Table Of ContentCOMPLEXITY OF VILLAMAYOR’S ALGORITHM
IN THE NON EXCEPTIONAL MONOMIAL CASE
9
0
0
2 Rocio Blanco∗
n
a
J
2 Abstract
2
We study monomial ideals, always locally given by a monomial, like a reasonable first step
] to estimate in general the number of monoidal transformations of Villamayor’s algorithm of
G resolution ofsingularities. Theresolution of amonomial ideal <X1a1·...·Xnan >isinteresting
A dueto its identification with theparticular toric problem <Zc−X1a1·...·Xnan >.
. In the special case, when all the exponents ai are greater than or equal to the critical
h
value c, we construct the largest branch of the resolution tree which provides an upper bound
t
a involving partial sums of Catalan numbers. This case will be called “minimal codimensional
m case”. Partial sums of Catalan numbers (starting 1,2,5,...) are 1,3,8,22,... These partial
[ sumsarewellknowninCombinatorics andcountthenumberofpathsstartingfrom therootin
all ordered trees with n+1 edges. Catalan numbers appear in many combinatorial problems,
4 counting the number of ways to insert n pairs of parenthesis in a word of n+1 letters, plane
v
trees with n+1 vertices, ..., etc.
6
The non minimal case, when there exists some exponent ai smaller than c, will be called
1 0
“case of higher codimension”. In this case, still unresolved, we give an example to state the
4
foremost troubles.
3
. Computation of examples has been helpful in both cases to study the behaviour of the
4
resolutioninvariant. ComputationshavebeenmadeinSingular(see[8])usingthedesingpackage
0
by G. Bodn´ar and J. Schicho, see [4].
7
0
:
v 1 Introduction
i
X
The existence of resolution of singularities in arbitrary dimension over a field of characteristic zero
r
a was solved by Hironaka in his famous paper [9]. Later on, different constructive proofs have been
given, among others, by Villamayor [13], Bierstone-Milman [1], Encinas-Villamayor [6], Encinas-
Hauser [5] and Wodarczyk [14].
This paper is devoted to study the complexity of Villamayor’s algorithm of resolution of singu-
larities. This algorithm appears originally in [13] and we will use the presentation given in [6]. In
this paper, the authors introduce a class of objects called basic objects B =(W,(J,c),E) where W
is a regular ambient space over a field k of characteristic zero, J ⊂ O is a sheaf of ideals, c is an
W
integer and E is a set of smooth hypersurfaces in W having only normal crossings. That is, they
consider the ideal J together with a positive integer c, or critical value defining the singular locus
Sing(J,c)={ξ ∈W| ord (J)≥c}, where ord (J) is the order of J in a point ξ.
ξ ξ
Let W ←π W be the monoidal transformation with center Z ⊂ Sing(J,c), π 1(Z) = Y is the
′ − ′
exceptional divisor. Let ξ be the generic point of Z, ord (J) = θ, the total transform of J in W
ξ ′
satisfies JOW′ =I(Y′)θ ·Jg where Jg is the weak transform of J, (see [6] for details).
∗ResearchpartiallysupportedbyanF.P.U.Fellowship,SpanishMinistryofEducationandCulture,AP2002-0009.
1
A transformation of a basic object (W,(J,c),E) ← (W ,(J ,c),E ) is defined by a monoidal
′ ′ ′
transformation W ←π W and defining J =I(Y )θ c·Jg, the controlled transform of J.
′ ′ ′ −
A sequence of transformations of basic objects
(W,(J,c),E)←(W(1),(J(1),c),E(1))←···←(W(N),(J(N),c),E(N)) (1)
is a resolution of (W,(J,c),E) if Sing(J(N),c)=∅.
Remark 1.1. Superscripts (k) in basic objects will denote the k-stage of the resolution process.
Subscripts will always denote the dimension of the ambient space W(k).
i i
Villamayor’s algorithm provides a log-resolution in characteristic zero. A log-resolution of J is
a sequence of monoidal transformations at regular centers as (1) such that each center has normal
crossings with the exceptional divisors E(i), and the total transform of J in W(N) is of the form
JOW(N) =I(H1)b1 ·...·I(HN)bN
with b ∈N for all 1≤i≤N and E(N) ={H ,...,H }.
i 1 N
In [6] it is shown that algorithmic principalization of ideals reduces to algorithmic resolution of
basicobjects. Thatis,startingwithc=maxord(J), the maximalorderofJ,weobtainaresolution
of (W,(J,c),E) as (1). At this step max ord(J(N))=c(N)< c. If c(N)> 1, we continue resolving
(W(N),(J(N),c(N)),E(N)) and so on, until have max ord(J( ))=c( )= 1. Finally, a resolution of
N N
(W( ),(J( ),1),E( )) provides a log-resolutionof J( ), and therefore a log-resolutionof J.
N N N N
In [6] it is also shown that algorithmic principalization of ideals leads to embedded desingulari-
zation of varieties. That is, given a closed subscheme X ⊂ W, the algorithmic principalization of
the ideal I(X) provides an embedded desingularization of X. See also [7] for more details.
Akeypointinthedefinitionofthealgorithmistouseinductiononthedimensionoftheambient
spaceW todefineanupper-semi-continuousfunctiont. Thesetofpointswherethisfunctionattains
its maximalvalue, Max t, is a regularclosedset, anddefines a regularcenter for the next monoidal
transformation.
A resolution of the basic object (W,(J,c),E) is achieved by a sequence of monoidal transfor-
mations as in (1), with centers Max t(k) for 0 ≤ k ≤ N −1. That is, the sequence of monoidal
transformations is defined by taking successively the center defined by the upper-semi-continuous
function. Thealgorithmstopsatsomestagebecausethemaximalvalueofthefunctiontdropsafter
monoidal transformations, that is, max t(0) >max t(1) >...>max t(N 1).
−
This function t will be the resolution invariant. We shall work with the invariant defined in [6],
using the language of mobiles developed in [5]. We remind briefly the main notions.
Let J ⊂ O be an ideal defining a singular algebraic set X ⊂ W. The ideal J factors into
W
J =M ·I, with M the ideal defining a normal crossing divisors, and I some ideal still unresolved.
ByinductiononthedimensionofW, wewillhavethis decompositionateverydimensionfromn
to1,thatisJ =M ·I ,forn≥i≥1,aredefinedinlocalflagsW ⊇W ⊇···⊇W ⊇···⊇W ,
i i i n n 1 i 1
−
where each J ,M ,I ∈ O are in dimension i. There is a critical value c at each dimension i,
i i i Wi i+1
(c =c),see[5]fordetails. Allthebasicobjects(W ,(J ,c ),E ),forn≥i≥1,willberesolved
n+1 i i i+1 i
during the process of the algorithm.
LetE betheexceptionaldivisorofpreviousmonoidaltransformations,andconsiderE =∪n E
i=1 i
where E applies to dimension i. Obviously, we start with E =∅.
i
For any point ξ ∈Sing(J,c), the function t will have n coordinates, with lexicographical order,
and it will be one of the following three types:
(a) t(ξ)=(t (ξ),t (ξ),...,t (ξ), ∞, ∞,...,∞)
n n 1 n r θ
− − i
(b) t(ξ)=(t (ξ),t (ξ),...,t (ξ),Γ(ξ),∞,...,∞) with t = ,m (2)
n n 1 n r i i
(c) t(ξ)=(tn(ξ),tn−1(ξ),...,tn−r(ξ),.........,t1(ξ)) (cid:20)ci+1 (cid:21)
− −
2
whereθ =ord (I ),m isthenumberofexceptionaldivisorsinE ,andΓistheresolutionfunction
i ξ i i i
corresponding to the so-called monomial case, following the notation of [6], pages 165−166. We
will recall the definition of Γ in equation (4).
For simplicity, let assume that we start with a polynomial ring, W = Spec(k[X ,...,X ]). In
1 n
O =k[X ,...,X ] the ideal J is locally given by a monomial with respect to a regular system of
W 1 n
parameters
J =<X1a1 ·...·Xnan >⊂OW with ai ∈N, for i=1,...,n.
Note that, in this situation, the center of the next monoidal transformation is combinatorial, it
is a linear combination of X ,...,X . And this is also true after monoidal transformations, since
1 n
Villamayor’salgorithmappliedtoamonomialidealprovidesalwayscombinatorialcenters,andafter
a monoidal transformation in a combinatorialcenter we obtain again a monomial ideal.
So, at any stage of the resolution process, W =∪iUi, where Ui ∼=Ank. Thereafter, we shall work
locally, so we will assume that W is an affine space.
To resolve the toric hypersurface {f =0}={Zc−Xa1 ·...·Xan =0} we note that its singular
1 n
locus Sing(<f >,c) is always included in {Z =0}, so we argue by induction onthe dimension and
reduce to the case where the corresponding ideal J is of the form
n
J =<X1a1 ·...·Xnan >⊂OW with 1≤a1 ≤a2 ≤...≤an, ai =d, d≥c, (3)
i=1
X
where c is the critical value. If a =0 for some i, then we may assume dim(W)<n.
i
After a monoidal transformation, we always consider the controlled transform of J with respect
to c, J = I(Y ) c ·J where J is the total transform of J and Y denotes the new exceptional
′ ′ − ∗ ∗ ′
divisor. For the toric problem J =<Zc−Xa1 ·...·Xan >, taking the origin as center of the next
1 n
monoidal transformation, at the i-th chart:
J∗ =<Zc·Xic−X1a1···Xid···Xnan >=<Xic·(Zc−X1a1···Xid−c···Xnan)>,
and we can only factorize c times the exceptional divisor.
Remark 1.2. We will denote as i-th chart the chart where we divide by X . When the center of the
i
monoidal transformation is the origin, this monoidal transformation is expressed:
k[Z,X1,...,Xn] → k[Z,X1,...,Xn,XZi,XX1i,...,XXi−i1,XXi+i1,...,XXni]
Z → Z
Xi
X → X
i i
X → Xj for j 6=i
j Xi
where k[Z,X1,...,Xn,XZi,XX1i,...,XXi−i1,XXi+i1,...,XXni]∼=
k[XZi,XX1i,...,XXi−i1,Xi,XXi+i1,...,XXni]. For simplicity, we will denote each XXji again as Xj, and XZi
as Z.
So we will apply the resolution algorithm to the basic object (W,(J,c),∅) for J =< Xa1 ·...·
1
Xan >, which is already a monomial ideal, but it is not necessarily supported on the exceptional
n
divisors.
2 Monomial case (exceptional monomial)
The monomial case is a special case in which J is a “monomial ideal” given locally by a monomial
that can be expressed in terms of the exceptional divisors. This case arises after several monoidal
transformations.
3
Thismeansthatwehaveabasicobject(W,(J,c),E)whereJ islocallydefinedbyonemonomial
supported on the hypersurfaces in E. In this case, the ideal J factors into J = M ·I with J = M
and I =1. We can also call it exceptional monomial.
Theorem 2.1. Let J ⊂ O be a monomial ideal as in equation (3). Let E = {H ,...,H }, with
W 1 n
H =V(X ), be a normal crossing divisor.
i i
Then an upper bound for the number of monoidal transformations to resolve (W,(J,c),E) is given
by
d−c+gcd(a ,...,a ,c)
1 n
.
gcd(a ,...,a ,c)
1 n
Proof. We may assume that the greatest common divisor of the exponents a and the critical value
i
cisequalto1,becauseboththesimplifiedproblemandtheoriginalproblemhavethe samesingular
locus. That is, if gcd(a ,...,a ,c) = k then d = k·d , c = k·c , a = k·b for all 1 ≤ i ≤ n and
1 n 1 1 i i
gcd(b1,...,bn,c1) = 1. The ideal J can be written as J = (J1)k where J1 =< X1b1 ·...·Xnbn >
therefore
Sing(J,c)={ξ ∈X| ord ((J )k)≥k·c }={ξ ∈X| ord (J )≥c }=Sing(J ,c ),
ξ 1 1 ξ 1 1 1 1
where X is the algebraic set defined by J.
For a point ξ ∈An, Γ(ξ)=(−Γ (ξ),Γ (ξ),Γ (ξ)) where
k 1 2 3
Γ (ξ)=min{p | ∃ i ,...,i ,a (ξ)+···+a (ξ)≥c, ξ ∈H ∩···∩H },
1 1 p i1 ip i1 ip
Γ2(ξ)=max ai1(ξ)+·c··+aip(ξ) | p=Γ1(ξ), ai1(ξ)+···+aip(ξ)≥c, ξ∈Hi1∩···∩Hip , (4)
Γ3(ξ)=max{n(i1,...,ip,0,...,0)∈Zn | Γ2(ξ)=ai1(ξ)+·c··+aip(ξ), ξ∈Hi1∩···∩oHip}
with lexicographicalorder in Zn.
The center Z of the next monoidal transformation is given by the set of points where Γ attains
its maximal value. It is easy to see that Z =∩n H .
i=n (r 1) i
− − n
So atthe j-th chart, the exponentof X after the monoidaltransformationis ( a )−c
j i=n r+1 i
and −
n P
a −c< min a =a
i i n r+1
! n r+1 i n −
i=n r+1 − ≤≤
X−
n
because a <c by construction of the center Z.
i=n r+2 i
This shows−that the order of the ideal drops after eachmonoidal transformationby at least one,
P
so in the worst case, we need d−(c−1) monoidal transformations to obtain an order lower than
c.
Remark 2.2. Note that it is necessary to consider the monomial case. On one hand, this case may
appear in dimension n, and also in lower dimensions, n−1,...,1, when we resolve any basic object
(W,(J,c),E) (where J is any ideal). So we need to resolve the monomial case in order to obtain a
resolution of the original basic object (W,(J,c),E).
Onthe other hand, the algorithmofresolutionleads to the monomialcase,since givenany ideal
J,thealgorithmprovidesalog-resolutionofJ. Anditisnecessarytocontinuetoaresolutionwithin
the monomial case.
Remark 2.3. The bound in theorem 2.1 is reached only for the following values of c:
1, a +...+a +1 for n≥j ≥2, d.
n j
Forthesevaluesofc,theorderoftheidealdropsaftereachmonoidaltransformationexactlybyone:
4
• If c = 1, the monoidal transformation is an isomorphism. The exponent of X after the
n
monoidal transformation is a −1.
n
• If c = a +...+a +1, for n ≥ j ≥ 2, the center of the monoidal transformation is Z =
n j
∩n H . At the l-th chart, for n ≥ l ≥ j − 1, the exponent of X after the monoidal
i=j 1 i l
− n n n
transformation is ( a )−c=( a )−( a )−1=a −1.
i=j 1 i i=j 1 i i=j i j 1
− − −
In particular,at the (j−1)-th chart,the exponent of X after the monoidaltransformation
P P P j 1
−
has droped exactly by one.
• If c=d, we finish after only one monoidal transformation.
Remark 2.4. If gcd(a ,...,a ,c)=k >1, then the bound of the theorem 2.1 is (d−c+k)/k.
1 n
As (d−c+k)/k<d−c+1, we can use in practice the bound for the case gcd(a ,...,a ,c)=1.
1 n
3 Case of one monomial
Toconstructanupperboundforthenumberofmonoidaltransformationsneededtoresolvethebasic
object (W,(J,c),E =∅), where J is locally defined by a unique monomial, we estimate the number
ofmonoidaltransformationsneededtoobtain(W ,(J ,c),E ),atransformationoftheoriginalbasic
′ ′ ′
object, with J = M (an exceptional monomial), and then apply theorem 2.1. In order to use
′ ′
theorem 2.1, we need an estimation of the order of M . This estimation will be valid at any stage
′
of the resolution process.
Lemma 3.1. Let (W,(J,c),∅) be a basic object where J is a monomial ideal as in equation (3). Let
J =M ·I be the factorization of J, where M =1, because of E =∅, and J =I. After N monoidal
transformations we have (W(N),(J(N),c),E(N)). Let ξ ∈W(N) be a point. Then
ord (M(N))≤(2N −1)(d−c)
ξ
where ord (M(N)) denotes the order at ξ of M(N), the (exceptional) monomial part of J(N).
ξ
Proof. It follows by induction on N:
• that if N =1, ord (M(1))=d−c.
ξ
At the beginning, the first center defined by this algorithmis always the origin, so at the i-th
chart:
J(1) =M(1)·I(1) =<Xid−c >·<X1a1·.bi..·Xnan >
with E(1) ={H }, where H =V(X ).
i i i
• We assume that the result holds for N =m−1.
J(m−1) =M(m−1)·I(m−1) =<Xib11···Xibss >·<Xiasi+s+11 ···Xianin >
with s b =d.Byinductivehypothesis,afterm−1monoidaltransformations,themaximal
i=1 i ′
order d of the (exceptional) monomial part M(m 1) satisfies
′ −
P
d ≤(2m 1−1)(d−c).
′ −
For N =m, there are two possibilities:
1. Ifmaxord(I(m 1))= n a ≥cthenthecenterofthenextmonoidaltransformation
− j=s+1 ij
contains only variables appearing in I(m 1).
P −
5
2. Ifmaxord(I(m 1))= n a <cthenthecenterofthenextmonoidaltransformation
− j=s+1 ij
contains variables appearing in I(m 1) and also variables appearing in M(m 1).
P − −
Case 1:
Ifthecenterofthemonoidaltransformationisassmallaspossible,thatisZ=∩n V(X ),
j=s+1 ij
at the i -th chart,
l
J(m)=M(m)·I(m)=<Xb1···Xbs ·Xd−Psj=1aij−c>·<Xais+1·ib·l·Xain>.
i1 is il is+1 in
s n
The exponent of X , d− a −c= a −c is as big as possible, so this is
il j=1 ij j=s+1 ij
the worst case, because the increase in the order of the exceptional monomial part after
P P
the monoidal transformation will be greater than that for another centers.
The highest order of M(m) is
s s s
b +d− a −c=d +d−c− a ≤d +d−c,
i ij ′ ij ′
i=1 j=1 j=1
X X X
so by inductive hypothesis
d +d−c≤(2m 1−1)(d−c)+d−c=2m 1(d−c)≤(2m−1)(d−c).
′ − −
Case 2: - At the i -th chart, for 1≤j ≤s
j
J(m) =M(m)·I(m) =<Xb1 ·ib·j·Xbs ·X(cid:3) >·<Xais+1···Xain >.
i1 is ij is+1 in
- At the i -th chart, for s+1≤l ≤n
l
J(m) =M(m)·I(m) =<Xb1···Xbs ·X△ >·<Xais+1 ·ib·l·Xain >.
i1 is il is+1 in
As above, if we are in the worst case, when the center of the monoidal transformation is
as small as possible, that is, the center is a point,
s
(cid:3)=△=d +d− a −c .
′ ij
j=1
X
Therefore in both cases the highest order of M(m) satisfies
≤2d +d−c ≤ 2(2m 1−1)(d−c)+d−c=(2m−1)(d−c) .
′ −
Remark 3.2. Due to its general character, this bound is large and far from being optimal.
Remark 3.3. The ideals M are supported on a normal crossing divisors D . Recall that their
i i
transformations after monoidal transformations, in the neighbourhood of a point ξ ∈W , are
i
D = Di∗+(θi−ci+1)·Y′ if(t′n(ξ′),...,t′i+1(ξ′)=(tn(ξ),...,ti+1(ξ)) , n≥i≥1,
i′ ∅ inothercase
(cid:26)
(Dn′=Dn∗+(θn−c)·Y′ always)
where D denotes the pull-back of D by the monoidal transformation π, Y denotes the new ex-
i∗ i ′
ceptional divisor, the point ξ ∈ W satisfies π(ξ ) = ξ, θ = ord (I ) and c is the corresponding
′ i′ ′ i ξ i i+1
critical value.
6
In what follows we will define the ideals J , n ≥ i > 1. We need some auxiliary definitions:
i 1
−
the companion ideals P and the composition ideals K , see [5] for details.
i i
We construct the companion ideals to ensure that Sing(P ,θ )⊂Sing(J ,c ),
i i i i+1
I if θ ≥c
i i i+1
Pi =( Ii+Mici+θ1i−θi if 0<θi <ci+1 (5)
where ξ ∈An is a point, θ =ord (I ) and c is the corresponding critical value.
k i ξ i i+1
Let J = M ·I be the factorization of an ideal J in W , where M ,I are ideals in W in the
i i i i i i i i
neighborhood of a point ξ ∈An. Let E be a normal crossing divisor in An.
k i k
The composition ideal K in W of the product J =M ·I , with respect to a control c , is
i i i i i i+1
P ·I (E ∩W ) if I 6=1,
K = i Wi i i i (6)
i 1 if I =1.
i
(cid:26)
The critical value for the following step of induction on the dimension is c =ord (K ).
i ξ i
TheconstructionofthecompositionidealK ensuresnormalcrossingwiththeexceptionaldivisor
i
E .
i
We say that an ideal K is bold regular if K =<Xa >, K ∈k[X], a∈N.
Finally, construct the junior ideal J
i 1
−
Coeff (K ) if K is not bold regular or 1
J = V i i (7)
i−1 1 otherwise
(cid:26)
where V is a hypersurface of maximal contact in W (see [5] page 830) and Coeff (K ) is the
i V i
coefficient ideal of K in V (see [5] page 829). The junior ideal J is an ideal in this suitable
i i 1
−
hypersurface V of dimension i−1.
If θcn ≥1 we are in the first case of equation(5), θncn−1 = θcnn−−12 =...= cjθ+j1 =1 and tj−1 =...=
t = ∞ for n−1 ≥ j ≥ 1, because D = ... = D = ∅ and P = I , and hence J is always
1 n 1 1 i i i 1
− −
given by a unique monomial.
Remark 3.4. ForanidealJ =<X1a1·...·Xnan >asinequation(3),ifweassumean ≥an 1 ≥...≥
a ≥c, then at every stage θn ≥ 1 , so we are always in the previous situation. The sing−ular locus
1 c
of (J,c) is always a union of hypersurfaces ∪r {X = 0}, 1 ≤ r ≤ n, and the center of the next
i=1 i
monoidaltransformationwill be the intersectionof some ofthese hypersurfaces. So we will callthis
case the minimal codimensional case.
Remark 3.5. If there exists some a < c, at a certain stage of the resolution process it may occur
i0
θcn < 1. Then we are in the second case of equation (5), the (exceptional) monomial part Mn can
appearinsomeJ forn−1≥j ≥1,and θj canbemuchgreaterthan1,whatincreasethenumber
j cj+1
of monoidal transformations. Now its singular locus is a union of intersections of hypersurfaces of
the type ∪ ({X =0}∩...∩{X =0}). This is the higher codimensional case.
lj l1 li
4 Bound in the minimal codimensional case
Remark 4.1. From now on, we always look to the points where the function t, defined in (2), is
maximal. Sothefollowingresultsconcerningthebehaviourofthefunctiontalwaysaffectthepoints
where it reaches its maximal value.
7
Proposition 4.2. Let (W,(J,c),E) be a basic object where J is a monomial ideal as in equation
(3), with a ≥ c for all 1 ≤ i ≤ n. We can factor J = J = M ·I , and after r−1 monoidal
i n n n
transformations, Jn(r−1) =Mn(r−1)·In(r−1). Let ξ ∈W(r−1) bea point where ordξ(In(r−1))=θn. After
each monoidal transformation π, the resolution function in a neighbourhood of ξ , where π(ξ ) = ξ
′ ′
and ordξ′(In(r))=θn′ <θn, is of the form
s
d− a
j=1 ij,s ,[1,0],...,[1,0] for some 1≤s≤n−1.
c
" P # !
Proof. After monoidal transformations,
J(r) =M(r)·I(r) =<Xb1···Xbs >·<Xais+1···Xain >
n n n i1 is is+1 in
with d− s a = n a ≥ c then, P(r) = I(r) and the (exceptional) monomial part does
j=1 ij j=s+1 ij n n
not appear in J(r) for all n≥l≥1.
P l P
We have θ 6= θ , then E(r) = Y +|E|g and m = s, we count all the exceptional divisors of
n′ n n ′ n
the previous steps and the new one. There are no exceptional divisors in lower dimension because
E(r) =(Y +|E|g)−E(r) =∅ and, in a similar way, we obtain E(r) =∅ for all n−1≥l ≥1.
n 1 ′ n l
−The normal crossing divisors D(r) = ∅ for all n − 1 ≥ i ≥ 1, so the corresponding ideals
i
M(r) =...=M(r) =1. In particular, M(r) =1, hence
n 1 1 n 1
− −
c′n =ordξ′(Kn(r))=ordξ′(Coeff(Kn(r)))=ordξ′(Jn(r)1)=ordξ′(In(r)1)=θn′ 1
− − −
withξ ∈W(r) suchthatπ(ξ )=ξ,becauseord(Coeff(K))=ord(K)whenK is amonomialideal,
′ ′
therefore θn′c′n−1 =1. By the same argument we obtain θc′nn′−−12 =...= θc′21′ =1.
Remark 4.3. After each monoidal transformation, the exceptional divisors at each dimension are:
Ej′ =(cid:26) E(Yjg′+(E1∪...∪En)g)−(En′+···+Ej′+1) iinf(ott′nh(eξr′)c,.a.s.,et′j+1(ξ′))=(tn(ξ),...,tj+1(ξ)) andθj′=θj
(En′=Eng if θn′=θn or En′=Y′+(E1∪...∪En)g otherwise)
for n > j ≥ 1, where Eg denotes the strict transform of E by the monoidal transformation π,
j j
Y′ denotes the new exceptional divisor, the point ξ′ ∈ Wi′ satisfies π(ξ′) = ξ, θj′ = ordξ′(Ij′) and
θ =ord (I ). We denote |E|=E ∪...∪E .
j ξ j 1 n
Hence, after the first monoidal transformation, since θ < θ we have E = Y and E =
n′ n n′ ′ n′ 1
··· = E = ∅. After the second monoidal transformation, at the chart where θ = θ we ob−tain
1′ n′′ n′
E = (E )g = ∅, E = Y , and E = ··· = E = ∅ and so on. We call this phenomena
n′′ n′ n′′ 1 ′′ n′′ 2 1′′
propagationbecauseev−eryexceptionaldiv−isorappearsintheresolutionfunctiontfirstlyindimension
n, then in dimension n−1, n−2, and so on.
Definition 4.4. We will call propagation, p(i,j), for 1 ≤ i ≤ j −1, 1 ≤ j ≤ n, to the number
of monoidal transformations needed to eliminate i exceptional divisors in dimension j, when we
remainconstant(t ,t ,...,t )andθ ,andtherearenoexceptionaldivisorsinlowerdimensions
n n 1 j+1 j
−
j−1,...,1. That is, passing from the stage
([θ ,m ],...,[θ ,m ],[θ ,i],[1,0],...,[1,0])
n n j+1 j+1 j
to the stage
i
([θ ,m ],...,[θ ,m ],[θ ,0],[1,0],...,[1,0],∞,...,∞).
n n j+1 j+1 j
z }| {
8
Lemma 4.5. Propagation Lemma Let (W,(J,c),E) be a basic object where J is a monomial
ideal as in equation (3) with a ≥c for all 1≤l ≤n. Let p(i,j) be the propagation of i exceptional
l
divisors in dimension j in the resolution process of (W,(J,c),E).
Then, for all 1≤j ≤n,
i+ i p (k,j−1) if 0≤i≤j−1
p (i,j)= k=1 (8)
0 if i=j
(cid:26) P
Proof. • If there are i exceptional divisors in dimension i, K is bold regular, t = ∞ then
i+1 i
p(i,i) = 0. We can not propagate these i exceptional divisors at this stage of the resolution
process.
If there are s exceptional divisors at this step of the resolution process, then there are n−s
variables in I . On the other hand, from dimension n to dimension i + 1 there are s − i
n
exceptional divisors.
WhenweconstructJ ,...,J ,addtothecorrespondingcompositionidealK thevariables
n 1 i+1 j
−
in I (E ∩W ), so in these dimensions there are (n−s)+(s−i)=n−i variables.
Wj j j
Whenwemakeinductiononthe dimension,ateachsteploseonevariable,soinn−i−1steps
obtainthatK ,thatcorrespondstothen−(n−i−1)=i+1position,isboldregular. And
i+1
the variables appearing in these i exceptional divisors do not appear in the center of the next
monoidal transformation.
• By induction on the dimension:
- If j =1, p(1,1)=0 by the previous argument.
- If j = 2, p(1,2)= 1 because when we propagate 1 excepcional divisor from dimension 2
to dimension 1, K is bold regular.
2′
([θ ,m ],...,[θ ,1],[1,0])
n n 2
↓Xi
([θ ,m ],...,[θ ,0],∞)
n n 2
Then p(1,2)=1=1+0=1+p(1,1).
- We assume that the result holds for j ≤s−1. For j =s:
([θ ,m ],...,[θ ,m ],[θ ,i],[1,0],...,[1,0])
n n s+1 s+1 s
↓
([θ ,m ],...,[θ ,m ],[θ ,i−1],[1,1],[1,0],...,[1,0])
n n s+1 s+1 s
↓
.
. p(1,s−1)
.
↓
([θn,mn],...,[θs+1,ms+1],[θs,i−1],[1,0],...,[1,0],∞)
↓
([θ ,m ],...,[θ ,m ],[θ ,i−2],[1,2],[1,0],...,[1,0])
n n s+1 s+1 s
We want ([θ ,m ]...[θ ,m ]) and θ remain constant. So after the first monoidal
n n s+1 s+1 s
transformation look to some suitable chart where m = i drops. As m drops then
s s
m =i−(i−1)=1 andpropagatethis exceptionaldivisorin dimensions−1,making
s 1
−
9
p(1,s−1) monoidal transformations. Otherwise, to keep ([θ ,m ]...[θ ,m ]) and
n n s+1 s+1
θ constant, the only possibility is to look to a suitable chart where m drops from i−1
s s
to i−2. But in this case this would provide the same resolution function that appears
after the propagation. As we wantto constructthe largestpossible sequence ofmonoidal
transformations, we follow the propagationphenomenon as above.
After more monoidal transformations:
↓
.
. p(2,s−1)
.
↓
↓
.
.
.
↓
i 1
−
([θ ,m ],...,[θ ,m ],[θ ,1],[1,0],...,[1,0],∞,...,∞)
n n s+1 s+1 s
↓
z }| {
([θ ,m ],...,[θ ,m ],[θ ,0],[1,i],[1,0],...,[1,0])
n n s+1 s+1 s
Then, for 1≤i≤s−1,
p(i,s)=1+p(1,s−1)+1+p(2,s−1)+···+1+p(i,s−1)
with p(l,s−1), 1≤l ≤i, defined by the induction hypothesis.
Remark 4.6. ComputationofexamplesinSingularwithdesing packagehasbeenusefultostatethis
behaviour of the exceptional divisors after monoidal transformations. The implementation of this
package is based on the results appearing in [3].
Theorem 4.7. Let (W,(J,c),∅) be a basic object where J is a monomial ideal as in equation (3)
with a ≥ c for all 1 ≤ i ≤ n. Then, the resolution function corresponding to (W,(J,c),∅) drops
i
after monoidal transformations in the following form:
10
