebras in (iii) and (iv) have the same integral closure in .□

3. On differential structures and Kollár's tuned ideals.

Here is smooth over a field , so for each non-negative integer there is a locally free sheaf of differential operators of order , say .

Definition 3.1. We say that a Rees algebra is a Diff-structure relative to the field , if:

i) .

ii) There is open covering of by affine open sets , and for any , and any , then provided .

Given a sheaf of ideals there is a natural definition of an extension, say (see Introduction). Note that (ii) can be reformulated by

ii') for each , and .

Fix a closed point , and a regular system of parameters at . The residue field, say is a finite extension of , and the completion

The Taylor development is the continuous -linear ring homomorphism:

that map to , . So for , , with .

Define, for each , . It turns out that

and that generate the -module (i.e. generate locally at ).

Theorem 3.2. For any Rees algebra over a smooth scheme , there is a Diff-structure, say such that:

i) .

ii) If and is a Diff-structure, then .

Furthermore, if is a closed point, and is a regular system of parameters at , and is locally generated by

then

(3.2.1)

generates locally at .

Remark 3.3. The local description in the Theorem shows that.

In fact, as , it is clear that . For the converse note that if , then has order at least at the local ring .

3.4. In general , and equality holds if is already a Diff-structure.

Let be a Diff-structure, in particular it is integral over a Rees subring, say for suitable (see 1.1). These ideals are called tuned ideals in [25], page 45.

The previous Theorem defines an operator that extends Rees algebras into Diff-structures. Another natural operator we have considered on Rees algebras it that defined by taking normalization. The next Theorem relates both notions of extensions.

Theorem 3.5. Let and be equivalent Rees algebras on a smooth scheme , then and are also equivalent (in the sense of 2).

(see Th 6.12 [33]).

Definition 3.6. Fix , a Rees algebra on , and let be a morphism of smooth schemes. We define the total transform of to be

Namely the Rees algebra defined by the total transforms of the ideals , .

Theorem 3.7. Let be a morphism of smooth schemes, then:

i) if is a Diff-structure on , the total transform is a Diff-structure on .

ii) .

(See Th 5.4 [33])

4. On differential structures and monoidal transformations.

Let us briefly recall some previous results, where now be the sheaf of ideals defining a hypersurface in the smooth scheme .

So , and for each positive integer there is an inclusion as sheaves of ideals in , and hence .

Recall that is the highest multiplicity at points of , if and only if and (i.e. if and only if and is a proper sheaf of ideals).

The closed set of interest is the set of -fold points of (i.e. ). Consider now a -permissible transformation, say

(i.e. the blow up of at a smooth sub-scheme ). In such case

where is the sheaf of functions vanishing along the exceptional hypersurface .

In this case is the sheaf of ideals defining a hypersurface , which is the strict transform of the hypersurface .

It is not hard to check that has at most order at points of (i.e. that . If, in addition, has no point of order , then we say that defines a -simplification of . At any rate, the closed set of interest is the set of -fold points .

If , let denote the monoidal transformation with center . So is - permissible, and set

So again has at most points of order , and if it does, define a -permissible transformation at some smooth center .

So for and as before, we define, by iteration, a -permissible sequence

and a factorization Where is the sheaf of ideals defining a hypersurface , which is the strict transform of .

From the point of view of resolution it is clear that our interest is to define a -permissible sequence so that has no -fold points.

We say that a -permissible sequence defines a -simplication of if the jacobian of has normal crossings, and (i.e. if has at most points of multiplicity ).

Hironaka attaches to the original data and the pair . The closed set assigned to this pair in is . In our case, the -fold points of the hypersurface .

We attached to the original data a Rees algebra (up to integral closure), namely . And to this Rees algebra a closed set in , namely , which is again .

Moreover, we extended   to a Diff-structure  , and   (Th. 3.2).

Let us focus on the -permissible transformation . The transform of Hironaka's pair is the pair . The transformation is also permissible for both and , defining transforms of Rees algebras, say and on .

Note that, in our setting, is the ideal defining defining , which is the strict transform of . The closed set assigned to is the set of -fold points of . On the other hand, , is such that is again the set of -fold points . A similar relation holds between pairs and the Rees algebras (transform of ), for any -permissible sequence.

The natural question is on how do the successive transforms of relate to the transforms of . The following theorem will address this question (see Th 7.6 [33]). It proves that the -operator on Rees algebras is, in a natural way, compatible with transformation.

Theorem 4.1. (J. Giraud) Let be a Rees algebra on a smooth scheme , and let be a permissible (moniodal) transformation for . Let and denote the transforms of and . Then:

1) .

2)

4.2. Hironaka considers the notion of Diff-structures in [23] and also in [24]. In this last paper he provides an interesting geometric interpretation of the elements of the integral closure of a Diff-structure, say , which we briefly discuss below.

Recall that given an ideal in a smooth scheme , and a positive integer , Hironaka defines a pair (actually a closely related notion of idealistic exponent). As mentioned in Section 1, there is a closed set in attached to the pair, and also a notion of permissible transforms of pairs.

We have assign a Rees algebra to , say ; and a closed set to , namely . We have also defined transformations of of Rees algebras, in accordance to transformations of pairs.

Here we have discussed integral closure of Rees algebras, and also a -operator on Rees algebras, as two different manners to extend a Rees algebra.

These two forms of extension of Rees algebras have a very particular geometric property. In fact, both extended algebras define the same closed set, and hence both admit the same transformations. Furthermore, the closed set defined by the transform of by a sequence of transformation, is the same closed set defined by the transform of the integral closure of . Theorem 4.1 asserts that the same holds for the transform of -extension of .

So given , it is quite natural to iterate both operators, by taking successively integral closure and Diff-structures, to obtain larger and larger extensions of with this geometric property.

The result of Hironaka in [24] says that the is the biggest extension of with this property. Namely that , and that the same equality of singular locus holds after any sequence of transformations. Theorem 4.1 can also be proved using this geometric characterization of . The approach in [33] is different, and does not make use the concept of infinitely near singular point, but rather on technics that will also be useful for [34].

5. Idealistic exponents versus basic objects.

Recall that two ideals, say and , in a normal domain have the same integral closure if they are equal for any extension to a valuation ring (i.e. if for any ring homomorphism on a valuation ring ). The notion extends naturally to sheaves of ideals.

Hironaka considers the following equivalence on pairs and over a smooth scheme .

Definition 5.1. The pairs and are idealistic equivalent on if and have the same integral closure.

Proposition 5.2. Let and be idealistic equivalent. Then:

1) .

Note, in particular, that any monoidal transform on a center defines transforms, say and on .

2)The pairs and are idealistic equivalent on .

If two pairs and be idealistic equivalent over , the same holds for the restrictions to any open subset of , and also for restrictions in the sense of etale topology, and even for smooth topology (i.e. pull-backs by smooth morphisms ).

Note that if and are idealistic equivalent, the they define the same closed set on (i.e. ), and the same holds for monoidal transformations, pull-backs by smooth schemes, and hence by concatenation of both kinds of transformations. When this last condition holds on the singular locus of two pairs we say that they define the same close sets.

Definition 5.3. Two pairs and are basically equivalent on , if the define the same close sets.

The proposition says that if two pairs are idealistic equivalent over , then they are basically equivalent.

An idealistic exponent, as defined by Hironaka in [23], is an equivalence class of pairs in the sense of idealistic equivalence. Whereas the notion of equivalence among basic objects (see [31] or [32]) is the second one. In fact, the key point for constructive desingularization was to define an algorithm of resolutions of pairs , so that two basically equivalent pairs undergo exactly the same resolution.

5.4. There are two notions of equivalence on the context of Rees algebras over . The first, already formulated in Section 2:

Definition 5.5. Two Rees algebras over , say and , are integrally equivalent, if both have the same integral closure.

Proposition 5.6. Let and be two integrally equivalent Rees algebras over Then:

1) .

Note, in particular, that any monoidal transform on a center defines transforms, say and on .

2) and are integrally equivalent on .

If and are integrally equivalent on , the same holds for any open restriction, and also for pull-backs by smooth morphisms .

On the other hand, as and are integrally equivalent, the they define the same closed set on (the same singular locus), and the same holds for further monoidal transformations, pull-backs by smooth schemes, and concatenations of both kinds of transformations.

When this condition holds on the singular locus of two Rees algebras over , we say that they define the same close sets.

Definition 5.7. Two Rees algebras over , say and , are basically equivalent, if both define the same closed sets.

The previous Proposition asserts that if and are integrally equivalent, then they are basically equivalent.

5.8. We assign to a pair over a smooth scheme the Rees algebra, say:

which is a graded subalgebra in .

Proposition 5.9. 1) Two pairs and are idealistically equivalent over a smooth scheme , if and only if the Rees algebras and are integrally equivalent.

2) Two pairs and are basically equivalent over , if and only if the Rees algebras and are basically equivalent.

6. Projection of differential structures and elimination of one variable.

6.1. The notion of Rees algebra parallels that of idealistic exponents in [23], and the notion of singular locus , is the natural analog for that defined for idealistic exponents.

We finally introduce a function, again a natural analog to that defined for idealistic exponents. Fix . Given , set

called the order of (weighted by ), where denotes the order at the local regular ring . As it follows that We also define

So, in general for any .

Proposition 6.2. 1) If is a Rees algebra generated over by , then

And if is any common multiple of all , then .

2) If and are graded structures with the same integral closure (e.g. if is a finite extension), then, for any

3) Set (the extension of to a differential structure), then for any .

6.3. Let be a Rees algebra, and fix a closed point . We assume that at a affine open neighborhood of the point, say , there is a finite set and , so that the restriction of to is

Let

be the localization of at . As , the order of at is at least . We say that is simple at the singular point , if for some positive index , has order . This amounts to saying that ; or equivalently, that for some , the element has order at .

Recall that locally at .

We may choose the system of parameters at , so that at the completion , say :

, and ; where is a unit of .

A similar result holds at a suitable étale neighborhood of . We may assume that is a monic polynomial of degree in , and of order in , where is regular.

Let be a smooth morphism defined at an étale neighborhood of , where is smooth, dim =dim -1. We say that is transversal to at , if the previous setting holds for , ; and for some , where has order at .

In these conditions, a transversal morphism , induces a finite morphism

Here we view as a hypersurface in , and locally at , is included in the -fold points of this hypersurface. So

In this setting the finite morphism is one to one over a closed subset of , namely on the image of the -fold points. Set

(6.3.1)

Since , induces a one to one map, say

for any transversal morphism .

Theorem 6.4. Let be a Diff-structure over a smooth scheme , and a closed point which we assume to be simple. Let be a smooth morphism defined at an étale neighborhood of , where is smooth, dim =dim -1. Assume that is transversal at . Then:

1) At a suitable neighborhood of , there is a Rees algebra over the smooth scheme , so that .

2) The morphism induces a one-to-one map from to . Furthermore, setting , and as before, then the one-to-one map is that described above.

The formulation of the theorem is independent of the choice of of order at . However given a finite morphisms as that in (6.3.1), and a smooth center , there is a unique and smooth center mapping isomorphically to via (and hence via ). Set .

So both in , and in , are regular centers.

Let now , and , denote the monoidal transformations at and respectively; and let denote the strict transform of . The hypersurface has at most points of multiplicity . Let denotes the closed set of points of multiplicity . After replacing by a suitable neighborhood of , we may assume that there is a finite morphism, say , compatible with .

As the regular center was chosen in , then a weighted transform, say

is defined, and . So locally at a point there is a finite morphism

where is a strict transform of . Let be the Diff-structure generated by . According to the previous Theorem, locally at there is an elimination algebra, say

On the other hand, , so there is also a weighted transform

The question now is to relate the Rees algebra with , locally at the point .

Proposition 6.5. With the setting as above:

1) There is a natural inclusion .

2) Over fields of characteristic zero both and define the same Diff-structure, up to integral closure.

Here is the transform of by one monoidal transformation. If we could guarantee that , we could identify the singular locus of (i.e. of ) with the singular locus of the transform of . If furthermore, this link between and is preserved by any sequence of monoidal transformations, then we have achieved a way of representing the singular locus of which is stable by monoidal transformations.

Part 2) in the previous Proposition ensures that this is the case over fields of characteristic zero, providing an alternative form of stability of elimination (see (C) in Introduction). This is not the case over fields of positive characteristic, but it is the starting point for new invariants in that context.

References

[1]    S.S. Abhyankar, Good points of a Hypersurface, Adv. in Math. 68 (1988) 87-256.         [ Links ]

[2]    D. Abramovich and A.J. de Jong, 'Smoothness, semistability and toroidal geometry', Journal of Algebraic Geometry 6 (1997) 789-801.         [ Links ]

[3]    D. Abramovich and J. Wang, 'Equivariant resolution of singularities in characteristic 0', Mathematical Research Letters 4 (1997) 427-433.         [ Links ]

[4]    J.M. Aroca, H. Hironaka and J.L. Vicente, 'The theory of maximal contact', Mem. Mat. Ins. Jorge Juan (Madrid) 29 (1975).         [ Links ]

[5]    M. Artin. 'Algebraic approximation of structures over complete local rings', Pub. Math. I.H.E.S. 36 (1969) 23-58.         [ Links ]

[6]    E. Bierstone and P. Milman, 'Canonical desingularization in characteristic zero by blowing-up the maxima strata of a local invariant', Inv. Math. 128 (1997) 207-302.         [ Links ]

[7]    E. Bierstone and P. Milman, 'Desingularization algorithms I. Role of exceptional divisors'. Mosc. Math. J. 3 (2003), no3, 751-805, 1197. MR2078560         [ Links ]

[8]    G. Bodnár and J. Schicho 'A Computer Program for the Resolution of Singularities', Resolution of singularities. A research book in tribute of Oscar Zariski (eds H. Hauser, J. Lipman, F. Oort, A. Quirós), Progr. Math. 181 (Birkhäuser, Basel, 2000) pp.231-238.         [ Links ]

[9]    G. Bodnár and J. Schicho, 'Automated resolution of singularities for hypersurfaces', J. Symbolic Comput. (4) 30 (2000) 401-428.         [ Links ]

[10]    F. Bogomolov and T. Pantev, 'Weak Hironaka Theorem', Mathematical Research Letters 3 (1996) 299-307.         [ Links ]

[11]    A. Bravo and O. Villamayor, 'Strengthening a Theorem of Embedded Desingularization,' Math. Res. Letters 8 (2001) 1-11.         [ Links ]

[12]    A. Bravo and O. Villamayor, 'A Strengthening of resolution of singularities in characteristic zero'. Proc. London Math. Soc. (3) 86 (2003) 327-357.         [ Links ]

[13]    S. Encinas and H. Hauser, 'Strong Resolution of Singularities'. Comment. Math. Helv. 77 2002, no. 4, 821-845.         [ Links ]

[14]    S. Encinas, A. Nobile and O. Villamayor, 'On algorithmic Equiresolution and stratification of Hilbert schemes'. Proc. London Math. Soc. (3) 86 (2003) 607-648.         [ Links ]

[15]    S. Encinas and O. Villamayor, 'Good points and constructive resolution of singularities', Acta Math. 181:1 (1998) 109-158.         [ Links ]

[16]    S. Encinas and O. Villamayor, 'A Course on Constructive Desingularization and Equivariance', Resolution of Singularities. A research textbook in tribute to Oscar Zariski (eds H. Hauser, J. Lipman, F. Oort, A. Quirós), Progr. in Math. 181 (Birkhäuser, Basel, 2000) pp. 147-227.         [ Links ]

[17]    S. Encinas and O. Villamayor, 'A new proof of desingularization over fields of characteristic zero'. Rev. Mat. Iberoamericana 19 (2003), no.2, 339-353.         [ Links ]

[18]    J. Giraud, 'Sur la theorie du contact maximal', Math. Zeit., 137 (1972), 285-310.         [ Links ]

[19]    J. Giraud. 'Contact maximal en caractéristique positive', Ann. Scien. de l'Ec. Norm. Sup. 4ème série, 8 (1975) 201-234.         [ Links ]

[20]    H. Hauser, Excellent surfaces and Their Taut Resolution in Resolution of Singularities. A research textbook in tribute to Oscar Zariski, Eds. H. Hauser, J. Lipman, F. Oort, A. Quirós. Progress in Math. vol 181, Birkhäuser 2000.         [ Links ]

[21]    H. Hauser, 'The Hironaka Theorem on Resolution of Singularities'(or: A proof we always wanted to understand), Bull. Amer. Math. Soc. (N.S.)40 (2003), no. 3,323-403.         [ Links ]

[22]    H. Hironaka, 'Resolution of singularities of an algebraic variety over a field of characteristic zero I-II', Ann. Math., 79 (1964) 109-326.         [ Links ]

[23]    H. Hironaka, 'Idealistic exponent of a singularity', Algebraic Geometry, The John Hopkins centennial lectures, Baltimore, John Hopkins University Press (1977), 52-125.         [ Links ]

[24]    H. Hironaka, 'Theory of infinitely near singular points', Journal Korean Math. Soc. 40 (2003), No.5, pp. 901-920         [ Links ]

[25]    J. Kollár, 'Resolution of Singularities- Seattle Lecture' arXiv:math.AG/ 0508332 v1 17 Aug 2005.         [ Links ]

[26]    J. Lipman, 'Introduction to Resolution of Singularities', Proc. Symp. in Pure Math. 29 (1975) pp. 187-230.         [ Links ]

[27]    J. Lipman, 'Equisingularity and Simultaneous Resolution of Singularities', Resolution of Singularities. A research textbook in tribute to Oscar Zariski, (eds H. Hauser, J. Lipman, F. Oort and A. Quirós), Progr. in Math. 181, (Birkhäuser, Basel, 2000) pp. 485-505.         [ Links ]

[28]    K. Matsuki, 'Notes on the inductive algorithm of resolution of singularities'. Preprint. arXiv:math.math.AG/0103120         [ Links ]

[29]    H. Matsumura. 'Commutative algebra', Mathematics Lecture Note Series, 56, Benjamin/Cummings Publishing Company, Inc., 2nd ed. edition, 1980.         [ Links ]

[30]    T. Oda, 'Infinitely very near singular points', Complex analytic singularities, Adv. Studies in Pure Math. 8 (North-Holland, 1987) pp. 363-404.         [ Links ]

[31]    O. Villamayor, 'Constructiveness of Hironaka's resolution', Ann. Scient. Ec. Norm. Sup. 4^e serie 22 (1989) 1-32.         [ Links ]

[32]    O. Villamayor, 'Patching local uniformizations', Ann. Scient. Ec. Norm. Sup., 25 (1992), 629-677.         [ Links ]

[33]    O. Villamayor, 'Rees algebras on smooth schemes: integral closure and higher differential operators'. ArXiv:math. AG/0606795.         [ Links ]

[34]    O. Villamayor, 'Hypersurface singularities in positive characteristic.' ArXiv:math.AG/0606796.         [ Links ]

[35]    J. Wlodarczyk, 'Simple Hironaka resolution in characteristic zero', to appear (2005).         [ Links ]

[36]    B. Youssin, 'Newton Polyhedra without coordinates', Mem. Amer. Math. Soc. 433 (1990), 1-74,75-99.         [ Links ]

Orlando Villamayor U.
Dpto. Matemáticas,
Facultad de Ciencias,
Universidad Autónoma de Madrid,
Canto Blanco 28049 Madrid, Spain
villamayor@uam.es

Recibido: 26 de diciembre de 2005
Aceptado: 7 de agosto de 2006