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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.2 Bahía Blanca jul./dic. 2005
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
![ordJ : V → ℤ](/img/revistas/ruma/v46n2/2a0112x.png)
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
![r ⊕r≥0IrW ,](/img/revistas/ruma/v46n2/2a0124x.png)
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
![V (J) ⊃ V (Dif f1(J)) ⊃ ⋅⋅⋅ ⊃ V (Dif fs-1(J)) ⊃ V (Dif fs(J))...](/img/revistas/ruma/v46n2/2a0152x.png)
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
![π V ← - V1 ∪ ∪ -1 Y π (Y ) = H](/img/revistas/ruma/v46n2/2a0162x.png)
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
![b JOV1 = I(H) J1,](/img/revistas/ruma/v46n2/2a0169x.png)
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
![b J1OV2 = I(H1) J2.](/img/revistas/ruma/v46n2/2a0189x.png)
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
![V ←π- V1 ←π1- V2 ←π2- ...Vr ←πr- Vr+1,](/img/revistas/ruma/v46n2/2a0197x.png)
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
![JO = I(H)a0I(H )a1 ⋅⋅⋅I(H )a0J . Vn 1 n-1 n](/img/revistas/ruma/v46n2/2a01105x.png)
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
![{ordJ ≥ b}](/img/revistas/ruma/v46n2/2a01130x.png)
denote the closed set (or say
).
Fix a closed point , and a regular system of parameters
at
. For any
, set
, and
![α -1- -1---∂-α1- -∂αn-- Δ = (α1! ⋅⋅⋅αn!)∂ α1x1 ⋅⋅⋅ ∂αnxn .](/img/revistas/ruma/v46n2/2a01138x.png)
If is locally generated by
, then
has order
at
, and
![(Dif fb-1(J ))x = 〈f, Δα(f )∕0 ≤ ∣α∣ < b〉.](/img/revistas/ruma/v46n2/2a01144x.png)
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
:
![u.f = f1 = Zb + a1Zb -1 + ⋅⋅⋅ + ab ∈ S[Z]](/img/revistas/ruma/v46n2/2a01155x.png)
, and
.
As is a field of characteristic zero,
, where
, and
![b ′ b-2 ′ f1 = Z 1 + a2Z1 + ⋅⋅⋅ + a b.](/img/revistas/ruma/v46n2/2a01161x.png)
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
![′ {ord f ≥ b}= ∩2 ≤i≤b{ord ai ≥ b - i}.](/img/revistas/ruma/v46n2/2a01169x.png)
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
![O [fW b](⊂ O [W ]). V V](/img/revistas/ruma/v46n2/2a01184x.png)
In general, if is affine, we define a Rees algebra as a subring of
generated by a finite set, say
![n1 n2 ns {f1W ,f2W ,...,fsW }.](/img/revistas/ruma/v46n2/2a01187x.png)
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
![{ordJ ≥ b}:= {x ∈ V ∕νx(Jx) ≥ b}](/img/revistas/ruma/v46n2/2a01200x.png)
where denote the order of
at the local regular ring
.
Given and
, then
![′ ′ {ordJ ≥ b} ∩ {ordJ ≥ b}= {ordK ≥ c}](/img/revistas/ruma/v46n2/2a01206x.png)
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
![b JOV1 = I(H) J1,](/img/revistas/ruma/v46n2/2a01217x.png)
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:
![⊕ k G = IkW , k≥0](/img/revistas/ruma/v46n2/2a01232x.png)
where and each
is a sheaf of ideals. And we assume that at any affine open set
, there is a finite set
![F = {f1W n1,...,fsW ns},](/img/revistas/ruma/v46n2/2a01236x.png)
and
, so that the restriction of
to
is
![O (U )[f W n1,...,f W ns](⊂ O (U)[W ]). V 1 s V](/img/revistas/ruma/v46n2/2a01241x.png)
To a Rees algebra we attach a closed set:
![Sing(G) := {x ∈ V∕ νx(Ik) ≥ k, for any k ≥ 1},](/img/revistas/ruma/v46n2/2a01243x.png)
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
![n1 ns r OV (U )[f1W ,...,fsW ] = ⊕r≥0IrW (⊂ OV (U )[W ]),](/img/revistas/ruma/v46n2/2a01252x.png)
is integral over the Rees sub-ring .
Proposition 1.2. Given an affine open , and
as above,
![Sing(G) ∩ U = ∩1 ≤i≤s{ord(fi) ≥ ni}.](/img/revistas/ruma/v46n2/2a01256x.png)
Proof. It is clear that for
,
. So
![Sing(G) ∩ U ⊂ ∩1 ≤i≤s{ord(fi) ≥ ni}.](/img/revistas/ruma/v46n2/2a01260x.png)
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
![IkOV1 = I(H)kI(k1).](/img/revistas/ruma/v46n2/2a01272x.png)
One can easily check that
![⊕ (1) k G1 = Ik W k≥0](/img/revistas/ruma/v46n2/2a01273x.png)
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
![OV (U )[f1W n1,...,fsW ns](⊂ OV (U)[W ]).](/img/revistas/ruma/v46n2/2a01282x.png)
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
![OV (U (l))[f ′W n1,...,f′W ns](⊂ OV (U (l))[W ]). 1 1 s 1](/img/revistas/ruma/v46n2/2a01291x.png)
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:
![⊕ G ⊙ G ′ = K ′kW k, k≥0](/img/revistas/ruma/v46n2/2a01299x.png)
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:
![′ ′ (G ⊙ G )1 = G1 ⊙ G 1.](/img/revistas/ruma/v46n2/2a01312x.png)
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
![b b OV (U )[f1W ,...,fsW ](⊂ OV (U)[W ]).](/img/revistas/ruma/v46n2/2a01349x.png)
ii) The restriction of is
![b′ b′ OV (U )[g1W ,...,grW ](⊂ OV (U )[W ]).](/img/revistas/ruma/v46n2/2a01351x.png)
iii) The restriction of to
is
![b b b′ b′ OV (U )[f1W ,...,fsW ,g1W ,...,grW ](⊂ OV (U )[W ]).](/img/revistas/ruma/v46n2/2a01354x.png)
iv) The restriction of the Rees algebra assigned to is generated by
![′ ′ {(fα11 ⋅⋅⋅fαss) ⋅ W bb ;(gβ11⋅⋅⋅gβss) ⋅ W bb∕α1 + ⋅⋅⋅ + αs = b′;β1 + ⋅⋅⋅ + βr = b}.](/img/revistas/ruma/v46n2/2a01356x.png)
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:
![T ay : k′[[x1,...,xn]] → k′[[x1,...,xn,T1,...,Tn]]](/img/revistas/ruma/v46n2/2a01384x.png)
that map to
,
. So for
,
, with
.
Define, for each ,
. It turns out that
![α Δ (OV,x) ⊂ OV,x,](/img/revistas/ruma/v46n2/2a01393x.png)
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
![n F = {gniW i,ni > 0,1 ≤ i ≤ m},](/img/revistas/ruma/v46n2/2a01410x.png)
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
![⊕ π-1(G) = I O ′ ⋅ W k. k V](/img/revistas/ruma/v46n2/2a01437x.png)
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
![π V ← - V1 ∪ ∪ Y π-1(Y ) = H](/img/revistas/ruma/v46n2/2a01466x.png)
(i.e. the blow up of at a smooth sub-scheme
). In such case
![JOW = I(H)bJ1, 1](/img/revistas/ruma/v46n2/2a01469x.png)
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
![J1OV2 = I(H1)bJ2.](/img/revistas/ruma/v46n2/2a01491x.png)
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
![π π1 π2 πr V ←- V1 ← - V2 ← - ...Vr ← - Vr+1,](/img/revistas/ruma/v46n2/2a01499x.png)
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:
![G = O [JbW b], (J,b) V](/img/revistas/ruma/v46n2/2a01655x.png)
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
![ordx(fn) = νx(fn)-∈ ℚ; n](/img/revistas/ruma/v46n2/2a01671x.png)
called the order of (weighted by
), where
denotes the order at the local regular ring
. As
it follows that
We also define
![ordx(G) = inf{ordx(fn); fnW n ∈ InW n}.](/img/revistas/ruma/v46n2/2a01678x.png)
So, in general for any
.
Proposition 6.2. 1) If is a Rees algebra generated over
by
, then
![ordx(G) = inf{ordx(gni); 1 ≤ i ≤ m}.](/img/revistas/ruma/v46n2/2a01685x.png)
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
![ordx(G) = ordx(G ′).](/img/revistas/ruma/v46n2/2a01693x.png)
3) Set (the extension of
to a differential structure), then for any
.
![′′ ordx(G) = ordx(G ).](/img/revistas/ruma/v46n2/2a01697x.png)
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
![OV (U )[f1W n1,...,fsW ns](⊂ OV (U)[W ]).](/img/revistas/ruma/v46n2/2a01706x.png)
Let
![⊕ k Gx = Ik ⋅ W (⊂ OV,x[W ])](/img/revistas/ruma/v46n2/2a01707x.png)
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
:
![nc nc-1 u.fc = Z + a1Z + ⋅⋅⋅ + anc ∈ S[Z]](/img/revistas/ruma/v46n2/2a01730x.png)
, 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
![π-: Spec(S[Z] ∕〈f (Z)〉) → Spec(S) 〉. c1](/img/revistas/ruma/v46n2/2a01757x.png)
Here we view as a hypersurface in
, and locally at
,
is included in the
-fold points of this hypersurface. So
![{ord fc1 ≥ nc1}:= V(Dif f ci-1(〈fc1(Z)〉)) ⊂ H.](/img/revistas/ruma/v46n2/2a01763x.png)
![π](/img/revistas/ruma/v46n2/2a01764x.png)
![Y](/img/revistas/ruma/v46n2/2a01765x.png)
![c1](/img/revistas/ruma/v46n2/2a01766x.png)
![]() | (6.3.1) |
Since ,
induces a one to one map, say
![1 to 1 Sing(G) - → π(Sing(G)),](/img/revistas/ruma/v46n2/2a01770x.png)
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
![⊕ G1 = I(n1) ⋅ W k(⊂ OV1[W ])](/img/revistas/ruma/v46n2/2a01820x.png)
is defined, and . So locally at a point
there is a finite morphism
![π′ : Spec(S ′[Z]∕〈f′ (Z)〉) → U1, c1](/img/revistas/ruma/v46n2/2a01823x.png)
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
![R ′⊂ O ′ [W ]. G1 U1,π(y)](/img/revistas/ruma/v46n2/2a01829x.png)
On the other hand, , so there is also a weighted transform
![(RG)1 ⊂ OU1[W ].](/img/revistas/ruma/v46n2/2a01831x.png)
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.
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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