Differential operators on smooth schemes and embedded singularities

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 desingular-ization 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. Koll´ar ([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]).

Contents

Part 1. Introduction. 1

1. Monoidal transformations and Hironaka’s topology. 5 2. Integral closure of Rees algebras and a notion of equivalence. 7 3. On differential structures and Koll´ar’s tuned ideals. 8 4. On differential structures and monoidal transformations. 10

5. Idealistic exponents versus basic objects. 12

6. Projection of differential structures and elimination of one variable. 13

References 16

Part 1. Introduction.

Let V be a smooth scheme over a field k of characteristic zero, and let X ⊂ V be a singular subscheme. Hironaka proves embedded desingularization ofX,

2000Mathematics subject classification. 14E15.

considering as invariants the Hilbert-Samuel functions at the points ofX. His proof is based on thereductionof Hilbert Samuel functions by monoidal transformations ([22]).

There is second theorem of Hironaka, used in his proof ofreduction of Hilbert Samuel functions, which is calledLog-resolutionof 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 X ⊂ V is defined by a sheaf of idealsJO_V, then desingularization is proved by considering the order of the ideal J at points in W, and hence avoiding the use of Hilbert Samuel functions.

LetV be a smooth scheme over a fieldk, and letJ ⊂ OV be a non-zero sheaf of ideals. Define a function

ordJ :V →Z

whereord_J(x) denotes the order ofJ_xat the local regular ringOV,x. Letbdenote the biggest value achieved by this function (the biggest order of J). The pair (J, b) isthe object of interest in Log principalization of ideals. There is a closed set attached to this pair inV, namely the set of points whereJ has orderb; and there is also a notion of transformation of such pairs by blowing up suitable regular centers.

We will attach to (J, b) a graded subring ofO_V[W] (sheaf of polynomial rings), namely a graded algebra (Rees algebra) of the form

⊕r≥0IrWr,

defined uniquely in terms ofJ andb.

Actually the Rees algebras that we will consider are closely related to Koll´ars notion oftuned ideals.

We will show that there is a closed set in V 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 integers the sheaf ofk-linear differential operators, say Dif f_ks, is coherent and locally free overV.

There is a natural identification, sayDif f_k0=OV, and for each s≥0 there is a natural inclusionsDif f_ks⊂Dif f_ks+1.

IfU is an affine open set inV, eachD ∈ Dif f_ks(U) is a differential operator: D : O_V(U) → O_V(U). We define an extension of a sheaf of idealsJ ⊂ O_V, say Dif f_ks(J), so that over the affine open set U, Dif f_ks(J)(U) is the extension of J(U) defined by adding all elementsD(f), for allD∈Dif f_ks(U) andf ∈J(U).

So Dif f0(J) = J, and Dif fs(J) ⊂ Dif fs+1(J) as sheaves of ideals in O_V. LetV(J)⊂V be the closed set defined byJ ∈ O_V. So

V(J)⊃V(Dif f1(J))⊃ · · · ⊃V(Dif fs−1(J))⊃V(Dif fs(J)). . .

It is simple to check that the order of the ideal at the local regular ringOV,xis

The previous observations say thatordJ :V →Zis an upper-semi-continuous function, and that the highest order ofJ(at pointsx∈V) isb, ifV(Dif fb(J)) =∅ andV(Dif fb−1(J))=∅. Let

V ←−π V1

∪ ∪

Y π−1(Y) =H

denote the blow up of W at a smooth irreducible sub-scheme Y, and H is the exceptional hypersurface. If Y ⊂V(Dif fb−1(J))) we say thatπis b-permissible. In such case

JOV₁ =I(H)bJ1,

whereI(H) is the sheaf of functions vanishing along the exceptional hypersurface H.

Ifπisb-permissible,J1has at most orderbat points ofW1(i.e. thatV(Diffb(J1)) =∅). If, in addition, J1 has no point of orderb, then we say that πdefines a

b-simplification ofJ.

IfV(Dif fb−1(J1))= ∅, letV1←−π1 V2 denote the monoidal transformation with

centerY1⊂V(Dif fb(J1)). We say thatπ1 isb-permissible, and set

J1O_V2 =I(H1)bJ2.

It turns out thatJ2has at most points of orderb. If it does, define ab-permissible transformation at some smooth irreducible centerY2⊂V(Dif fb−1(J2))).

ForJ andbas before, we define, by iteration, ab-permissible sequence

V ←−π V1←−π1 V2←−π2 . . . Vr←−πr Vr+1,

and a factorizationJn−1OVn =I(Hn)bJn.

LetHi⊂Vn denote the strict transform of exceptional hypersurfaceHi⊂Vi−1. Note that:

1) {H, H1, . . . , Hn−1} are the irreducible components of the exceptional locus

ofV ←V_n.

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

JOV_n=I(H)a0_I(H1)a1_{· · ·I(H}

n−1)a0_J n.

We say that this b-permissible sequence defines a b-simplication ofJ ⊂ OV if

∪H_i has normal crossings, and V(Dif fb−1(J_n)) = ∅ (i.e. J_n has order at most b−1 atW_n).

When k is a field of characteristic zero, and b is the highest order of a sheaf of ideals J ⊂ O_V, Hironaka proves that there is ab-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 ofb-simplification is existential.

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

provide resolution en ´etale topology, they are compatible with change of base field etc. (see [32]).

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

The key point forb-simplification, already used in Hironaka’s proof, is a form of induction. In fact, Hironaka provesb-simplification, by induction on the dimension of the ambient spaceV. To simplify matters, assume thatJ is locally principal, and letb denote the highest order of J along points in V, which is now smooth over a field of characteristic zero. Let

{ord_J ≥b}

denote the closed set{x∈V /ordJ(x)≥b} (or say =b).

Fix a closed point x ∈ {ordJ ≥ b}, and a regular system of parameters

{x1, x2, . . . , xn} atOV,x. For anyα= (α1, . . . , αn)∈Nn, set|α|=α1+· · ·+αn,

and

∆α= ( 1 α1!· · ·

1 α_n!)

∂α1

∂α1_x1· · ·

∂αn ∂αn_xn.

IfJx is locally generated byf ∈ OV,x, thenf has orderb atOV,x, and

(Dif fb−1(J))x=f,∆α(f)/0≤ |α|< b.

The key point is that, the order of (Dif fb−1(J))_xatOV,xis one. This holds when kis a field of characteristic zero.

Recall thatV(Dif fb−1(< f >)) ={ord_J ≥b} locally atx. One way to check that (Dif fb−1(J))xhas order one atOV,x, is to check this at the completion ˆOV,x, say R = k[[x1, .., xn]]. We may choose the system of parameters so that, for a

suitable unitu:

u.f=f1=Zb+a1Zb−1+· · ·+ab∈S[Z]

S=k[[x1, .., xn−1]], andZ=xn.

Askis a field of characteristic zero,S[Z] =S[Z1], where Z1=Z+1_ba1, and f1=Z1b+a2Z1b−2+· · ·+ab.

Then:

A) Z1 ∈ Dif fb−1(f) (in fact ∂

b−1_f

∂b−1_Z ∈ Dif fb−1(f)). In particular the ideal

Dif fb−1(f) has order one atx, and the closed set {ord_J ≥b} is locally included in a smooth scheme of dimensionn−1.

B)(Elimination.) {ord f ≥b}(⊂V(Z1)) can be described as

{ord f ≥b}=∩2_≤i≤b{ord a_i≥b−i}.

C) (Stability of elimination.) Both A), and the description in B), are preserved by anyb-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{ord f ≥b} defined in terms off, is also described as ∩2≤i≤b{ord a_i ≥b−i}, where now the a_i involve one variable less.

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, thatV is affine, thatf is global inOV, and thatbis the highest order ofJ =f. We reformulate the studyb-sequences of transformations overJ. In doing so we replaceJ by a graded ring subring ofOV[W]. In this case we consider the subring

OV[f Wb](⊂ OV[W]).

In general, if V is affine, we define a Rees algebra as a subring ofOV[W] gen-erated by a finite set, say

{f1Wn1, f2Wn2, . . . , fsWns}.

These subrings can also be expressed as _k≥0IkWk, I0 = OV, and each Ik is an ideal. We say that _k≥0I_kWk hasdifferential structure, say Diff-structure, if D(I_N)⊂I_N−r for 0≤r≤N, andD∈Dif f_kr.

Diff-structures appear in [23] and [24](see 4.2), and they are closely related to the notion of tuned idealsintroduced by J Koll´ar.

It is easy to show that any Rees algebra spans a smallest Diff-structure contain-ing it. Diff-structures are known to have important geometric properties, which make them objects of particular interest. In this paper we report on a character-istic free form ofelimination defined 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 formulatestability 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 ofelimination in the setting of Diff-structures.

1. Monoidal transformations and Hironaka’s topology.

Fix a smooth schemeV over a fieldk, an idealJ ⊂ OV, and a positive integer b. Hironaka attaches to these data, say (J, b), a closed set, say

{ord_J≥b}:={x∈V /ν_x(J_x)≥b}

where νx(Jx) denote the order ofJ at the local regular ringOV,x. Given (J, b) and (J, b), then

{ord_J ≥b} ∩ {ord_J ≥b}={ord_K ≥c}

where K=Jb+Jb, and c=b·b. Set formally (J, b)(J, b) = (K, c).

V ←−π V1

∪ ∪

Y π−1(Y) =H,

(1.0.1)

be the blow up ofV at a smooth sub-schemeY. Note that

JO_V1 =I(H)bJ1,

whereI(H) is the sheaf of functions vanishing along the exceptional hypersurface H.

We call (J1, b) the transformof (J, b) by the permissible monoidal

transforma-tion.

Ifπ is permissible for both (J, b) and (J, b), then it is permissible for (K, c). Moreover, if (J1, b), (J1, b), and (K1, c) denote the transforms, then (J1, b)

(J1, b) = (K1, c).

We now define a Rees algebra over V to be a graded noetherian subring of

OV[W], say:

G=

k≥0

I_kWk,

whereI0=OV and eachIk is a sheaf of ideals. And we assume that at any affine open setU ⊂V, there is a finite set

F ={f1Wn1, . . . , fsWns},

n_i≥1 andf_i∈ OV(U), so that the restriction ofGto U is

OV(U)[f1Wn1, . . . , fsWns](⊂ OV(U)[W]).

To a Rees algebraG we attach a closed set:

Sing(G) :={x∈V /ν_x(I_k)≥k, for anyk≥1},

whereν_x(I_k) denotes the order of the idealI_k at the local regular ringO_V,x.

Remark 1.1. Rees algebras are related to Rees rings. A Rees algebra is a Rees ring if, given any affine open setU ⊂V, andF={f1Wn1, . . . , fsWns}as above,

all degreesn_i are one.

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

OV(U)[f1Wn1, . . . , fsWns] =⊕r≥0IrWr(⊂ OV(U)[W]),

is integral over the Rees sub-ringO_V(U)[I_NWN](⊂ O_V(U)[WN]).

Proposition 1.2. Given an affine open U ⊂ V, andF ={f1Wn1, . . . , fsWns}

as above,

Sing(G)∩U =∩1_≤i≤s{ord(f_i)≥ni}.

Proof. It is clear thatν_x(f_i)≥n_i forx∈Sing(G), 0≤i≤s. So

Sing(G)∩U ⊂ ∩1≤i≤s{ord(fi)≥ni}.

weighted homogeneous of degreeN, provided eachYj has weightnj. The reverse inclusion is now clear.

A monoidal transformation (1.0.1) is said to bepermissibleforGifY ⊂Sing(G). In such case, for each index k ≥1, there is a sheaf of ideals, say I_k(1) ⊂ O_V1, so that

IkOV₁ =I(H)kI_k(1). One can easily check that

G1=

k≥0

I_k(1)Wk

is a Rees algebra overV1, which we call thetransformofG.

Let G =_k≥0I_kWk be a Rees algebra onV, U ⊂V an affine open set, and letF={f1Wn1, . . . , fsWns}be such that the restriction ofG toU is

OV(U)[f1Wn1_{, . . . , f}

sWns_{](⊂ OV(U)[W]).}

Proposition 1.3. Let V ←V1 be a permissible transformation ofG. There is an

open covering of π−1(U)by affine setsU(l), so that: 1)fi=I(H∩U(l))ni_f

i for suitablefi∈ OV1(U(l)).

2) The restriction of G1 toU(l) is

OV1(U (l)

)[f1Wn1, . . . , fsWns](⊂ OV1(U (l)

)[W]).

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

Given two Rees algebras over V, sayG=_k≥0IkWk andG =_k≥0JkWk, set K_k=I_k+J_k in O_V, and define:

G G₌

k≥0

K_kWk,

as the subalgebra ofO_V[W] generated by{K_kWk, k≥0}. One can check that:

1)Sing(GG) =Sing(G)∩Sing(G). In particular, ifπin (1.0.1) is permissible forG G, it is also permissible forG and forG.

2) Set π as in 1), and let (G G)1, G1, andG1 denote the transforms atV1.

Then:

(G G)1=G1 G1.

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

We say that two Rees algebras overV, sayG=L_k≥0IkWkandG=Lk≥0JkWk,

areequivalent, if both have the same integral closure inOV[W]. IfG andG are equivalent, then:

2) Setπas in 1), and letG1 andG1 denote the transforms atV1. ThenG1 and

G

1are equivalent overV1.

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 V, and (J, b) as in 1, we consider the Rees algebra generated overO_V byJ Wb (as graded subring ofO_V[W]).

Proposition 2.1. If G andG are the Rees algebras corresponding to Hironaka’s pairs (J, b) and (J, b), then G G is equivalent to the Rees algebra assigned to (J, b)(J, b).

Proof. Fix an affine open setU in V, {f1, . . . , fs} ∈ OV(U) generators ofJ(U),

and{g1, . . . , gr} ∈ OV(U) generators ofJ(U). Then:

i) The restriction ofG toU is

OV(U)[f1Wb, . . . , fsWb](⊂ OV(U)[W]). ii) The restriction ofG is

OV(U)[g1Wb, . . . , grWb](⊂ OV(U)[W]).

iii) The restriction ofG G to U is

OV(U)[f1Wb, . . . , fsWb, g1Wb, . . . , grWb](⊂ OV(U)[W]).

iv) The restriction of the Rees algebra assigned to (J, b)(J, b) is generated by

{(fα1

1 · · ·fsαs)·Wbb

; (g1β1· · ·gβss)·Wbb

/α1+· · ·+αs=b;β1+· · ·+βr=b}. One can finally check that both algebras in (iii) and (iv) have the same integral closure inOV(U)[W].

3. On differential structures and Koll´ar’s tuned ideals.

Here V is smooth over a field k, so for each non-negative integer r there is a locally free sheaf of differential operators of orderr, sayDif f_kr.

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

i)I_n ⊃I_n+1.

ii) There is open covering of V by affine open sets {Ui}, and for any D ∈ Dif f(r)(U_i), and anyh∈I_n(U_i), thenD(h)∈I_n−r(U_i) providedn≥r.

Given a sheaf of idealsI∈ OV there is a natural definition of an extension, say Dif f(r)(I) (see Introduction). Note that (ii) can be reformulated by

ii’)Dif f(r)(In)⊂In−r for eachn, and 0≤r≤n.

Fix a closed point x ∈ V, and a regular system of parameters {x1, . . . , xn} at OV,x. The residue field, say k is a finite extension of k, and the completion

ˆ

The Taylor development is the continuousk-linear ring homomorphism:

T ay:k[[x1, . . . , xn]]→k[[x1, . . . , xn, T1, . . . , Tn]]

that map xi to xi +Ti, 1 ≤ i ≤ n. So for f ∈ k[[x1, . . . , xn]], T ay(f(x)) =

α∈NngαTα, withgα∈k[[x1, . . . , xn]].

Define, for eachα∈Nn, ∆α(f) =gα. It turns out that

∆α(O_V,x)⊂ O_V,x,

and that{∆α, α∈(N)n,0≤ |α| ≤c}generate theOZ,x-moduleDif f_kc(OZ,x) (i.e. generateDif f_kc locally atx).

Theorem 3.2. For any Rees algebraG over a smooth scheme V, there is a Diff-structure, sayG(G) such that:

i)G ⊂G(G).

ii) If G ⊂ G andG is a Diff-structure, thenG(G)⊂ G.

Furthermore, ifx∈V is a closed point, and{x1, . . . , xn} is a regular system of parameters at OV,x, andG is locally generated by

F={g_niWni_{, n}

i>0,1≤i≤m},

then

F={∆α(gni)Wn

i−α_/gn

iWni∈ F, α= (α1, α2, . . . , αn)∈(N)n, and0≤ |α|< ni≤ni} (3.2.1) generates G(G)locally at x.

Remark 3.3. The localdescriptionintheTheorem shows thatSing(G) =Sing(G(G)). In fact, as G ⊂G(G), it is clear thatSing(G)⊃Sing(G(G)). For the converse note that ifν_x(g_ni)≥n_i, then ∆α(g_ni) has order at leastn_i− |α|at the local ring

OV,x.

3.4. In generalG ⊂G(G), and equality holds ifG is already a Diff-structure. LetG=_k≥0I_rWrbe a Diff-structure, in particular it is integral over a Rees subring, sayO_V[I_NWN] for suitableN (see 1.1). These idealsI_N are calledtuned idealsin [25], page 45.

The previous Theorem defines an operator Gthat 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 G andG be equivalent Rees algebras on a smooth schemeV, then G(G)andG(G)are also equivalent (in the sense of 2).

(see Th 6.12 [33]).

Definition 3.6. FixG=I_k·Wk, a Rees algebra onV, and letV ←−V be a morphism of smooth schemes. We define the total transformofG to be

π−1(G) =I_kO_V·Wk.

Theorem 3.7. Let V−→π V be a morphism of smooth schemes, then:

i) ifG is a Diff-structure onV, the total transformπ−1(G) is a Diff-structure onV.

ii)Sing(π−1(G)) =π−1(Sing(G)). (See Th 5.4 [33])

4. On differential structures and monoidal transformations.

Let us briefly recall some previous results, where nowJ ⊂ OV be the sheaf of ideals defining a hypersurfaceX in the smooth schemeV.

SoDif f0(J) =J, and for each positive integersthere is an inclusionDif fs(J)

⊂Dif fs+1(J) as sheaves of ideals inOV, and henceV(Diffs(J))⊃V(Diffs+1(J)). Recall thatbis the highest multiplicity at points ofX, if and only ifV(Dif fb(J)) =∅andV(Dif fb−1(J))=∅(i.e. if and only ifDif fb(J) =O_V andDif fb−1(J) is a proper sheaf of ideals).

The closed set of interest is the set ofb-fold points ofX (i.e. V(Dif fb−1(J))). Consider now ab-permissible transformation, say

V ←−π V1

∪ ∪

Y π−1(Y) =H

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

JOW₁=I(H)bJ1,

whereI(H) is the sheaf of functions vanishing along the exceptional hypersurface H.

In this caseJ1 is the sheaf of ideals defining a hypersurfaceX1⊂V1, which is

the strict transform of the hypersurfaceX.

It is not hard to check that J1 has at most orderb at points of V1 (i.e. that V(Dif fb(J1)) =∅). If, in addition, J1 has no point of order b, then we say that πdefines a b-simplificationof J. At any rate, the closed set of interest is the set ofb-fold points X1.

IfV(Dif fb−1(J1))= ∅, letV1←−π1 V2denote the monoidal transformation with centerY1⊂V(Dif fb−1(J1)). So π1 isb- permissible, and set

J1O_V2 =I(H1)bJ2.

So again J2 has at most points of orderb, and if it does, define a b-permissible

transformation at some smooth centerY2⊂V(Dif fb−1(J2))).

So forJ andb as before, we define, by iteration, ab-permissible sequence

V ←−π V1←−π1 V2←−π2 . . . Vr←−πr Vr+1,

and a factorizationJn−1OVn =I(Hn)bJn.WhereJn is the sheaf of ideals defining a hypersurfaceXi⊂Vi, which is the strict transform ofX.

We say that ab-permissible sequence defines ab-simplication ofJ⊂ OW if the jacobian of V ← Vr+1 has normal crossings, and V(Dif fb−1(Jr+1)) = ∅ (i.e. if Xr+1has at most points of multiplicityb−1).

Hironaka attaches to the original data J and b the pair (J, b). The closed set assigned to this pair inV is{ordJ ≥b}=V(Dif fb−1(J)). In our case, theb-fold points of the hypersurfaceX.

We attached to the original data a Rees algebra (up to integral closure), namely

G=OV[J Wb]. And to this Rees algebra a closed set inV, namelySing(G), which is againV(Dif fb−1(J)).

Moreover, we extended Gto a Diff-structure G(G), and Sing(G) =Sing(G(G)) (Th. 3.2).

Let us focus on theb-permissible transformationπ. The transform of Hironaka’s pair is the pair (J1, b). The transformation πis also permissible for both G and

G(G), defining transforms of Rees algebras, sayG1 andG(G)1onV1.

Note that, in our setting,J1is the ideal defining definingX1, which is the strict

transform of X. The closed set assigned to (J1, b) is the set of b-fold points of

X1. On the other hand, G1 =OV₁[J1Wb], is such that Sing(G1) is again the set of b-fold points X1. A similar relation holds between pairs (Ji, b) and the Rees algebrasGi(transform ofG), for anyb-permissible sequence.

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

Theorem 4.1. (J. Giraud) Let G be a Rees algebra on a smooth scheme V, and letV ←−V1 be a permissible (moniodal) transformation forG. LetG1 andG(G)1

denote the transforms ofG andG(G). Then: 1)G1⊂G(G)1.

2)G(G1) =G(G(G)1).

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, sayG(G), which we briefly discuss below. Recall that given an ideal J in a smooth scheme V, and a positive integer b, Hironaka defines a pair (J, b) (actually a closely related notion of idealistic exponent). As mentioned in Section 1, there is a closed set in V attached to the pair, and also a notion ofpermissibletransforms of pairs.

We have assign a Rees algebra to (J, b), sayG=O_V[J Wb]; and a closed set to

G, namely Sing(G). 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 aG-operator on Rees algebras, as two different manners to extend a Rees algebra.

the transform of the integral closure ofG. Theorem 4.1 asserts that the same holds for the transform ofG-extension ofG.

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

The result of Hironaka in [24] says that the G(G) is the biggest extension of

G with this property. Namely that Sing(G) = Sing(G(G)), 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 ofG(G). 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, sayIandJ, in a normal domainRhave the same integral closure if they are equal for any extension to a valuation ring (i.e. if IS = J S for any ring homomorphismR →S on a valuation ring S). The notion extends naturally to sheaves of ideals.

Hironaka considers the following equivalence on pairs (J, b) and (J, b) over a smooth schemeV.

Definition 5.1. The pairs (J, b) and (J, b) are idealisticequivalent on V if Jb and (J)b have the same integral closure.

Proposition 5.2. Let (J, b)and(J, b)be idealistic equivalent. Then: 1)Sing(J, b) =Sing(J, b).

Note, in particular, that any monoidal transform V ← V1 on a center Y ⊂

Sing(J, b) =Sing(J, b)defines transforms, say (J1, b)and((J)1, b)on V1.

2)The pairs(J1, b)and((J)1, b)are idealisticequivalent on V1.

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

Note that if (J, b) and (J, b) are idealistic equivalent, the they define the same closed set onV (i.e. Sing(J, b) =Sing(J, b)), 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 theydefine the same close sets.

Definition 5.3. Two pairs (J, b) and (J, b) arebasicallyequivalent onV, if the define the same close sets.

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

constructive desingularization was to define an algorithm of resolutions of pairs (J, b), 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 overV. The first, already formulated in Section 2:

Definition 5.5. Two Rees algebras over V, say G = _k≥0IkWk and G =

k≥0JkWk, areintegrally equivalent, if both have the same integral closure.

Proposition 5.6. Let G andG be two integrally equivalent Rees algebras overV Then:

1)Sing(G) =Sing(G).

Note, in particular, that any monoidal transform V ← V1 on a center Y ⊂ Sing(G) =Sing(G)defines transforms, say (G)1 and(G)1 onV1.

2)(G)1 and(G)1 are integrally equivalent on V1.

IfG andG areintegrallyequivalent onV, the same holds for any open restric-tion, and also for pull-backs by smooth morphismsW →V.

On the other hand, as (G)1 and (G)1are integrally equivalent, the they define

the same closed set onV1(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 V, we say that theydefine the same close sets.

Definition 5.7. Two Rees algebras over V, say G = _k≥0I_kWk and G =

k≥0JkWk, are basically equivalent, if both define the same closed sets.

The previous Proposition asserts that ifG=_k≥0IkWkandG=_k≥0JkWk are integrally equivalent, then they are basically equivalent.

5.8. We assign to a pair (J, b) over a smooth schemeV the Rees algebra, say:

G(J,b)=OV[JbWb],

which is a graded subalgebra inO_V[W].

Proposition 5.9. 1) Two pairs(J, b)and(J, b)are idealistically equivalent over a smooth schemeV, if and only if the Rees algebrasG(_J,b)andG(J_,b)are integrally

equivalent.

2) Two pairs(J, b)and(J, b)are basically equivalent overV, if and only if the Rees algebras G(_J,b) andG(J_,b) are basically equivalent.

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

We finally introduce a function, again a natural analog to that defined for idealistic exponents. Fixx∈Sing (G). GivenfnWn∈InWn, set

ord_x(f_n) = νx(fn) n ∈Q;

called the order of f_n (weighted by n), where ν_x denotes the order at the local regular ringO_Z,x. Asx∈Sing (G) it follows thatord_x(f_n)≥1.We also define

ord_x(G) =inf{ord_x(f_n);f_nWn∈I_nWn}.

So, in generalord_x(G)≥1 for anyx∈Sing (G).

Proposition 6.2. 1) IfG is a Rees algebra generated over O_Z byF ={g_niWni_, n_i>0,1≤i≤m}, then

ordx(G) =inf{ordx(gni); 1≤i≤m}.

And if N is any common multiple of all n_i,1≤i≤m, thenord_x(G) =ν(IN)

N .

2) If G and G are graded structures with the same integral closure (e.g. if

G ⊂ G _{is a finite extension), then, for any x∈Sing(G)(=Sing(G))}

ordx(G) =ordx(G).

3) SetG(G) =G=I_n·Wn (the extension of G to a differential structure), then for any x∈Sing(G)(=Sing(G)).

ordx(G) =ordx(G).

6.3. LetG be a Rees algebra, and fix a closed point x∈ Sing(G). We assume that at a affine open neighborhood of the point, sayU ⊂V, there is a finite set

F={f1Wn1, . . . , fsWns}, ni≥1 andfi∈ OV(U), so that the restriction ofG to

U is

OV(U)[f1Wn1, . . . , fsWns](⊂ OV(U)[W]).

Let

Gx=I_k·Wk(⊂ OV,x[W])

be the localizationof G at x. As x∈Sing(G), the order ofI_k at O_V,x is at least k. We say that G is simple at the singular pointx, if for some positive indexk, I_k has orderk. This amounts to saying thatord_x(G) = 1; or equivalently, that for somef_cWnc ∈ F_{, the elementf}

chas ordernc at OV,x.

Recall thatV(Dif fnc−1_{(< f}

c>))⊂Sing(G) locally atx.

We may choose the system of parameters {x1, . . . , xn} at x, so that at the completion ˆO_V,x, sayR=k[[x1, .., xn]]:

u.f_c=Znc_+a1Znc−1₊· · ·_+a

nc ∈S[Z] S=k[[x1, .., xn−1]], andZ=xn; whereuis a unit ofR.

A similar result holds at a suitable ´etale neighborhood ofx. We may assume that fc1is a monic polynomial of degreec1inS[Z], and of orderc1inS[Z]<MS,Z>⊂R, whereS is regular.

the previous setting holds for R= ˆOV,x, S= ˆO_V,π(x); and for somefcWnc ∈ F_, where fc has ordernc atOV,x.

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

π:Spec(S[Z]/f_c1(Z))→Spec(S).

Here we view H =Spec(S[Z]/f_c1(Z)) as a hypersurface inV, and locally at x,Sing(G) is included in thec1-fold points of this hypersurface. So

{ord f_c1 ≥n_c1}:=V(Dif fci−1_(f

c1(Z)))⊂H.

In this setting the finite morphism π is one to one over a closed subset of Y, namely on the image of the c1-fold points. Set

Spec(S[Z]/ < f_c1(Z)>) −→π Spec(S)

∪ ∪

{ord fc1 ≥nc1}

1to 1

−→ π({ord fc1≥nc1})

(6.3.1)

SinceSing(G)⊂ {ord f_c1≥n_c1},πinduces a one to one map, say

Sing(G)1−→to 1π(Sing(G)), for any transversal morphismπ:Spec(S[Z])→Spec(S).

Theorem 6.4. Let G be a Diff-structure over a smooth scheme V, and x ∈ Sing(G) a closed point which we assume to be simple. Let π : V → V be a smooth morphism defined at an ´etale neighborhood ofx, whereV is smooth, dim V=dimV-1. Assume that πis transversal at x. Then:

1) At a suitable neighborhood ofπ(x), there is a Rees algebraRGover the smooth scheme V, so thatπ(Sing(G)) =Sing(RG).

2) The morphismπinduces a one-to-one map fromSing(G)toSing(RG). Fur-thermore, setting S =O_V,π(x), and S[Z] as before, then the one-to-one map is

that described above.

The formulation of the theorem is independent of the choice of f_c1 of order n_c1 at OV,x. However given a finite morphisms as that in (6.3.1), and a smooth centerY1⊂Sing(RG), there is a unique and smooth centerY ⊂Sing(G) mapping

isomorphically toY1viaπ(and hence viaπ). SetY1=π(Y).

So bothY in V, andπ(Y) inSpec(S), are regular centers.

Let now V ←V1, andSpec(S)←U1, denote the monoidal transformations at

Y andπ(Y) respectively; and letHdenote the strict transform ofH. The hyper-surfaceH has at most points of multiplicityn_c1. LetF(⊂H) denotes the closed set of points of multiplicityn_c1. After replacingV1by a suitable neighborhood of

F, we may assume that there is a finite morphism, sayH→U1, compatible with

π.

As the regular centerY was chosen inSing(G), then a weighted transform, say

G1=I_n(1)·Wk(⊂ O_V1[W])

is defined, and Sing(G1)⊂F. So locally at a pointy∈Sing(G1) there is a finite morphism

where f_c1 is a strict transform offc1. LetG1 be the Diff-structure generated by G1. According to the previous Theorem, locally at π(y) there is an elimination algebra, say

RG

1 ⊂ OU1,π(y)[W].

On the other hand,Y1 =π(Y)⊂Sing(RG), so there is also a weighted

trans-form

(R_G)1⊂ OU1[W].

The question now is to relate the Rees algebra (R_G)1 with RG

1, locally at the

pointπ(y).

Proposition 6.5. With the setting as above: 1) There is a natural inclusion(R_G)1⊂ RG

1.

2) Over fields of characteristic zero both(RG)1 and R_G

1 define the same

Diff-structure, up to integral closure.

Here (RG)1is the transform ofRG by one monoidal transformation. If we could guarantee thatSing(R_G)1=π(Sing(G1), we could identify the singular locus of G1(i.e. ofG₁) with the singular locus of the transform ofR_G. If furthermore, this link betweenGandR_G is preserved by any sequence of monoidal transformations, then we have achieved a way of representing the singular locus ofGwhich 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. Matem´aticas, Facultad de Ciencias,

Universidad Aut´onoma de Madrid, Canto Blanco 28049 Madrid, Spain [email protected]