monomialization of strongly prepared morphisms from nonsingular n-folds to surfaces

a

s

to aand

s

Journal of Algebra 275 (2004) 275–320

www.elsevier.com/locate/jalgebr

Monomialization of strongly prepared morphismfrom nonsingularn-folds to surfaces

Steven Dale Cutkosky∗,1 and Olga Kashcheyeva

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Received 20 March 2003

Communicated by Craig Huneke

1. Introduction

Monomialization of morphisms is the problem of transforming a mapping inmonomial mapping by blowing up a chain of nonsingular subvarieties in its domainimage. Some articles on this and related problems are [1–7].

Consider the following basic example. LetΦ :Ank →Am

k be a morphism of affine spaceover fieldk. ThenΦ is given by a collection of polynomialsf1, . . . , fm in n variables:

y1 = f1(x1, . . . , xn),

...

ym = fm(x1, . . . , xn).

The simplest structure ofΦ is obtained whenf1, . . . , fm are monomials and

y1 = xa111 · · ·xa1n

n ,

...

ym = xam11 · · ·xamn

n .

If, moreover,Φ is a dominant morphism the matrix(aij ) is forced to satisfy thenondegeneracy condition rank(aij )=m.

* Corresponding author.E-mail address:[email protected] (S.D. Cutkosky).

1 The author was partially supported by NSF.

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/S0021-8693(03)00366-1

276 S.D. Cutkosky, O. Kashcheyeva / Journal of Algebra 275 (2004) 275–320

arod

st

es

wsnenthe

In case of a general dominant morphismΦ :X → Y between twok-varietiesX andYwe would like to get such a nice description locally.

Definition 1.1. Suppose thatΦ :X → Y is a dominant morphism of nonsingulk-varieties.Φ is called monomial if for all pointsp ∈ X there exist an étale neighborhoU of p, uniformizing parameters(x1, . . . , xn) on U , regular parametersy1, . . . , ym inOY,Φ(p) and a matrix(aij ) of nonnegative integers with rankm such that

y1 = xa111 · · ·xa1n

n ,

...


n .

The natural question arises.

Question. Suppose thatΦ :X → Y is a dominant morphism ofk-varieties. Does there exia monomialization ofΦ? Or, more precisely, given a dominant morphismΦ :X → Y doesthere exist a monomial morphismΦ1 :X1 → Y1 such that the following diagram commut

X1Φ1

Y1

XΦ

Y

and all vertical maps are products of blowups of nonsingular subvarieties inX andY?

The answer is “yes” over a characteristic zero field whenY is a curve or whenY is asurface and dim(X) 3.

Suppose thatk is an algebraically closed field of characteristic zero. IfΦ :X → C is adominant morphism from ak-variety to a curve the existence of monomialization follofrom resolution of singularities. IfΦ :P → S is a dominant morphism of surfaces oproof of monomialization (overC) is given by Akbulut and King [3]. And the last knowcase, whenΦ :X → S is a dominant morphism from a 3-fold to a surface, is done byfirst author in [4]. This proof of monomialization breaks down into two key steps:

(1) obtain a diagram

X1Φ1

Y1

XΦ

Y

S.D. Cutkosky, O. Kashcheyeva / Journal of Algebra 275 (2004) 275–320 277

cts of

a

sm.

a

3.2.ample

ietyagood

l

t

whereΦ1 is a strongly prepared morphism and the vertical maps are the produblowups of nonsingular subvarieties;

(2) monomialize the strongly prepared morphismΦ1 :X1 → Y1.

The natural next case to consider is monomialization of morphisms fromn-folds tosurfaces. A proof would follow from the two steps above whenX is ann-fold andY isa surface. In this paper we complete step (2). Our main result is

Theorem 1.2. Suppose thatΦ :X → S is a strongly prepared morphism fromnonsingularn-foldX to a nonsingular surfaceS.

Then there exists a finite sequence of quadratic transformsπ :S1 → S and monoidaltransforms, centered at nonsingular varieties of codimension2, π2 :X1 →X such that theinduced morphismΦ :X1 → S1 is monomial.

From here we deduce that it is possible to toroidalize a strongly prepared morphi

Theorem 1.3. Suppose thatΦ :X → S is a strongly prepared morphism fromnonsingularn-foldX to a nonsingular surfaceS.

Then there exists a finite sequence of quadratic transformsπ1 :S1 → S and monoidaltransforms, centered at nonsingular varieties of codimension2, π2 :X1 →X such that theinduced morphismΦ :X1 → S1 is toroidal.

The definition of a strongly prepared morphism is given in Section 3, DefinitionThe class of strongly prepared morphisms is rather restrictive. However, a natural exof such a morphism can be obtained as follows.

Let Φ :X → S be a monomial mapping from ann-fold to a surface andπ :X1 → X

be a finite sequence of blowups of points. Then the composition mapΦ π :X1 → S isstrongly prepared, but not necessarily monomial.

2. Notations

We will suppose thatk is an algebraically closed field of characteristic zero. By a varwe will mean a separated, integral finite typek-scheme. A point of a variety will meanclosed point. By a generic point on a variety we will mean a point which satisfies acondition which holds on an open set of points.

Suppose thatZ is a variety andp ∈Z is a point. Thenmp will denote the maximal ideaof OZ,p.

Suppose thatP(x) =∑∞i=0 cix

i ∈ kx is a series. Givene ∈ N, Pe(x) will denote thepolynomialPe(x) =∑e

i=0 cixi . Given a seriesf (x1, . . . , xn) ∈ kx1, . . . , xn, ordf will

denote the order off (with ord0= ∞).If x ∈ Q, [x] will denote the greatest integern ∈ N such thatn x. The greates

common divisor ofa1, . . . , an ∈ N will be denoted by(a1, . . . , an).


d

arod

lar

t

Definition 2.1. A reduced divisorD on a nonsingular varietyX of dimensionn is called asimple normal crossing divisor (SNC divisor) if all components ofD are nonsingular anthe following condition holds.

Suppose thatp ∈ X is a point andD1, . . . ,Ds are the components ofD containingp.Then s n and there exist regular parameters(x1, . . . , xn) in OX,p such thatD1, . . . ,Ds have atp local equationsx1 = 0, . . . , xs = 0, respectively.

Definition 2.2. Codimension-2 subvarietiesC1, . . . ,Cm of a nonsingular dimension-nvarietyX make simple normal crossings (SNCs) ifCi is nonsingular for alli = 1, . . . ,mand the following condition holds.

Suppose thatp ∈ X is a point andC1, . . . ,Cs are the subvarieties containingp. Thens

( n2

)and there exist regular parameters(x1, . . . , xn) in OX,p such that for all

i = 1, . . . , s, xli = xki = 0 are local equations ofCi atp and 1 li < ki n.

Definition 2.3. Suppose thatΦ :X → Y is a dominant morphism of nonsingulk-varieties.Φ is called monomial if for all pointsp ∈ X there exist an étale neighborhoU of p, uniformizing parameters(x1, . . . , xn) on U , regular parametersy1, . . . , ym inOY,Φ(p) and a matrix(aij ) of nonnegative integers with rankm such that

y1 = xa111 · · ·xa1n

n ,

...


n .

3. Monomialization

Definition 3.1. Suppose thatΦ :X → S is a dominant morphism from a nonsinguvariety X to a nonsingular varietyS with reduced SNC divisorsDS on S andEX onX such thatΦ−1(DS)red =EX . Let sing(Φ) be the locus of singular points ofΦ. We willsay thatΦ is quasi-prepared (with respect toDS ) if sing(Φ)⊂EX .

Suppose thatΦ :X → S is a quasi-prepared morphism from a nonsingularn-fold X toa nonsingular surfaceS. If p ∈ EX we will say thatp is a 1,2, . . . , n point depending onif p is contained in 1,2, . . . , n components ofEX · q ∈ DS will be called a 1 or 2 poindepending on ifq is contained in 1 or 2 components ofDS .

Regular parameters(u, v) ∈OS,q for q ∈ DS are permissible if:

(1) q is a 1 point andu= 0 is a local equation ofDS , or(2) q is a 2 point anduv = 0 is a local equation ofDS .


lar

rs

lar

t

Definition 3.2. Suppose thatΦ :X → S is a quasi-prepared morphism from a nonsingun-fold X to a nonsingular surfaceS. We will say thatΦ is strongly prepared atp ∈X (withrespect toDS) if there exist permissible parameters(u, v) atΦ(p) and regular paramete(x1, . . . , xn) in OX,p such that one of the following forms holds:

(1) 1 k n− 1: p is ak point,u= 0 is a local equation ofEX and

u = (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1,

wherem> 0, a1, . . . , ak > 0 with (a1, . . . , ak)= 1, b1, . . . , bk 0 andP is a series;(2) 2 k n: p is ak point,u = 0 is a local equation ofEX and

u = (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk ,

wherem> 0, a1, . . . , ak > 0 with (a1, . . . , ak)= 1, b1, . . . , bk 0, rank( a1 ··· akb1 ··· bk

)= 2andP is a series;

(3) 2 k n: p is ak point,uv = 0 is a local equation ofEX and

u= xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk ,

wherea2, . . . , ak−1, b2, . . . , bk−1 0, a1, bk > 0 andai + bi > 0 for all i = 2, . . . ,k − 1.

In this case the regular parameters(x1, . . . , xn) in OX,p are called∗-permissibleparameters atp for (u, v) and the permissible parameters(u, v) are called prepared.

Φ :X → S is strongly prepared if it is strongly prepared at every pointp ∈X.

We will now assume thatΦ :X → S is a strongly prepared morphism from a nonsingun-fold X to a nonsingular surfaceS.

Lemma 3.3. Suppose thatOX,p →R is finite étale and there existx1, . . . , xn ∈ R such that(x1, . . . , xn) are regular parameters inRq for all primesq ⊂ R such thatq ∩OX,p =mp .Then there exists an étale neighborhoodU of p such that(x1, . . . , xn) are uniformizingparameters onU .

Proof. There exists an affine neighborhoodV1 = Spec(A) of p ∈X and a finite étaleextensionB of A such thatB ⊗A Amp

∼= R. SetU1 = Spec(B). Let π :U1 → V1 be thenatural map. There exists an open neighborhoodU2 of π−1(p) such that(x1, . . . , xn) areuniformizing parameters onU2. LetZ =U1 −U2 andW = π(Z). SetU =U1 −π−1(W),thenU → V = V1 − W is finite étale. Thus there exists an étale neighborhoodU of pwhere(x1, . . . , xn) are uniformizing parameters.Lemma 3.4. Suppose thatΦ :X → S is strongly prepared atp ∈ EX . Then there exisprepared parameters(u, v) for Φ(p) and∗-permissible parameters(x1, . . . , xn) for (u,v)


f
at p such that(x1, . . . , xn) are uniformizing parameters on an étale neighborhood op
and one of the following forms holds:

(1) 1 k n− 1: p is a k point,u= 0 is a local equation ofEX and

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1, (I)

wherem> 0, a1, . . . , ak > 0 with (a1, . . . , ak) = 1, b1, . . . , bk 0 and eitherP ≡ 0or P is a polynomial of order max1ikbi/ai such that ifΦ(p) is a 1 point thenm ordP ;

(2) 2 k n: p is a k point,u= 0 is a local equation ofEX and

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk , (II)

wherem> 0, a1, . . . , ak > 0 with (a1, . . . , ak) = 1; b1, . . . , bk 0, rank( a1 ··· akb1 ··· bk

)= 2and eitherP ≡ 0 or P is a polynomial of order max1ikbi/ai such that ifΦ(p)

is a1 point thenm ordP ;(3) 2 k n: p is a k point,uv = 0 is a local equation ofEX and

u = xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk , (III)

wherea2, . . . , ak−1, b2, . . . , bk−1 0, a1, bk > 0 and ai + bi > 0 for all i = 2, . . . ,k − 1.

Proof. Let (u, v) be prepared parameters forΦ(p).Suppose first that there exist regular parameters(x1, . . . , xn) ∈ OX,p such that

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1.

Then there existy1, . . . , yk ∈OX,p and unitsα1, . . . , αk ∈ OX,p such thatxi = αiyi for alli = 1, . . . , k.

Setγ = (αa11 · · ·αak

k )1/a1 andR = OX,p[γ ]. R is finite étale overOX,p . Let L be thequotient field ofR.

Sete = max1ik[bi/ai] and

yk+1 = αb11 · · ·αbk

k xk+1 + P(αa11 · · ·αak

k ya11 · · ·yakk )− Pe(α

a11 · · ·αak

k ya11 · · ·yakk )

yb11 · · ·ybkk

.

Then

yk+1 = v −Pe(αa11 · · ·αak

k ya11 · · ·yakk )

yb11 · · ·ybkk

∈ Rq ∩L=Rq

for all maximal idealsq of R. Thusyk+1 ∈⋂Rq =R.


t

Sety1 = γy1 andyk+1 = γ−b1yk+1, so that

u= (ya11 y

a22 · · ·yakk

)m, v = Pe

(ya11 y

a22 · · ·yakk

)+ yb11 y

b22 · · ·ybkk yk+1.

For i = k+2, . . . , n chooseyi ∈OX,p such thatyi ≡ xi modm2pOX,p. Theny1, y2, . . . ,

yk, yk+1, yk+2, . . . , yn ∈ R are regular parameters at all maximal ideals ofR. SinceR isfinite étale overOX,p , by Lemma 3.3 there exists an étale neighborhoodU of p such that(y1, y2, . . . , yk, yk+1, yk+2, . . . , yn) are uniformizing onU.

To finish the analysis of this case whenΦ(p) is a 1 point and ordPe <∞, we only needto ensure thatm ordP in the formula forv. Suppose thatPe(t) =∑e

i=1λi(xa11 · · ·xakk )i .

Set

v = v −[e/m]∑i=1

λimui and P = Pe −

[e/m]∑i=1

λim(xa11 · · ·xakk

)imso that regular parameters(u, v) at Φ(p) and regular parameters(y1, y2, . . . , yk, yk+1,

yk+2, . . . , yn) atp satisfy the conditions of the lemma.Suppose now that there exist regular parameters(x1, . . . , xn) in OX,p such that

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk ,

where by permutingx1, . . . , xk we can assume thatf = a1b2 − a2b1 > 0. Then there exisy1, . . . , yk ∈ OX,p and unitsα1, . . . , αk ∈ OX,p such thatxi = αiyi for all i = 1, . . . , k.

Setγ = (αa11 · · ·αak

k )1/f andR = OX,p[γ ]. R is finite étale overOX,p . Let L be thequotient field ofR.

Sete = max1ik[bi/ai] and

ω = αb11 · · ·αbk

k + P(αa11 · · ·αak

k ya11 · · ·yakk )−Pe(α

a11 · · ·αak

k ya11 · · ·yakk )

yb11 · · ·ybkk

.

Thenω is a unit and

ω = v − Pe(αa11 · · · αak

k ya11 · · ·yakk )

yb11 · · ·ybkk

∈ Rq ∩L =Rq

for all maximal idealsq of R. Thusω ∈⋂Rq =R.Let R1 =R[ω−1/f ]. R1 is finite étale overR and, therefore, it is finite étale overOX,p .Sety1 = γ b2ω−a2/f y1 andy2 = γ−b1ωa1/f y2, so that

u= (ya11 y

a22 y

a33 · · ·yakk

)m, v = Pe

(ya11 y

a22 y

a33 · · ·yakk

)+ yb11 y

b22 y

b33 · · ·ybkk .

For i = k + 1, . . . , n chooseyi ∈ OX,p such thatyi ≡ xi mod m2pOX,p . Then

y1, y2, y3, . . . , yn ∈ R1 are regular parameters at all maximal ideals ofR1. SinceR1 is


rsf

finite étale overOX,p , by Lemma 3.3 there exists an étale neighborhoodU of p such that(y1, y2, y3 · · ·yn) are uniformizing onU .

To finish the analysis of this case ifΦ(p) is a 1 point we changev to v andPe to P

in the same way as we did above, so that regular parameters(u, v) at Φ(p) and regularparameters(y1, y2, y3, . . . , yn) atp satisfy the conditions of the lemma.

Finally suppose that there exist regular parameters(x1, . . . , xn) in OX,p such that

u= xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk .

Then there existy1, . . . , yk ∈OX,p and unitsα1, . . . , αk ∈ OX,p such thatxi = αiyi for alli = 1, . . . , k.

Setγ = (αa11 · · ·αak−1

k−1 )1/a1, ω = (α

b22 · · ·αbk

k )1/bk andR = OX,p[γ,ω]. R is finite étaleoverOX,p.

Sety1 = γy1 andyk = ωyk , so that

u= ya11 y

a22 · · ·yak−1

k−1 , v = yb22 · · ·ybk−1

k−1 ybkk .

For i = k + 1, . . . , n chooseyi ∈ OX,p such thatyi ≡ xi modm2pOX,p . Theny1, y2, . . . ,

yk−1, yk, yk+1, . . . , yn ∈ R are regular parameters at all maximal ideals ofR. SinceR isfinite étale overOX,p , by Lemma 3.3 there exists an étale neighborhoodU of p such that(y1, y2, . . . , yk−1, yk, yk+1, . . . , yn) are uniformizing onU .

This completes the proof.Suppose thatp ∈ EX and (u, v) are permissible parameters forΦ(p). (u, v) will be

called strongly prepared atΦ(p) and∗-permissible parameters(x1, . . . , xn) for (u, v) atpwill be called strongly permissible if they satisfy the conditions of Lemma 3.4.

Definition 3.5. Suppose thatΦ :X → S is strongly prepared with respect toDS . Supposethatp ∈ EX . We will say thatp is a good point forΦ if there exist permissible paramete(u, v) atΦ(p) and∗-permissible parameters(x1, . . . , xn) at p for (u, v) such that one othe following forms holds:

(1a) 1 k n− 1: p is ak point,u= 0 is a local equation ofEX and

u= (xa11 · · ·xakk

)m, v = α

(xa11 · · ·xakk

)t + (xa11 · · ·xakk

)txk+1, (G.Ia)

wherem> 0, t 0, a1, . . . , ak > 0 with (a1, . . . , ak)= 1 andα ∈ k;(1b) 2 k n− 1: p is ak point,u= 0 is a local equation ofEX and

u= xa11 · · ·xakk , v = x

b11 · · ·xbkk xk+1, (G.Ib)

wherea1, . . . , ak > 0, b1, . . . , bk 0, and rank( a1 ··· akb ··· b

)= 2;
1 k


eeII)

rs

f

(2) 2 k n: p is ak point,u = 0 is a local equation ofEX and

u = xa11 · · ·xakk , v = x

b11 · · ·xbkk , (G.II)

wherea1, . . . , ak > 0, b1, . . . , bk 0, and rank( a1 ··· akb1 ··· bk

)= 2;(3) 2 k n: p is k point,uv = 0 is a local equation ofEX and

u = xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk , (G.III)

wherea2, . . . , ak−1, b2, . . . , bk−1 0, a1, bk > 0, andai + bi > 0 for all i = 2, . . . ,k − 1.

p will be called a bad point ifp is not a good point.

Remark 3.6. If p ∈ EX is a good point then following the proof of Lemma 3.4 wcan always find strongly prepared parameters(u, v) at Φ(p) and strongly permissiblparameters(x1, . . . , xn) at p for (u, v) such that one of the forms (G.Ia), (G.Ib), (G.or (G.III) holds.

Lemma 3.7. Suppose thatp ∈ X is a 1 point and(u, v) are strongly prepared parameteat Φ(p), (x1, . . . , xn) are strongly permissible parameters atp for (u, v) such that(I)holds and

u = xa11 , v = P(x1)+ x

c11 x2, ordP = d1.

Suppose that(u, v) are also strongly prepared parameters atΦ(p) and(y1, . . . , yn) arestrongly permissible parameters atp for (u, v) such that(I) holds and

u= ya21 , v =Q(y1)+ y

c21 y2, ordQ = d2.

If Φ(p) is a 1 point thena1 = a2, c1 = c2 andd1 = d2.If Φ(p) is a 2 point thend1, d2 <∞ andc1 − d1 = c2 − d2, a1 + d1 = a2 + d2.

Proof. This follows from the discussion before Definition 18.7 in [4].Suppose thatE is a component ofEX , p ∈ E, f ∈ OX,p andx = 0 is a local equation

of E atp. Then we define

νE(f )= maxn such thatxn | f .

Suppose thatp ∈ X is a 1 point andE is the component ofEX containingp. Supposethat(u, v) are strongly prepared parameters atΦ(p) such thatu = 0 is a local equation oE atp and(x1, . . . , xn) are strongly permissible parameters for(u, v) atp with

u= xa, v = P(x1)+ xcx2, ordP = d.
1 1


,eterssible

e

or

Then νE(v) = d if d < ∞ and νE(v) = c if d = ∞. Thus, in view of Lemma 3.7c − νE(v) anda + νE(v) are independent of the choice of strongly prepared param(u, v) at Φ(p) and they are also independent of the choice of strongly permisparameters for(u, v) atp.

Definition 3.8. Let p ∈ X, E ⊂ EX , regular parameters(u, v) in OS,Φ(p) and regularparameters(x1, . . . , xn) in OX,p be as above with

u = xa1, v = P(x1)+ xc1x2.

DefineA(Φ,p) = c − νE(v).If A(Φ,p) > 0 defineC(Φ,p) = (c− νE(v), a + νE(v)).

Notice that ifp ∈ X is a 1 point thenp is a good point if and only ifA(Φ,p) = 0 or,equivalently, ordP = d c.

Lemma 3.9. Suppose thatp ∈ EX is a1 point andE is the component ofEX containingp.Then there exists an open neighborhoodU of p such thatA(Φ,p′) = A(Φ,p) for all

p′ ∈ E ∩U and ifA(Φ,p) > 0 thenC(Φ,p′)= C(Φ,p) for all p′ ∈E ∩U .

Proof. There exist strongly prepared parameters(u, v) atΦ(p) and strongly permissiblparameters(x1, . . . , xn) atp for (u, v) such that

u= xa1, v = P(x1)+ xc1x2, and ordP = d.

If U is an étale neighborhood ofp where(x1, . . . , xn) are uniformizing parameters then fanyp′ ∈ U ∩E there existα2, . . . , αn ∈ k such that(x1, x2 = x2 + α2, . . . , xn = xn + αn)

are strongly permissible parameters atp′ for strongly prepared parameters(u, v) atΦ(p′)andv = v if Φ(p′) is a 2 point ora c, v = v + α2u

c/a if Φ(p′) is a 1 point anda | c.Suppose that the first assumption holds andv = v then atp′

u= xa1, v = P(x1)− α2xc1 + xc1(x2 + α2)= P1(x1)+ xc1x2.

Thus if d < c then ordP1 = d andA(Φ,p′) = c − d = A(Φ,p) > 0, C(Φ,p′) = (c − d,

a + d)= C(Φ,p). If d c then ordP1 c andA(Φ,p′)= 0 =A(Φ,p).Suppose that the second assumption holds andv = v + α2u

c/a then atp′

u= xa1, v = P(x1)+ xc1(x2 + α2)= P(x1)+ xc1x2.

ThusA(Φ,p′)=A(Φ,p) andC(Φ,p′) = C(Φ,p) if A(Φ,p) > 0. Now we can defineA(Φ,E) = A(Φ,p) for p ∈ E a 1 point. IfA(Φ,E) > 0 define

C(Φ,E) = C(Φ,p).We will call E ⊂ EX a good component ofEX if A(Φ,E) = 0.E will be called a bad

component if it is not a good component or, equivalently, ifA(Φ,E) > 0.


d

t

rs

d

Lemma 3.10. Suppose2 k n. Suppose thatp ∈ X is a k point andE1, . . . ,Ek are thecomponents ofEX containingp. Suppose that(u, v) are strongly prepared atΦ(p) and(x1, . . . , xn) are strongly permissible parameters for(u, v) at p with xi = 0 being a localequation ofEi for i = 1, . . . , k.

If

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1, (I)

or

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk , (II)

then A(Φ,Ei) = bi − νEi (v) for i = 1, . . . , k and if A(Φ,Ei) > 0, thenC(Φ,Ei) =(bi − νEi (v), aim+ νEi (v)).

If

u= xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk , (III)

thenA(Φ,Ei)= 0 for all i = 1, . . . , k.

Proof. Suppose thatp ∈ X is a k point satisfying (I), (u, v) are strongly prepareparameters atΦ(p) and(x1, . . . , xn) are strongly permissible parameters for(u, v) at psuch that

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1, and ordP = d.

After possibly permutingx1, . . . , xk we can assume thati = 1 and prove only thaA(Φ,E1)= b1 − νE1(v) andC(Φ,E1)= (b1 − νE1(v), a1m+ νE1(v)) if A(Φ,E1) > 0.

Suppose thatU is an étale neighborhood ofp where(x1, . . . , xn) are uniformizingparameters andp′ ∈ U ∩ E1 is a 1 point. Then there existα2, . . . , αk ∈ k − 0 andαk+1, . . . , αn ∈ k such that(x1, x2 = x2 − α2, . . . , xn = xn − αn) are regular parameteatp′.

Set γ = ((x2 + α2)a2 · · · (xk + αk)

ak )1/a1, for i = 2, . . . , k set fi = a1bi − aib1 andω = ((x2 + α2)

f2 · · · (xk + αk)fk )1/a1. Notice thatγ a1,ωa1 ∈ OU,p′ are units inOU,p′ and

thereforeOU,p′ [γ,ω] is finite étale overOU,p′ .

Setx1 = γ x1 andxk+1 = xk+1ω + αk+1ω − αk+1(αf22 · · ·αfk

k )1/a1; then

u = xa1m1 ,

v = P(xa11

)+ xb11 xk+1 + αx

b11 , whereα = αk+1

(αf22 · · ·αfk

k

)1/a1 ∈ k.

Assume first thatΦ(p) is a 2 point ora1m b1, then (u, v) are strongly prepareparameters atΦ(p′). If a1d b1 then (x1, . . . , xk, xk+1, xk+2, . . . , xn) are stronglypermissible parameters atp′ for (u, v) and


rs

at

rs

rs

t

rs

u= xa1m1 , v = P1

(x1)+ x

b11 xk+1

with ordP1 = a1d if a1d < b1 and ordP1 b1 if a1d = b1. ThusA(Φ,E1) =A(Φ,p′) =b1 − νE1(v) in this case and ifA(Φ,p′) > 0 or, equivalently, ifa1d < b1 thenC(Φ,E1)=C(Φ,p′)= (b1 − a1d, a1m+ a1d)= (b1 − νE1(v), a1m+ νE1(v)).

If a1d > b1 then setxk+1 = xk+1 + P(xa11 )/x

b11 to get strongly permissible paramete

(x1, . . . , xn) for (u, v) atp′, so that

u= xa1m1 , v = x

b11 xk+1 + αx

b11 ,

andA(Φ,E1) =A(Φ,p′)= 0 = b1 − νE1(v).Assume now thatΦ(p) is a 1 point anda1m | b1, then (u, v = v − αub1/(a1m)) are

strongly prepared parameters atΦ(p′).If a1d b1 then(x1, . . . , xk, xk+1, xk+2, . . . , xn) are strongly permissible parameters

p′ for (u, v) and

u = xa1m1 , v = P

(xa11

)+ xb11 xk+1 = P1(x1)+ x

b11 xk+1.

ThusA(Φ,E1)= A(Φ,p′)= b1−a1d = b1−νE1(v) in this case and ifA(Φ,p′) > 0 thenC(Φ,E1)= C(Φ,p′)= (b1 − a1d, a1m+ a1d)= (b1 − νE1(v), a1m+ νE1(v)).

If a1d > b1 then setxk+1 = xk+1 + P(xa11 )/x

b11 to get strongly permissible paramete

(x1, . . . , xn) for (u, v) atp′ so that

u= xa1m1 , v = x

b11 xk+1,

andA(Φ,E1) =A(Φ,p′)= 0 = b1 − νE1(v).Suppose thatp ∈ X is ak point satisfying (II),(u, v) are strongly prepared paramete

atΦ(p) and(x1, . . . , xn) are strongly permissible parameters for(u, v) atp such that

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk , and ordP = d.

After possibly permutingx1, . . . , xk we can assume thati = 1 and prove only thaA(Φ,E1)= b1 − νE1(v) andC(Φ,E1)= (b1 − νE1(v), a1 + νE1(v)) if A(Φ,E1) > 0.

Suppose thatU is an étale neighborhood ofp where(x1, . . . , xn) are uniformizingparameters andp′ ∈ U ∩ E1 is a 1 point. Then there existα2, . . . , αk ∈ k − 0 andαk+1, . . . , αn ∈ k such that(x1, x2 = x2 − α2, . . . , xn = xn − αn) are regular parameteatp′.

Notice that since rank( a1 ··· akb1 ··· bk

)= 2 there existsj ∈ 2, . . . , k such thata1bj − ajb1 =0. So, after possibly permutingx2, . . . , xk we can assume thatj = 2, that is,a1b2 −a2b1 =0.

Setγ = ((x2 + α2)a2 · · · (xk + αk)

ak )1/a1, for i = 2, . . . , k, setfi = a1bi − aib1, ω =((x3 + α3)

f3 · · · (xk + αk)fk )1/a1 andδ = (x2 + α2)

f2/a1. Notice thatγ a1,ωa1, δa1 ∈ OU,p′are units inOU,p′ and thereforeOU,p′ [γ,ωδ] is finite étale overOU,p′ .

Setx1 = γ x1 andx2 = δω − (αf2 · · ·αfk )1/a1; then
2 k


t

rs

g

.

ntsd

f

u= xa1m1 ,

v = P(xa11

)+ xb11 x2 + αx

b11 , whereα = (

αf22 · · ·αfk

k

)1/a1 ∈ k − 0.Now the same analysis as above withx2 playing the role ofxk+1 above shows tha

A(Φ,E1)= b1 − νE1(v) andC(Φ,E1)= b1 − νE1(v), a1m+ νE1(v)) if A(Φ,E1) > 0.Suppose thatp ∈ X is ak point satisfying (III),(u, v) are strongly prepared paramete

atΦ(p) and(x1, . . . , xn) are strongly permissible parameters for(u, v) atp such that

u= xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk .

Sincea1, bk > 0 andaj + bj > 0 for all j = 2, . . . , k − 1, after possibly permutinx1, . . . , xk andu,v we can assume thati = 1 and

u= xa11 · · ·xak−1

k−1 ,

v = P(xa11 · · ·xak−1

k−1

)+ xb11 · · ·xbkk , wherea1, bk > 0, and P(t) ≡ 0.

In this notations the proof of the required statement repeats the proof for case (II)Theorem 3.11. Suppose thatΦ :X → S is strongly prepared. Then the locus of bad poiin X is a Zariski closed set of pure codimension1, consisting of the union of all bacomponents ofEX .

Proof. Let Z be the union of all bad components ofEX and letp be a good point onEX ,q be a bad point onEX . Then it suffices to show thatq ∈ Z while p ∈ EX −Z.

Suppose thatp is a goodk point, (u, v) are strongly prepared parameters atΦ(p),(x1, . . . , xn) are strongly permissible parameters for(u, v) at p such that one of thefollowing forms holds:

(1a) u= (xa11 · · ·xakk

)m, v = α

(xa11 · · ·xakk

)t + (xa11 · · ·xakk

)txk+1;

(1b) u= xa11 · · ·xakk , v = x

b11 · · ·xbkk xk+1;

(2) u= xa11 · · ·xakk , v = x

b11 · · ·xbkk ;

(3) u= xa11 · · ·xak−1

k−1 , v = xb21 · · ·xbkk .

In all these casesx1 = 0, . . . , xk = 0 are local equations of the components ofEX

containingp. So we can assume that for alli = 1, . . . , k, xi = 0 is a local equation oEi ⊂E, a component ofEX.

By Lemma 3.10, we can computeA(Φ,Ei) as follows:

(1a) A(Φ,Ei) = ait − νEi (v) = ait − ait = 0,

(1b) A(Φ,Ei) = bi − νEi (v) = bi − bi = 0,

(2) A(Φ,Ei) = bi − νEi (v) = bi − bi = 0,


(3) A(Φ,Ei) = 0.

Thus all components ofEX containingp are good, sop does not lie inZ.Suppose thatq is a badk point, (u, v) are strongly prepared parameters atΦ(q),

(x1, . . . , xn) are strongly permissible parameters for(u, v) atq and (I) holds; that is,

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1, whereP(t) = 0.

Let x1 = 0, . . . , xk = 0 be local equations of the componentsE1, . . . ,Ek ⊂ EX contain-ing q . By Lemma 3.10, we can computeA(Φ,Ei) for all i = 1, . . . , k as follows:

A(Φ,Ei)= bi − νEi (v) =bi − ai ordP > 0 if ai ordP < bi;0 otherwise.

To prove the statement of the theorem we need to find suchj ∈ 1, . . . , k thatEj is abad component or, equivalently,A(Φ,Ej ) > 0.

Assume the contrary. Letai ordP bi for all i = 1, . . . , k then

v = xb11 · · ·xbkk

(xk+1 + P(x

a11 · · ·xakk )

xb11 · · ·xbkk

)and

P(xa11 · · ·xakk )

xb11 · · ·xbkk

∈ OX,q .

If ordP(x

a11 ···xakk )

xb11 ···xbkk

1 then set

xk+1 = xk+1 + P(xa11 · · ·xakk )

xb11 · · ·xbkk

so that(x1, . . . , xk, xk+1, xk+2, . . . , xn) are strongly permissible parameters for(u, v) at qand

u= (xa11 · · ·xakk

)m, v = x

b11 · · ·xbkk xk+1.

Thus one of the forms (G.Ia) or (G.Ib) holds forq and, therefore,q is a good point whileq was originally chosen to be a bad point.

If

ordP(x

a11 · · · xakk )

xb11 · · · xbkk

= 0

then there existα ∈ k − 0 andy ∈ mqOX,q such that

P(xa11 · · · xakk )

xb1 · · · xbk = y + α.

1 k


e

t

a

Set xk+1 = xk+1 + y so that (x1, . . . , xk, xk+1, xk+2, . . . , xn) are strongly permissiblparameters for(u, v) at q and

u= (xa11 · · ·xakk

)m, v = x

b11 · · ·xbkk

(xk+1 + α

).

Now if rank( a1 ··· akb1 ··· bk

)< 2 then (G.Ia) holds and, therefore,q is a good point. This

contradicts the choice ofq.If rank

( a1 ··· akb1 ··· bk

) = 2 then after possibly permutingx1, . . . , xk we can assume thaf = a1b2 − a2b1 = 0. Set

x1 = x1(xk+1 + α)−a2/f , x2 = x2(xk+1 + α)a1/f ,

then (x1, x2, x3, . . . , xk, xk+1, xk+2, . . . , xn) are∗-permissible parameters for(u, v) at qand

u= (xa11 x

a22 x

a33 · · ·xakk

)m, v = x

b11 x

b22 x

b33 · · ·xbkk .

Thus (G.II) holds forq and, therefore,q is a good point whileq was originally chosento be a bad point.

This shows that ifq is a bad point onEX and (I) holds atq then there exists suchcomponentE of EX containingq thatA(Φ,E) > 0. SoE ⊂Z and, therefore,q ∈Z.

Suppose thatq is a badk point, (u, v) are strongly prepared parameters atΦ(q),(x1, . . . , xn) are strongly permissible parameters for(u, v) atq and (II) holds; that is,

u = (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk , whereP(t) = 0.

Let x1 = 0, . . . , xk = 0 be local equations of the componentsE1, . . . ,Ek ⊂ EX contain-ing q . By Lemma 3.10, we can computeA(Φ,Ei) for all i = 1, . . . , k as follows:

A(Φ,Ei)= bi − νEi (v) =bi − ai ordP > 0 if ai ordP < bi;0 otherwise.

To prove the statement of the theorem we need to find suchj ∈ 1, . . . , k thatEj is abad component or, equivalently,A(Φ,Ej ) > 0.

Assume the contrary. Letai ordP bi for all i = 1, . . . , k; then

v = xb11 · · ·xbkk

(1+ P(x

a11 · · ·xakk )

xb11 · · ·xbkk

).

Since rank( a1 ··· akb1 ··· bk

)= 2 there existsl ∈ 1, . . . , k such thatal ordP > bl , so

P(xa11 · · · xakk )

xb11 · · · xbkk

∈ mqOX,q.

After possibly permutingx1, . . . , xk , we can assume thatf = a1b2 − a2b1 = 0. Set


a

x1 = x1

(1+ P(x

a11 · · ·xakk )

xb11 · · ·xbkk

)−a2/f

, x2 = x2

(1+ P(x

a11 · · ·xakk )

xb11 · · ·xbkk

)a1/f

,

so that(x1, x2, x3, . . . , xn) are∗-permissible parameters for(u, v) atq and

u= (xa11 x

a22 x

a33 · · ·xakk

)m, v = x

b11 x

a22 x

a33 · · ·xbkk .

Thus (G.II) holds forq and, therefore,q is a good point whileq was originally chosen tobe a bad point.

This shows that ifq is a bad point onEX and (II) holds atq then there exists suchcomponentE of EX containingq thatA(Φ,E) > 0. SoE ⊂Z and, therefore,q ∈Z. Lemma 3.12. Suppose thatΦ :X → S is strongly prepared,q ∈ DS andp ∈ Φ−1(q) is ak point onX such that one of the forms(I), (II) or (III) holds atp. ThenmqOX,p is notinvertible if and only if one of the following holds:

(1a) 1 k n− 1:u= (

xa11 · · ·xakk

)m, v = (

xa11 · · ·xakk

)txk+1, (N.Ia)

wherem> 0, a1 . . . , ak > 0 with (a1, . . . , ak) = 1 and0 t < m;(1b) 2 k n− 1:

u= (xa11 · · ·xakk

)m, v = (

xb11 · · ·xbkk

)xk+1, (N.Ib)

wherea1, . . . , ak > 0 with (a1, . . . , ak) = 1, b1, . . . , bk 0, rank( a1 ··· akb1 ··· bk

) = 2 andmin1ikbi/ai<m;

(1c) 2 k n− 1:u= (

xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1, (N.Ic)


) = 2 andmin1ikbi/ai< ordP < max1ikbi/ai, min1ikbi/ai<m;

(2a) 2 k n:u= (

xa11 · · ·xakk

)m, v = (

xb11 · · ·xbkk

), (N.IIa)


) = 2 andmin1ikbi/ai<m< max1ikbi/ai;

(2b) 2 k n:u= (

xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk , (N.IIb)


) = 2 andmin1ikbi/ai< ordP < max1ikbi/ai, min1ikbi/ai<m;


.

(3) 2 k n:

u = xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk , (N.III)

wherea2, . . . , ak−1, b2, . . . , bk−1 0, a1, bk > 0, andai + bi > 0 for all i = 2, . . . ,k − 1.

Proof. Suppose that (I) holds atp. First consider the case when rank( a1 ··· akb1 ··· bk

)< 2. Then

there existt 0 such that

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ (xa11 · · ·xakk

)txk+1.

If d = ordP t then v = (xa11 · · ·xakk )dγ , whereγ ∈ OX,p is a unit. So eitheru is a

multiple ofv if d m or v is a multiple ofu if m d. Thus we may assume that ordP > t .Set

xk+1 = xk+1 + P(xa11 · · ·xakk )

(xa11 · · ·xakk )t

∈ OX,p

to get strongly permissible parameters(x1, . . . , xk, xk+1, xk+2, . . . , xn) for (u, v) atp suchthat

u = (xa11 · · ·xakk

)m, v = (

xa11 · · ·xakk

)txk+1.

So(u, v)OX,p is not invertible if and only ift < m and we get case (N.Ia) of the lemmaSuppose now that rank

( a1 ··· akb1 ··· bk

)= 2. If P(t) ≡ 0 in the formula forv then

u= (xa11 · · ·xakk

)m, v = x

b11 · · ·xbkk xk+1,

and v is not a multiple ofu if and only if aim > bi for somei ∈ 1, . . . , k, that is ifm> min1ikbi/ai. Thus we meet case (N.Ib) of the lemma.

Suppose thatP(t) = 0 thend = ordP max1ikai/bi. If d = max1ikbi/aithen

v = P(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1 = (

xb11 · · ·xbkk

)(P(xa11 · · ·xakk )

xb11 · · ·xbkk

+ xk+1

),

where the smallest degree term of

P(xa11 · · · xakk )

xb1 · · · xbk
1 k


is xa1d−b11 · · ·xakd−bk

k . xa1d−b11 · · ·xakd−bk

k ∈ mpOX,p sinced bi/ai for all i = 1, . . . , kandd = bi/aj for at least onej ∈ 1, . . . , k due to maximality of the rank

( a1 ··· akb1 ··· bk

). So

by setting

xk+1 = xk+1 + P(xa11 · · ·xakk )

xb11 · · ·xbkk

we return to the situation whenP(t) ≡ 0 in the formula forv.Therefore, we may restrict our considerations tod = ordP < max1ikbi/ai. If

d min1ikbi/ai thenv = (xa11 · · ·xakk )dγ , where

γ = P(xa11 · · ·xakk )

(xa11 · · ·xakk )d

+ xb1−a1d1 · · ·xbk−akd

k xk+1 ∈ OX,p

is a unit. So eitherv is a multiple ofu if m d or u is a multiple ofv if d m. On theother hand, if ordP > min1ikbi/ai, we denote byci the minimum ofaid andbi topresentv as follows:

v = xc11 · · ·xckk

(P(x

a11 · · ·xakk )

xc11 · · ·xckk

+ xb1−c11 · · ·xbk−ck

k xk+1

)= x

c11 · · ·xckk γ,

whereγ ∈mpOX,p andxi γ for all i = 1, . . . , k. Thenu cannot be a multiple ofv, andvis a multiple ofu if and only if

m min1ik

ci

ai

= min

1ikmin

d,

bi

ai

= min

1ik

bi

ai

.

Thus, with the assumptionm > min1ikbi/ai, (u, v)OX,p is not invertible and this iscase (N.Ic) of the lemma.

Suppose that (II) holds atp. Assume first thatP(t) ≡ 0 in the formula forv so that

u= (xa11 · · ·xakk

)m, v = x

b11 · · ·xbkk .

Thenv is not a multiple ofu if and only if there existi ∈ 1, . . . , k such thataim > bi ,that is, ifm> min1ikbi/ai. The symmetric condition foru not being a multiple ofvgives the restrictionm< max1ikbi/ai. Thus we meet case (N.IIa) of the lemma.

Let P(t) = 0 thend = ordP max1ikbi/ai. If d = max1ikbi/ai then

v = P(xa11 · · ·xakk

)+ xb11 · · ·xbkk = (

xb11 · · ·xbkk

)(P(xa11 · · ·xakk )

xb11 · · ·xbkk

+ 1

)= x

b11 · · ·xbkk γ,

where the smallest degree term of

P(xa11 · · · xakk )

xb1 · · · xbk
1 k


is xa1d−b11 · · ·xakd−bk

k . So γ ∈ OX,p is a unit sinced bi/ai for all i = 1, . . . , k andd = bj/aj for at least onej ∈ 1, . . . , k due to maximality of the rank

( a1 ··· akb1 ··· bk

).

After possibly permuting regular coordinatesx1, . . . , xn, we can assume thatf =a1b2 − a2b1 = 0. Then by setting

x1 = x1γ−a2/f , x2 = x2γ

a1/f ,

we get strongly permissible parameters(x1, x2, x3, . . . , xn) for (u, v) atp such that

u= (xa11 x

a22 x

a33 · · ·xakk

)m, v = x

b11 x

b22 x

b33 · · ·xbkk .

Thus we may assume that ordP < max1ikbi/ai.If d min1ikbi/ai thenv = (x

a11 · · ·xakk )dγ , where

γ = P(xa11 · · ·xakk )

(xa11 · · ·xakk )d

+ xb1−a1d1 · · ·xbk−akd

k ∈ OX,p

is a unit. So eitherv is a multiple ofu if m d or u is a multiple ofv if d m. On theother hand if ordP > min1ikbi/ai, we denote byci the minimum ofaid andbi topresentv as follows:

v = xc11 · · ·xckk

(P(x

a11 · · ·xakk )

xc11 · · ·xckk

+ xb1−c11 · · ·xbk−ck

k

)= x

c11 · · ·xckk γ,

whereγ ∈ mpOX,p andxi γ for all i = 1, . . . , k. Thenu cannot be a multiple ofv andvis a multiple ofu if and only if

m min1ik

ci

ai

= min

1ikmin

d,

bi

ai

= min

1ik

bi

ai

.

Thus, with the assumptionm> min1ikbi/ai, (u, v)OX,p is not invertible and this iscase (N.IIb) of the lemma.

Suppose that (III) holds atp. Then

u = xa11 · · ·xak−1

k−1 , v = xb21 · · ·xbkk , with a1, bk > 0.

SomqOX,p is not invertible and this describes case (N.III) of the lemma.Lemma 3.13. Suppose thatΦ :X → S is strongly prepared. Letπ1 :S1 → S be the blowup ofS at a pointq ∈ DS . LetU be the largest open set ofX such that the rational mapX → S1 is a morphismΦ1 :U → S1 onU .

ThenΦ1 is strongly prepared with respect toπ−11 (DS), and if all points ofU are good

for Φ then all points ofU are good forΦ1 also.


ionsn

orns

d

t

d

Proof. This follows from the analysis of the proof of Lemma 3.12. In fact, the conclusof the lemma are clear ifu|v in OX,p. If v | u and (1) or (2) holds in Definition 3.2, we camake a change of variables in thexi , replacingxi with γixi , whereγi is a unit series for1 i k, and making an appropriate change ofxk+1 to get an expression of the form (1)(2) of Definition 3.2, withu andv interchanged. Ifp is a good point, the new expressioof v andu will have the good expressions of (1a) or (2) of Definition 3.5.Theorem 3.14. Suppose thatΦ :X → S is strongly prepared,p ∈ X is a 1 point and therational mapΦ1 fromX to the blow upS1 of q = Φ(p) is a morphism in a neighborhooof p.

ThenA(Φ,p) A(Φ,p) and ifA(Φ1,p) =A(Φ,p) > 0 thenC(Φ1,p) < C(Φ,p).

Proof. Let (x1, . . . , xn) be strongly permissible parameters atp for strongly preparedparameters(u, v) atΦ(p), then

u= xa1, v = P(x1)+ xb1x2

with ordP = d ∞.Suppose first thatP(x) ≡ 0. Thenb a since(u, v)OX,p is principal and there exis

strongly prepared parameters(u1, v1) atΦ1(p) such that

u= u1, v = u1v1.

Thus

u1 = xa1, v1 = xb−a1 x2,

andA(Φ1,p) = 0 =A(Φ,p).Now suppose thatP(x) = 0, sod b. If d a then there existα ∈ k and strongly

prepared parameters(u1, v1) atΦ1(p) such that

u = u1, v = u1(v1 + α).

Thus

u1 = xa1 , v1 = P(x1)

xa1− α + xb−a

1 x2,

andA(Φ1,p) (b − a) − (d − a) = b − d = A(Φ,p), where the equality holds if anonly if d > a. In this caseC(Φ1,p) = (b− d, d − a + a)= (b− d, d) < (b− d, d + a)=C(Φ,p).

If d < a then there exist strongly prepared parameters(u1, v1) at Φ1(p) and stronglypermissible parameters(x1, x2, x3, . . . , xn) atp such that

u= u1v1, v = u1


f

rem

d

f

and

u1 = xd1 , v1 = P(x1)

xd1

+ xb+a−2d1 x2 with ordP = a.

ThusA(Φ1,p) = (b + a − 2d) − (a − d) = b − d = A(Φ,p) andC(Φ1,p) = (b − d,

a − d + d)= (b − d, a) < (b − d, d + a)= C(Φ,p). Suppose thatΦ :X → S is strongly prepared. We will denote byZ(Φ) the locus of bad

points inX. If q ∈ DS denote byNq(Φ) the locus of points inX whereΦ does not factorthrough the blowup ofq . ThenN(Φ) will denote the union ofNq(Φ) for all q ∈DS .

We will denote byB2(X) the set of all 2 points inX. LetB2(X) be the Zariski closureof B2(X). We will also say that a codimension-2 subvarietyC ⊂ X is a 2-variety ifC = E1 ∩E2 for some componentsE1 andE2 of EX.

Remark 3.15. B2(X) is the union of all 2-varieties onX.

Lemma 3.16. Suppose thatΦ :X → S is strongly prepared andq ∈ DS is such thatZ(Φ) ∩ Nq(Φ) = ∅. ThenZ(Φ) ∩ Nq(Φ) is a Zariski dosed set of pure codimension2,consisting of the union of all2-varieties inΦ−1(q) with a generic point in the form o(N.IIb).

Suppose thatC is a component ofZ(Φ) ∩ Nq(Φ) and π :X1 → X is the blowup ofC with exceptional varietyE = π−1(C)red. ThenΦ1 = Φ π is strongly prepared andA(Φ1,E) < A(Φ).

Proof. Z(Φ) andNq(Φ) are both closed, so to prove the first statement of the theoit suffices to show that any bad pointp ∈ Nq(Φ) lies on a 2-varietyC such that ageneric pointp′ ∈ C is in the form of (N.IIb) andp′ ∈ Φ−1(q). Notice also that ifp ∈Z(Φ) ∩Nq(Φ) then either (N.Ic) or (N.IIb) holds atp.

Suppose thatp ∈ Nq(Φ) is ak point and (N.Ic) holds atp, (u, v) are strongly prepareparameters atq and(x1, . . . , xn) are strongly permissible parameters atp for (u, v), then

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1.

After possibly permutingx1, . . . , xk we can assume thatb1/a1 < ordP < b2/a2 andb1/a1 <m.

Suppose thatE1, . . . ,Ek are the components ofEX containingp with local equationsx1 = 0, . . . , xk = 0, respectively. LetU be an étale neighborhood ofp where(x1, . . . , xn)

are uniformizing parameters.Set C = E1 ∩ E2 and fix a 2 pointp′ ∈ U ∩ C away from the vanishing locus o

xk+1. Then there existα3, . . . , αk+1 ∈ k − 0 andαk+2, . . . , αn ∈ k such that (x1, x2, x3 =x3 − α3, x4 = x4 − α4, . . . , xn = xn − αn) are regular parameters atp′.

Since f = a1b2 − a2b1 = 0, we can setγ = ((x3 + α3)a3 · · · (xk + αk)

ak )1/f andω = ((x3+α3)

b3 · · · (xk+αk)bk (xk+1+αk+1))

1/f . Thenγ f ,ωf ∈OU,p′ are units inOU,p′and, therefore,OU,p′ [γ,ω] is finite étale overOU,p′ .


e

d

,

e

Setx1 = γ b2ω−a2x1 andx2 = γ−b1ωa1x2 so that(x1, . . . , xn) are strongly permissiblparameters atp′ for strongly prepared parameters(u, v) at q and

u= (xa11 x

a22

)m, v = P

(xa11 x

a22

)+ xb11 x

b22 .

Thus (N.IIb) holds at a generic pointp′ of C andΦ(p′)= q .If π :X1 → X is the blow up ofC thenΦ1 = Φ π is strongly prepared abovep.

If p lies in the intersection of more than 2 componentsE1, . . . ,Ek of EX thenπ−1(p)

does not contain any 1 point. Assume thatp is a 2 point. If s ∈ π−1(p) is a 1 pointthen(x1, x2, x3, . . . , xn) are strongly permissible parameters ats wherex2 is defined byx2 = x1(x2 + α) for some nonzeroα ∈ k.

After setting x1 = x1(x2 + α)a2/(a1+a2) and x3 = x3(x2 + α)f/(a1+a2), the followingequalities hold:

u= (xa11 x

a22

)m = x(a1+a2)m1 ,

v = P(xa11 x

a22

)+ xb11 x

b22 x3 = P

(x(a1+a2)1

)+ x(b1+b2)1 x3.

If (a1 + a2)ordP (b1 + b2) then s is a good point,A(Φ1,E) = A(Φ1, s) = 0,andA(Φ) > 0 since the locus of bad pointsZ(Φ) is not empty. So, assume that(a1 +a2)ordP < (b1 + b2). Sinceb1 − a1 ordP < 0,

A(Φ1,E) =A(Φ1, s) = b1 + b2 − (a1 + a2)ordP < b2 − a2 ordP

=A(Φ,E2) A(Φ).

Suppose thatp ∈ Nq(Φ) is ak point and (N.IIb) holds atp, (u, v) are strongly prepareparameters atq , and(x1, . . . , xn) are strongly permissible parameters atp for (u, v); then

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk .

After possibly permutingx1, . . . , xk , we can assume thatb1/a1 < ordP < b2/a2 andb1/a1 <m.

Suppose thatE1, . . . ,Ek are the components ofEX containingp with local equationsx1 = 0, . . . , xk = 0, respectively. LetU be an étale neighborhood ofp where(x1, . . . , xn)

are uniformizing parameters.SetC = E1 ∩ E2 and fix a 2 pointp′ ∈ U ∩ C. Then there existα3, . . . , αk ∈ k − 0

andαk+1, . . . , αn ∈ k such that(x1, x2, x3 = x3 − α3, x4 = x4 − α4, . . . , xn = xn − αn) areregular parameters atp′.

Sincef = a1b2 − a2b1 = 0, we can setγ = ((x3 + α3)a3 · · · (xk + αk)

ak )1/f andω =((x3 + α3)

b3 · · · (xk + αk)bk )1/f . Thenγ f ,ωf ∈ OU,p′ are units inOU,p′ and, therefore

OU,p′ [γ,ω] is finite étale overOU,p′ .Setx1 = γ b2ω−a2x1 andx2 = γ−b1ωa1x2 so that(x1, . . . , xn) are strongly permissibl

parameters atp′ for strongly prepared parameters(u, v) at q and

u= (xa1x

a2)m

, v = P(xa1xa

)+ xb1x

b2.
1 2 1 2 1 2


ll

r of

Thus (N.IIb) holds at a generic pointp′ of C andΦ(p) = q .If π :X1 → X is the blow up ofC thenΦ1 = Φ π is strongly prepared abovep.

If p lies in the intersection of more than 2 componentsE1, . . . ,Ek of EX thenπ−1(p)

does not contain any 1 point. Assume thatp is a 2 point. If s ∈ π−1(p) is a 1 pointthen (x1, x2, x3, . . . , xn) are strongly permissible parameters ats wherex2 is defined byx2 = x1(x2 + α) for some nonzeroα ∈ k.

After setting x1 = x1(x2 + α)a2/(a1+a2) and x2 = (x2 + α)f/(a1+a2) − αf/(a1+a2), thefollowing equalities hold:

u= (xa11 x

a22

)m = x(a1+a2)m1 ,

v = P(xa11 x

a22

)+ xb11 x

b22 = P

(xa1+a21

)+ xb1+b21 x2 + αf/(a1+a2)x

b1+b21 .

If (a1 + a2)ordP (b1 + b2) then s is a good point andA(Φ1,E) = A(Φ1, s) = 0 <

A(Φ). So, assume that(a1 + a2)ordP < (b1 + b2).Sinceb1 − a1 ordP < 0,

A(Φ1,E) =A(Φ1, s) = b1 + b2 − (a1 + a2)ordP < b2 − a2 ordP

=A(Φ,E2) A(Φ). Theorem 3.17. Suppose thatΦ :X → S is strongly prepared andZ(Φ)∩N(Φ) = ∅. ThenZ(Φ) ∩N(Φ) is a Zariski closed set of pure codimension2, consisting of the union of a2-varieties with a generic point in the form of(N.IIb).

Suppose thatC is a component ofZ(Φ) ∩ N(Φ) and π :X1 → X is the blowup ofC with exceptional varietyE = π−1(C)red. ThenΦ1 = Φ π is strongly prepared andA(Φ1,E) < A(Φ).

Proof. This theorem follows from Lemma 3.16 due to finiteness of the numbe2-varieties inX. Remark 3.18. With the notation of Lemma 3.16,Φ(Z(Φ) ∩ Nq(Φ)) = q and eachcomponent ofZ(Φ) ∩ Nq(Φ) is the intersection of a good componentE1 with a badcomponentE2.

Lemma 3.19. Suppose thatp ∈ Z(Φ)∩Nq(Φ) is a2point and(u, v) are strongly preparedparameters atq , (x1, . . . , xn) are strongly permissible parameters at p for(u, v) such that(N.IIb) holds and

u= (xa11 x

a22

)m, v = P

(xa11 x

a22

)+ xb11 x

b22 ,

whereb1/a1 < ordP < b2/a2 andd = ordP . q is a1-point.Suppose that(u, v) are also strongly prepared parameters atq and (y1, . . . , yn) are

strongly permissible parameters atp for (u, v) such that(N.IIb) holds and


en

nt

sd

tdts

e

u=(ya′

11 y

a′2

2

)m′, v = Q

(ya′

11 y

a′2

2

)+ y

b′1

1 yb′

22 ,

whereb′1/a

′1 < ordQ< b′

2/a′2 andd ′ = ordQ.

Thena1 = a′1, a2 = a′

2, b1 = b′1, b2 = b′

2, d = d ′, m=m′.

Proof. In order to decide whetherq is a 1 or 2 point, we will compare the varieties givby local equationsu = 0 anduv = 0 onX. According to the assumption ond,uv can bepresented as

uv = xa1m+b11 x

a2m+a2d2

(αx

a1d−b11 + x

b2−a2d2 + x1x2

P(xa11 x

a22 )− α(x

a11 x

a22 )d

xa1d+11 x

a2d+12

),

where 0= α ∈ k, xa1d−b11 , x

b2−a2d2 ∈ mpOX,p ,

P(xa11 x

a22 )− (x

a11 x

a22 )d

xa1d+11 x

a2d+12

∈ OX,p.

Thusuv = 0 defines a variety with at least 3 irreducible components at the 2 poip.Thereforeuv = 0 cannot be a local equation ofDS . Soq is a 1 point.

This implies that every permissible change of coordinates atq will translateu into αu

for some unit seriesα ∈ OX,p. Thus

(xa11 x

a22

)m = α(ya′

11 y

a′2

2

)m′, whereα is a unit series.

The powers of irreducible factors on the left-hand sidea1m anda2m are equal to the powerof irreducible factors on the right-hand sidea′

1m′ anda′

2m′, possibly in reverse order. An

since(a1, a2)= 1 and(a′1, a

′2)= 1, we can claim thatm=m′ anda1, a2 = a′

1, a′2.

Denote byE1 and E2 the components ofEX containingp with local equationsx1 = 0 andx2 = 0, respectively. Then by Lemma 3.10,A(Φ,E1) = 0 andA(Φ,E2) =b2 − a2 ordP > 0. SoE1 is a good component whileE2 is a bad component.

Sinceu = 0 is a local equation ofEX, y1 = 0 andy2 = 0 are local equations ofE1andE2, possibly in reverse order. Then by Lemma 3.10 the invariantA of the componenof EX with local equationy1 = 0 is equal to 0. Soy1 = 0 is a local equation of the goocomponentE1, while y2 = 0 is a local equation ofE2. From here and equality of the sea1, a2 anda′

1, a′2 it follows thata1 = a′

1 anda2 = a′2.

Suppose thatU is an étale neighborhood ofp where (x1, . . . , xn) and (y1, . . . , yn)

are uniformizing parameters. Fix a 1 pointp′ ∈ U ∩ E2. Following the proof of Lemma3.10, we can find strongly permissible parameters(x1, . . . , xn) and strongly permissiblparameters(y1, . . . , yn) atp′ such that

u = xa2m2 , v = P(x2)+ x

b22 x1 with ordP = a2d,

and


e

lyngly

u= ya′

2m′

2 , v = Q(y2)+ yb′

22 y1 with ordQ= a′

2d′.

So by Lemma 3.7,b2 = b′2, a2d = a′

2d′, and therefore,d = d ′.

To show thatb1 = b′1 we fix a 1 pointp′ ∈ U ∩ E1. Then following the proof of

Lemma 3.10 we find strongly permissible parameters(x1, . . . , xn) and strongly permissiblparameters(y1, . . . , yn) atp′ such that

u= xa1m1 , v = α1x

b11 + x

b11 x2 with α1 ∈ k,

and

u= ya′

1m

1 , v = α2yb11 + y

b′1

1 y1 with α2 ∈ k.

So by Lemma 3.7,b1 = b′1.

If α,β are real numbers, define

S(α,β) = max(α,β), (β,α)

,

where the maximum is in the lexicographic ordering.

Definition 3.20. Suppose thatΦ :X → S is strongly prepared andp ∈EX is a 2 point suchthat (II) holds atp. Define

σ(p) =S(|b1 − a1 ordP |, |b2 − a2 ordP |) if p ∈ N(Φ) ∩Z(Φ);

0 otherwise.

If C is a 2-variety inEX containing a 2 pointp in the form of (II), setσ(C) = σ(p).Setσ(C)= 0, otherwise.

Finally, defineσ(Φ) = maxσ(C) | C ⊂EX is a 2-variety.

Remark 3.21. In view of Lemmas 3.16 and 3.19, at every 2 pointp ∈ N(Φ) ∩ Z(Φ),where (II) holds,b1 − a1 ordP andb2 − a2 ordP are independent of the choice of strongprepared parameters(u, v) atΦ(p) and they are also independent of the choice of stropermissible parameters for(u, v) at p. So,σ(p) is well defined at every 2 pointp ∈ EX

in the form of (II). To justify the definition ofσ(C) for a 2-varietyC we will prove thefollowing lemma.

Lemma 3.22. Suppose thatp ∈ EX is a2 point in the form of(N.IIb) andC is a2-varietycontainingp.

Then there exists an open neighborhoodU of p such thatσ(p) = σ(p′) for allp′ ∈ U ∩C.


e

s.

at

nce

d

of

d

nts

Proof. There exist strongly prepared parameters(u, v) atΦ(p) and strongly permissiblparameters(x1, . . . , xn) atp such that

u= (xa11 x

a22

)m, v = P

(xa11 x

a22

)+ xb11 x

b22 , (∗)

andb1/a1 < d = ordP < b2/a2, b1/a1 <m.Let U be an étale neighborhood ofp where(x1, . . . , xn) are uniformizing parameter

Sincex1 = x2 = 0 are local equations ofC, for anyp′ ∈ U ∩ C there existα3, . . . , αn ∈ ksuch that(x1, x2, x3 = x3 + α3, . . . , xn = xn + αn) are strongly permissible parametersp′ for strongly prepared parameters(u, v) atΦ(p′). Then the same equations(∗) hold atp′ and, therefore,σ(p′)= S(|b1 − a1 ordP |, |b2 − a2 ordP |) = σ(p). Theorem 3.23. Suppose thatΦ :X → S is strongly prepared. Then there exists a sequeof blowups of2-varietiesX1 → X such that the induced mapΦ1 :X1 → S is stronglyprepared,A(Φ1,E) < A(Φ1) = A(Φ) if E is an exceptional component ofEX1 forX1 → X andZ(Φ1)∩N(Φ1) = ∅.

Proof. Z(Φ)∩N(Φ) = ∅ if and only if σ(Φ) = 0.Suppose thatσ(Φ) > 0 andC ⊂ Z(Φ) ∩ N(Φ) is a 2-variety such thatσ(C) = σ(Φ).

Let π :X1 → X be the blowup ofC. Then by Theorem 3.17,Φ1 = Φ π is stronglyprepared andA(Φ1,E) < A(Φ), so we will show that at every 2 points ∈ π−1(C) in theform of (II) σ(s) < σ(Φ).

Suppose thatp ∈ C is a k point and (N.Ic) holds atp, (u, v) are strongly prepareparameters atΦ(p) and(x1, . . . , xn) are strongly permissible parameters atp for (u, v)such that

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk xk+1,

andx1 = x2 = 0 are local equations ofC. Then there will not be any 2 point in the form(II) in π−1(p).

Suppose thatp ∈C is a k point and (N.IIb) holds atp, (u, v) are strongly prepareparameters atΦ(p) and (x1, . . . xn) are strongly permissible parameters atp for (u, v)

such that

u= (xa11 · · ·xakk

)m, v = P

(xa11 · · ·xakk

)+ xb11 · · ·xbkk ,

andx1 = x2 = 0 are local equations ofC.Then after possibly permutingx1 andx2 we can assume thatb1/a1 < ordP < b2/a2

andb1/a1 <m.Suppose thatE1, . . . ,Ek are the components ofEX containingp with local equations

x1 = 0, . . . , xk = 0, respectively. Ifp lies in the intersection of more than 3 componeE1, . . . ,Ek of EX there will not be any 2 point inπ−1(p).

Assume first thatp is a 2 point with

u= (xa1x

a2)m

, v = P(xa1x

a2)+ x

b1xb2 and d = ordP.
1 2 1 2 1 2


ble

n

Thenσ(Φ)= σ(C) = σ(p) = S(a1d − b1, b2 − a2d).Suppose that a 2 points ∈ π−1(p) has∗-permissible parameters(x1, x2, x3, . . . , xn)

such thatx2 = x1x2; then

u= (xa1+a21 x

a22

)m, v = P

(xa1+a21 x

a22

)+ xb1+b21 x

b22 .

Sinced < b2/a2, following the proof of Lemma 3.4 we can find strongly permissiparameters(y1, . . . , yn) at s such that

u= (ya1+a21 y

a22

)m, v = P

(ya1+a21 y

a22

)+ yb1+b21 y

b22 .

Thenσ(s) > 0 if and only if (u, v)OX1,s is not invertible and (N.IIb) holds atp. Let itbe the case; thend > (b1 + b2)/(a1 + a2) and(a1 + a2)d − (b1 + b2) < a1d − b1 sincea2d − b2 < 0. Thus

σ(s) = S((a1 + a2)d − (b1 + b2), b2 − a2d

)< S(a1d − b1, b2 − a2d)= σ(Φ).

Suppose that a 2 points ∈ π−1(p) has∗-permissible parameters(x1, x2, x3, . . . , xn)

such thatx1 = x1x2; then

u= (xa11 x

a1+a22

)m, v = P

(xa11 x

a1+a22

)+ xb11 x

b1+b22 .

If d (b1 + b2)/(a1 + a2), there exist strongly permissible parameters(y1, . . . , yn) at ssuch that

u = (ya11 y

a1+a22

)m, v = y

b11 y

b1+b22 .

Thusσ(s)= 0 in this case.Assume thatd < (b1 + b2)/(a1 + a2), then following the proof of Lemma 3.4 we ca

find strongly permissible parameters(y1, . . . , yn) at s such that

u= (ya11 y

a1+a22

)m, v = P

(ya11 y

a1+a22

)+ yb11 y

b1+b22 .

Suppose thatσ(s) > 0; that is, (N.IIb) holds ats. Then sinceb1 − a1d < 0, (b1 + b2)−(a1 + a2)d < b2 − a2d and

σ(s)= S(a1d − b1, (b1 + b2)− (a1 + a2)d) < S(a1d − b1, b2 − a2d)= σ(Φ).

Assume now thatp is a 3 point with

u= (xa11 x

a22 x

a33

)m, v = P

(xa11 x

a22 x

a33

)+ xb11 x

b22 x

b33 , and d = ordP.

Suppose thats ∈ π−1(p) is a 2 point, thens has regular parameters(x1, x2, x3, . . . , xn)

defined byx2 = x1(x2 + α) for some nonzeroα ∈ k and


e

eters

u= (xa1+a21 (x2 + α)a2x

a33

)m,

v = P(xa1+a21 (x2 + α)a2x

a33

)+ xb1+b21 (x2 + α)b2x

b33 .

If rank(a1+a2b1+b2

a3b3

)< 2 thenσ(s) = 0 since (II) cannot hold ats. So, consider the cas

when rank(a1+a2b1+b2

a3b3

)= 2.Seth = (a1 + a2)b3 − a3(b1 + b2) and

x1 = x1(x2 + α)(a2b3−a3b2)/h, x3 = x3(x2 + α)(a1b2−a2b1)/h,

to get∗-permissible parameters(x1, x2, x3, x4, . . . , xn) at s with

u= (xa1+a21 x

a33

)m, v = P

(xa1+a21 x

a33

)+ xb1+b21 x

b33 .

If

d max

b1 + b2

a1 + a2,b3

a3

there exist strongly permissible parameters(y1, . . . , yn) at s such that

u = (ya1+a21 y

a33

)m, v = y

b1+b21 y

b33 .

Thusσ(s)= 0 in this case.Assume that

d < max

b1 + b2

a1 + a2,b3

a3

,

then following the proof of Lemma 3.4, we can find strongly permissible param(y1, . . . , yn) at s such that

u= (ya1+a21 y

a33

)m, v = P

(ya1+a21 y

a33

)+ yb1+b21 y

b33 .

Suppose thatσ(s) > 0, so (N.IIb) holds ats and

min

b1 + b2

a1 + a2,b3

a3

< d < max

b1 + b2

a1 + a2,b3

a3

.

If

b1 + b2

a1 + a2< d <

b3

a3


a

t

at

t of

then according to the proof of Lemma 3.16, the 2-varietyE1∩E3 lies inZ(Φ)∩N(Φ) andσ(E1 ∩E3) = S(a1d − b1, b3 − a3d). Thus, sincea2d − b2 < 0, (a1 + a2)d − (b1 + b2) <

a1d − b1 and

σ(s)= S((a1 + a2d − (b1 + b2), b3 − a3d

)< S(a1d − b1, b3 − a3d)

= σ(E1 ∩E3) σ(Φ).

If

b3

a3< d <

b1 + b2

a1 + a2

then notice thatσ(s) > 0 implies thatb3/a3 < m. So, according to the proof of Lemm3.16, the 2-varietyE2 ∩E3 lies inZ(Φ)∩N(Φ) andσ(E2 ∩E3)= S(a3d −b3, b2−a2d).Thus, sinceb1 − a1d < 0, (b1 + b2)− (a1 + a2)d < b2 − a2d and

σ(s) = S(a2d − b3, (b1 + b2)− (a1 + a2)d

)< S(a3d − b3, b2 − a2d)

= σ(E2 ∩E3) σ(Φ).

By Theorem 3.17, induction on the number of 2-varietiesC ⊂ X with σ(C) = σ(Φ) andinduction onσ(Φ) we achieve the conclusions of the theorem.Theorem 3.24. Suppose thatΦ :X → S is strongly prepared andq ∈ S. Suppose also thaN(Φ) does not contain any bad point. IfNq(Φ) = ∅ thenNq(Φ) is a pure codimension-2subscheme which makes SNCs withB2(X).

Suppose thatC is a component ofNq(Φ) and π :X1 → X is the blowup ofC,E = π−1(C)red and Φ1 = Φ π . ThenΦ1 is strongly prepared,Z(Φ1) ∩ N(Φ1) = ∅andA(Φ1,E)= 0.

Proof. Suppose thatp ∈ Nq(Φ) is a k point, (u, v) are strongly prepared parametersq and(x1, . . . , xn) are strongly permissible parameters for(u, v) at p. Let U be an étaleneighborhood ofp where(x1, . . . , xn) are uniformizing parameters. Denote byE1, . . . ,Ek

the components ofEX containingp with local equationsx1 = 0, . . . , xk = 0, respectively.The assumption thatN(Φ) does not contain bad points implies that there is no poin

the form (N.Ic) or (N.IIb) inNq(Φ).Suppose that (N.Ia) holds atp:

u = (xa11 · · ·xakk

)m, v = (

xa11 · · ·xakk

)txk+1.

If p′ ∈ E1 ∩ U then there existα2, . . . , αn ∈ k such that(x1, x2 = x2 − α2, . . . , xn =xn − αn) are regular parameters atp′ and

u= (xa11 (x2 + α2)

a2 · · · (xk + αk)ak)m

,

v = (xa1(x2 + α2)

a2 · · · (xk + αk)ak)t(xk+1 + αk+1).
1


d

From here it follows thatmqOX,p′ is not invertible if and only if(xk+1 + αk+1) isnot a unit, i.e., ifαk+1 = 0. ThusNq(Φ) ∩ E1 ∩ U = V (x1, xk+1) and by symmetryNq(Φ) ∩ U = V (x1, xk+1) ∪ V (x2, xk+1) ∪ · · · ∪ V (xk, xk+1). So, in the neighborhooof p, Nq(Φ) is a union of codimension-2 varieties which make SNCs withB2.

Let π :X1 → X be the blowup of a componentC of Nq(Φ) passing throughp, thenΦ1 = Φ π is strongly prepared abovep. If p lies in more than 1 component ofEX ,π−1(p) does not contain any 1 point.

Assume thatp is a 1 point and the equationsu= xa1m1 , v = x

a1t1 x2 hold atp. ThenC is

defined byx1 = x2 = 0 in the neighborhood ofp.If s ∈ π−1(p) is a 1 point thens has∗-permissible parameters(x1, x2, x3, . . . , xn),

wherex2 = x1(x2 + α) for someα ∈ k, and

u= xa1m1 , v = αx

a1t+11 + x

a1t+11 x2.

Thuss is a good point andA(Φ1,E)=A(Φ, s) = 0.Finally, notice thatZ(Φ1) ∩ N(Φ1) ⊂ (π−1(N(Φ)) ∪ E) ∩ (Z(Φ1)) ⊂ π−1(N(Φ) ∩

Z(Φ)), sinceE is a good component ofEX1. SoZ(Φ1)∩N(Φ1)= ∅.Suppose that (N.Ib) holds atp:

u= (xa11 · · ·xakk

)m, v = x

b11 · · ·xbkk xk+1.


u= (xa11 (x2 + α2)

a2 · · · (xk + αk)ak)m

,

v = xb11 (x2 + α2)

b2 · · · (xk + αk)bk (xk+1 + αk+1).

ThusmqOX,p′ is not invertible if and only if at least one of the following holds:

(1) (xk+1 + αk+1) is not a unit andaim > bi for somei such thatαi = 0.(2) (xj + αj ) is not a unit and

min

b1

a1,bj

aj

<m< max

b1

a1,bj

aj

for somej ∈ 2, . . . , k.

Fix i ∈ 1, . . . , k and denote byJi the set of allj which satisfy the inequality

min

bi,bj<m< max

bi,bj,

ai aj ai aj


2

then

Nq(Φ)∩E1 ∩U =

⋃j∈J1

V (x1, xj )∪ V (x1, xk+1), if b1 < a1m,

⋃j∈J1

V (x1, xj ) if b1 a1m,

and by symmetry

Nq(Φ)∩U =(

k⋃i=1

⋃j∈Ji

V (xi, xj )

)∪(⋃

i∈IV (xi, xk+1)

)

with I = i | bi < aim. So, in the neighborhood ofp, Nq(Φ) is a union of codimension-varieties which make SNCs withB2.

Let π :X1 → X be the blowup of a componentC of Nq(Φ) passing throughp, thenΦ1 = Φ π is strongly prepared abovep. If p lies in more than 2 components ofEX ,π−1(p) does not contain any 1 point.

Suppose thatp is a 2 point and

u= (xa11 x

a22

)m, v = x

b11 x

b22 x3 with

b1

a1<

b2

a2.

If C is defined byx1 = x3 = 0 or x2 = x3 = 0 in the neighborhood ofp then there is no 1point inπ−1(p). So we may assume thatb1/a1 <m< b2/a2 andC = V (x1, x2).

If s ∈ π−1(p) is a 1 point thenOX1,s has regular parameters(x1, x2, x3, . . . , xn), wherex2 = x1(x2 + α) for some nonzeroα ∈ k, and

u= x(a1+a2)m1 (x2 + α)a2m, v = x

b1+b21 (x2 + α)b2x3.

Set x1 = x1(x2 + α)a2/(a1+a2) and x3 = x3(x2 + α)(a1b2−a2b1)/(a1+a2), so that(x1, x2, x3,

x4, . . . , xn) are∗-permissible parameters atp satisfying the equalities

u= x(a1+a2)m1 , v = x

b1+b21 x3.

Thuss is a good point andA(Φ1,E)=A(Φ, s) = 0.Arguing as above we also conclude thatZ(Φ1)∩N(Φ1)= ∅.Suppose that (N.IIa) holds atp:

u= (xa11 · · ·xakk

)m, v = x

b11 · · ·xbkk .


u = (xa1(x2 + α2)

a2 · · · (xk + αk)ak)m

, v = xb1(x2 + α2)

b2 · · · (xk + αk)bk .
1 1


ThusmqOX,p′ is not invertible if and only if(xj + αj ) is not a unit and

min

b1

a1,bj

aj

<m< max

b1

a1,bj

aj

for somej ∈ 2, . . . , k.

Fix i ∈ 1, . . . , k and denote byJi the set of allj which satisfy the inequality

min

bi

ai,bj

aj

<m< max

bi

ai,bj

aj

,

then

Nq(Φ) ∩E1 ∩U =⋃j∈J1

V (x1, xj )

and, by symmetry,

Nq(Φ) ∩U =k⋃

i=1

⋃j∈Ji

V (xi, xj ).

So, in the neighborhood ofp, Nq(Φ) is a union of 2-varieties; in particular,Nq(Φ) makesSNCs withB2.

Let π :X1 → X be the blowup of a componentC of Nq(Φ) passing throughp, thenΦ1 = Φ π is strongly prepared abovep. If p lies in more than 2 components ofEX ,π−1(p) does not contain any 1 point.

Assume thatp is a 2 point and

u= (xa11 x

a22

)m, v = x

b11 x

b22 with

b1

a1<m<

b2

a2.

ThenC is defined byx1 = x2 = 0 in the neighborhood ofp.If s ∈ π−1(p) is a 1 point thenOX1,s has regular parameters(x1, x2, x3, . . . , xn), where

x2 = x1(x2 + α) for some nonzeroα ∈ k, and

u= x(a1+a2)m1 (x2 + α)a2m, v = x

b1+b21 (x2 + α)b2. (∗∗)

Setf = a1b2−a2b1, x1 = x1(x2+α)a2/(a1+a2) andx2 = (x2+α)f/(a1+a2)−αf/(a1+a2).Then(x1, x2, x3, x4, . . . , xn) are∗-permissible parameters atp satisfying the equalities

u= x(a1+a2)m1 , v = αf/(a1+a2)x

b1+b21 + x

b1+b21 x2.

Thuss is a good point andA(Φ1,E)=A(Φ, s) = 0.Arguing as above, we also conclude thatZ(Φ1)∩N(Φ1)= ∅.Suppose that (N.III) holds atp:


.

.

-

t

ion

u= xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk .

So if we setm= (a1, . . . , ak−1) > 0, ai = ai/m for all i = 1, . . . , k−1 andak = b1 = 0,we obtain the required statement by going through the analysis of the previous caseRemark 3.25. Suppose thatC is a component ofNq(Φ) passing through a pointp ∈ EX ,wherep is a 2 point in the form of (N.IIa) or (N.III). Letπ :X1 → X be the blowup ofC.Then formula(∗∗) shows that at every 1 points ∈ π−1(p), mΦ(p)OX1,s is invertible. Inparticular, (N.Ia) cannot hold ats.

Moreover, since every 1 points ∈ π−1(C) can only lie inπ−1(p) for some 2 pointp ∈C, this implies that (N.Ia) does not hold at any 1 points ∈ π−1(C).

Suppose thatΦ :X → S is strongly prepared andq ∈ DS . Suppose thatp ∈ Ng(Φ)

is a 1 point,(u, v) are strongly prepared parameters atq , and(x1, . . . , xn) are stronglypermissible parameters atp such that

u= xm1 , v = xt1x2.

Let U be an étale neighborhood ofp where(x1, . . . , xn) are uniformizing parametersSuppose thatC ⊂Nq(Φ) is a codimension-2 variety containingp. ThenC = V (x1, x2)

in the neighborhood ofp and for every pointp′ ∈ C ∩U there exist(α3, . . . , αn) ∈ k andstrongly permissible parameters(x1, x2, x3 = x3 − α3, . . . , xn = xn − αn) such that

u= xm1 , v = xt1x2.

For any 1 pointp ∈Nq(Φ) defineΩq(p)=m− t > 0. If C ⊂Nq(Φ) is a codimension2 variety, define

Ωq(C)=Ωq(p) if there exists a 1 pointp ∈C;0 otherwise.

SetΩq(Φ)= maxΩq(C) | C is a codimension-2 variety inNq(Φ).

Remark 3.26. In view of Lemma 3.7,Ωq(p) is well defined for every 1 pointp ∈Nq(Φ).If C ⊂ Nq(Φ) is a variety of codimension 2,Ωq(C) is also well defined sinceΩq(p) =Ωq(p

′) for all 1 pointsp andp′ in C.

Lemma 3.27. Suppose thatΦ :X → S is strongly prepared andq ∈ DS . Suppose also thaN(Φ) does not contain any bad point.

Then there exists a sequence of blowups of nonsingular varieties of codimens2,X1 → X, which are not2-varieties, such that the induced mapΦ1 :X1 → S is stronglyprepared,Z(Φ1)∩N(Φ1)= ∅, A(Φ1,E)= 0 if E is an exceptional component ofEX1 forX1 → X andNq(Φ1) contains only2-varieties.


to

t

r

t

d

d to

d

Proof. Nq(Φ) contains only 2-varieties if and only ifΩq(Φ) = 0.Suppose thatΩq(Φ) > 0 andC ∈ Nq(Φ) is a codimension-2 variety withΩq(C) =

Ωq(Φ). Let π :X1 → X be the blowup ofC. Then by Theorem 3.24 we only needverify thatΩq(s) <Ωq(Φ) for every 1 points ∈ Nq(Φ1)∩E.

Suppose thatp ∈ C is such that there exist a 1 points ∈ π−1(p). Thenp is necessarilya 1 point. If (u, v) are strongly prepared parameters atq and (x1, . . . , xn) are stronglypermissible parameters atp, thens has∗-permissible parameters(x1, x2, x3, . . . , xn) suchthatx2 = x1(x2 + α) for someα ∈ k and

u= xm1 , v = xt+11 (x2 + α).

ThusmqOX1,s is not invertible if and only ifα = 0 and in this case

Ωq(s) =m− (t + 1) < m− t =Ωq(p) =Ωq(C) =Ωq(Φ).

By induction on the number of codimension-2 varieties inNq(Φ) such thatΩq(Φ) =Ωq(C) and induction onΩq(Φ) we achieve the conclusions of the lemma.Definition 3.28. Suppose thatΦ :X → S is strongly prepared andq ∈ S. Suppose also thap ∈Φ−1(q) is a 2 point. Define

ω(p) =S(|b1 − a1m|, |b2 − a2m|) if (N.IIa) holds atp;

S(a1, b2) if (N.III) holds atp;0 otherwise.

If C is a 2-variety inΦ−1(q) containing a 2 pointp in one of the forms (N.IIa) o(N.III), setω(C) = ω(p). Setω(C) = 0, otherwise.

Defineωq(Φ)= maxω(C) | C ⊂Φ−1(q) is a 2-variety.

Remark 3.29. It is not hard to see thatω(p) is well defined for every 2 pointp ∈ Φ−1(q).Using arguments similar to the proof of Lemma 3.22, we can also show thatω(C) is well-defined for every 2-varietyC ⊂Φ−1(q).

Lemma 3.30. Suppose thatΦ :X → S is strongly prepared andq ∈ S. Suppose also thaN(Φ) does not contain any bad point andNq(Φ) consists of a union of2-varieties.

Then there exists a sequence of blowups of2-varietiesX1 → X such that the inducemapΦ1 :X1 → S is strongly prepared,Z(Φ1) ∩ N(Φ1) = ∅, A(Φ1,E) = 0 if E is anexceptional component ofEX1 for X1 → X andNq(Φ1) = ∅.

Proof. Under the assumptions of the lemmaNq(Φ)= ∅ if and only if ωq(Φ) = 0.Suppose thatωq(Φ) > 0 andC ⊂ Nq(Φ) is a 2-variety such thatω(C) = ωq(Φ). Let

π :X1 → X be the blowup ofC. Then by Theorem 3.24 and Remark 3.25 we only neeverify thatω(s) < ωq(Φ) at every 2 points ∈ π−1(C).

Suppose thatp ∈ C is a k point and (N.IIa) holds atp, (u, v) are strongly prepareparameters atΦ(p) and(x1, . . . , xn) are strongly permissible parameters atp for (u, v)such that


nts

se

u= (xa11 · · ·xakk

)m, v = x

b11 · · ·xbkk

andx1 = x2 = 0 are local equations ofC. Then after possibly permutingx1 andx2 we canassume thatb1/a1 <m< b2/a2.

Suppose thatE1, . . . ,Ek are the components ofEX containingp with local equationsx1 = 0, . . . , xk = 0, respectively. Ifp lies in the intersection of more than 3 componeE1, . . . ,Ek of EX, there will not be any 2 point inπ−1(p).

Assume first thatp is a 2 point with

u= (xa11 x

a22

)m, v = x

b11 x

b22 .

Thenωq(Φ)= ω(C) = ω(p) = S(a1m− b1, b2 − a2m).Suppose that a 2 points ∈ π−1(p) has strongly permissible parameters(x1, x2, x3, . . . ,

xn) such thatx2 = x1x2, then

u = (xa1+a21 x

a22

)m, v = x

b1+b21 x

b22 .

So,ω(s) > 0 if and only if (u, v)OX1,s is not invertible and (N.IIa) holds atp. Let itbe the case, thenm> (b1 + b2)/(a1 + a2) and(a1 + a2)m− (b1 + b2) < a1m− b1 sincea2d − b2 < 0. Thus

ω(s) = S((a1 + a2)m− (b1 + b2), b2 − a2m

)< S(a1m− b1, b2 − a2m)= ωq(Φ).

Suppose that a 2 points ∈ π−1(p) has strongly permissible parameters(x1, x2, x3, . . . , xn)

such thatx1 = x1x2, then

u = (xa11 x

a1+a22

)m, v = x

b11 x

b1+b22 .

So,ω(s) > 0 if and only if(u, v)OX1,s is not invertible, that is, ifm< (b1 + b2)/(a1 + a2).Then(b1 + b2)− (a1 + a2)m < b2 − a2m, sinceb1 − a1m< 0, and

ω(s) = S(a1m− b1, (b1 + b2)− (a1 + a2)m

)< S(a1m− b1, b2 − a2m)= ωq(Φ).

Assume now thatp is a 3 point with

u= (xa11 x

a22 x

a33

)m, v = x

b11 x

b22 x

b33 .

Suppose thats ∈ π−1(p) is a 2 point, thens has regular parameters(x1, x2, x3, . . . , xn)

defined byx2 = x1(x2 + α) for some nonzeroα ∈ k and

u= (xa1+a21 (x2 + α)a2x

a33

)m, v = x

b1+b21 (x2 + α)b2x

b33 .

If rank(a1+a2 a3b1+b2 b3

)< 2 thenω(s) = 0 since(u, v)OX1,s is invertible. So, consider the ca

when rank(a1+a2 a3

)= 2.
b1+b2 b3


eters

d

Seth = (a1 + a2)b3 − a3(b1 + b2) and

x1 = x1(x2 + α)(a2b3−a3b2)/h, x3 = x3(x2 + α)(a1b2−a2b1)/h,

to get∗-permissible parameters(x1, x2, x3, x4, . . . , xn) at s with

u = (xa1+a21 x

a33

)m, v = x

b1+b21 x

b33 .

Following the proof of Lemma 3.4, we can find strongly permissible param(y1, . . . , yn) at s such that

u = (ya1+a21 y

a33

)m, v = y

b1+b21 y

b33 .

Assume thatω(s) > 0, then

min

b1 + b2

a1 + a2,b3

a3

<m< max

b1 + b2

a1 + a2,b3

a3

.

If

b1 + b2

a1 + a2<m<

b3

a3

then, according to the proof of Theorem 3.24, the 2-varietyE1 ∩ E3 lies in Nq(Φ) andω(E1∩E3)= S(a1m−b1, b3−a3m). Thus, sincea2m−b2 < 0, (a1+a2)m−(b1+b2) <

a1m− b1 and

ω(s) = S((a1 + a2)m− (b1 + b2), b3 − a3m

)< S(a1m− b1, b3 − a3m)

= ω(E1 ∩E3) ωq(Φ).

If

b3

a3<m<

b1 + b2

a1 + a2

then, according to the proof of Theorem 3.24, the 2-varietyE2 ∩ E3 lies in Nq(Φ) andω(E2∩E3)= S(a3m−b3, b2−a2m). Thus, sinceb1−a1m< 0, (b1+b2)−(a1+a2)m <

b2 − a2m and

ω(s) = S(a3m− b3, (b1 + b2)− (a1 + a2)m

)< S(a3m− b3, b2 − a2m)

= ω(E2 ∩E3) ωq(Φ).

Suppose thatp ∈ C is a k point and (N.III) holds atp, (u, v) are strongly prepareparameters atΦ(p) and(x1, . . . , xn) are strongly permissible parameters atp for (u, v)such that


a

30,rieties

e

u= xa11 · · ·xak−1

k−1 , v = xb22 · · ·xbkk .

If we setm = (a1, . . . , ak−1) > 0, ai = ai/m for all i = 1, . . . , k1, andak = b1 = 0, weobtain the required statement by going through the analysis of the previous case.

By Theorem 3.24, induction on the number of 2-varietiesC ∈ Nq(Φ) with ω(C) =wq(Φ), and induction onωq(Φ), we achieve the conclusions of the lemma.Theorem 3.31. Suppose thatΦ :X → S is a strongly prepared morphism fromnonsingularn-foldX to a nonsingular surfaceS.

Then there exists a finite sequence of quadratic transformsπ1 :S1 → S and monoidaltransformsπ2 :X1 → X centered at nonsingular varieties of codimension2 and such thatthe induced morphismΦ :X1 → S1 is monomial.

Proof. Φ :X → S is monomial if and only if all points ofX are good forΦ, that is ifA(Φ)= 0.

Suppose thatA(Φ) > 0 andE is a component ofEX such thatC(Φ,E) = C(Φ). SinceA(Φ,E) > 0,Φ(E) is a pointq ∈DS .

Let π1 :S1 → S be the blowup ofq . Then, by Theorem 3.23, Lemmas 3.27, 3.and 3.13, there exists a sequence of blowups of nonsingular codimension-2 vaπ2 :X1 → X such thatΦ1 :X1 → S is strongly prepared,A(Φ1,E) < A(Φ1) = A(Φ) ifE is the exceptional divisor forΦ1 and the induced morphismΦ2 :X1 → S1 is stronglyprepared.

Thus, if E is the strict transform ofE onX1, by Theorem 3.14,C(Φ2, E) < C(Φ1) =C(Φ).

By induction on the number of componentsE of EX with C(Φ,E) = C(Φ) andinduction onC(Φ), we achieve the conclusion of the theorem.

4. Toroidalization

Definition 4.1. A normal varietyX with a SNC divisorEX on X is called toroidal iffor every pointp ∈X there exists an affine toric varietyXσ , a pointp′ ∈ Xσ and anisomorphism ofk algebrasOX,p

∼= OXσ ,p′ such that the ideal ofEX corresponds to thideal ofXσ − T (whereT is the torus inXσ ). Such a pair is called a model atp.

A dominant morphismΦ :X → Y of toroidal varieties with SNC divisorsEX on X

andDY on Y satisfyingΦ−1(DY ) ⊂ EX is called toroidal atp ∈ X if there exist localmodels(Xσ ,p

′) at p, (Yτ , q ′) at q = Φ(p) and a toric morphismΨ :Xσ → Yτ such thatthe following diagram commutes:

OX,p OXσ ,p′∼

OY,q

Φ∗

OYτ ,q ′ .

Ψ ∗

∼


t

a

tg

ety

Φ :X → Y is called toroidal (with respect toEX andDY ) if Φ is toroidal at every poinp ∈X.

From now on we will assume thatΦ :X → S is a strongly prepared morphism fromnonsingularn-fold X to a nonsingular surfaceS, and all points ofEX are good forΦ. Wewill also say thatp is a toroidal point forΦ if Φ is toroidal atp. A point p which is nottoroidal forΦ will be called nontoroidal.

Lemma 4.2. Suppose thatΦ :X → S is a morphism from a nonsingularn-fold X to anonsingular surfaceS, DS is a SNC divisor onS such thatEX = Φ−1(DS) is a SNCdivisor onX andp ∈ X is a k point. ThenΦ is toroidal atp if and only if there exisregular parameters(x1, . . . , xn) in OX,p and(u, v) in OS,p such that one of the followinforms holds:

(1z) 1 k n − 1: u = 0 is a local equation ofDS , x1 · · ·xk = 0 is a local equation ofEX and

u= xa11 · · ·xakk , v = xk+1, (T.Iz)

wherea1, . . . , ak > 0;(1n) 1 k n− 1: uv = 0 is a local equation ofDS , x1 · · ·xk = 0 is a local equation of

EX and

u= (xa11 · · ·xakk

)m, v = α

(xa11 · · ·xakk

)t + (xa11 · · ·xakk

)txk+1, (T.In)

wherea1, . . . , ak > 0, m, t > 0 andα ∈ k − 0;(2) 2 k n: uv = 0 is a local equation ofDS , x1 · · ·xk = 0 is a local equation ofEX

and

y1 = xa11 · · ·xakk , y2 = x

b11 · · ·xbkk , (T.II)

where a1, . . . , ak , b1, . . . , bk 0, ai + bi > 0 for all i = 1, . . . , k, andrank

( a1 ··· akb1 ··· bk

)= 2.

Proof. Let Xσ,l be then-dimensional nonsingular affine toric variety Speck[z1, . . . , zn,

z−1l+1, . . . , z

−1n ] (wherel ∈ 1, . . . , n) with the torusTn = Speck[z1, . . . , zn, z

−11 , . . . , z−1

n ]and Zσ,l = Xσ,l − Tn. Let Yτ be the 2-dimensional nonsingular affine toric variSpeck[y1, y2] with the torusT2 = Speck[y1, y2, y

−11 , y−1

2 ] and Zτ = Yτ − T2. Thenany dominant toric morphismΨ :Xσ,l → Xτ satisfyingZσ,l = Ψ−1(Zτ ) is given by theequations

y1 = za11 · · ·zann , y2 = z

b11 · · ·zbnn ,

wherea1, . . . , an, b1, . . . , bn are integers,ai, bi 0 andai + bi > 0 for all i = 1, . . . , l,rank

( a1 ··· anb ··· b

)= 2.
1 n


g

rs

at

We will describe the mapΨ :Xσ,l → Yτ locally.If p′ ∈ Xσ,l is ak point (with respect toZσ,l), thenk l and, after possibly permutin

z1, . . . , zl , we can find nonzeroαk+1, . . . , αn ∈ k such thatz1, . . . , zk, zk+1 = zk+1 −αk+1,

. . . , zn = zn − αn are regular parameters atp′ andz1 · · ·zk = 0 is a local equation ofZσ,l

atp′.Assume first thatb1 = · · · = bk = 0, thena1, . . . , ak > 0, (y1, y2 = y2 − α

bk+1k+1 · · ·αbn

n )

are regular parameters atq ′ = Ψ (p′) and y1 = 0 is a local equation ofZτ at q ′. Setzk = zk((zk+1 + αk+1)

ak+1 · · · (zn + αn)an)1/ak and zk+1 = y2 to get regular paramete

(z1, . . . , zk1, zk, zk+1, zk+2, . . . , zn) in OXσ ,l,p′ such that

y1 = za11 · · ·zak−1

k−1 zakk , y2 = zk+1.

Assume now that at least one ofb1, . . . , bk is greater than 0 and rank( a1 ··· akb1 ··· bk

) = 1.Let m = (a1, . . . , ak) and ai = ai/m, thenbi = ai t for somet > 0 andai, bi > 0 for alli = 1, . . . , k, (y1, y2) are regular parameters atq ′ = Ψ (p′) andy1y2 = 0 is a local equationof Zτ atp′. Set

α = αbk+1− t

m ak+1

k+1 · · ·αbn− tm an

n ,

zk = zk((zk+1 + αk+1)

ak+1 · · · (zn + αn)an)1/ak ,

zk+1 = (zk+1 + αk+1)bk+1− t

m ak+1 · · · (zn + αn)bn− t

m an

to get regular parameters(z1, . . . , zk−1, zk, zk+1, zk+2, . . . , zn) in OXσ ,l,p′ such that

y1 = (za11 · · ·zak−1

k−1 zakk

)m, y2 = α

(za11 · · ·zak−1

k−1 zakk

)t + (za11 · · ·zak−1

k−1 zakk

)tzk+1.

Finally assume that rank( a1 ··· akb1 ··· bk

) = 2. Then, after possibly permutingz1, . . . , zk , wecan suppose thatf = ak−1bk − akbk−1 = 0. (y1, y2) are regular parameters atq ′ = Ψ (p′)andy1y2 = 0 is a local equation ofZτ atq ′ in this case. Set

zk−1 = zk−1(zk+1 + αk+1)(ak+1bk−akbk+1)/f · · · (zn + αn)

(anbk−akbn)/f ,

zk = zk(zk+1 + αk+1)(ak−1bk+1−ak+1bk−1)/f · · · (zn + αn)

(ak−1bn−anbk−1)/f

to get regular parameters (z1, . . . , zk−2, zk−1, . . . , zn) in OXσ,l,p′ such that

y1 = za11 · · ·zak−2

k−2 zak−1k−1 z

akk , y2 = z

b11 · · ·zbk−2

k−2 zbk−1k−1 z

bkk .

By the definition,Φ is toroidal atp if and only if there existk l n, a k pointp′ ∈ Xσ,l , and a toric morphismΨ :Xσ,l → Yτ such thatΦ has the same local descriptionp as the morphismΨ has atp′, that is, if one of the forms (T.Iz), (T.In) or (T.II) holds.


cans

tersf

rse

s atf

st

Remark 4.3. If p ∈ EX is a toroidal point, arguing as tn the proof of Lemma 3.4 wealways find strongly permissible parameters(x1, . . . , xn) at p such that one of the form(T.Iz), (T.In) or (T.II) holds.

Remark 4.4. Suppose thatΦ :X → S is a strongly prepared morphism withZ(Φ) = ∅and ak point p ∈ X is not toroidal forΦ. Then there exist strongly prepared parame(u, v) at q =Φ(p) and strongly permissible parameters(x1, . . . , xn) atp such that one othe following forms holds:

(1a) 1 k n− 1: u= 0 is a local equation ofDS , x1 · · ·xk = 0 is a local equation ofEX

and

u= (xa11 · · ·xakk

)m, v = α

(xa11 · · ·xakk

)t + (xa11 · · ·xakk

)txk+1, (NT.Ia)

wherea1, . . . , ak > 0,m, t > 0 andα ∈ k;(1b) 2 k n−1: u= 0 is a local equation ofDS , x1 · · ·xk = 0 is a local equation ofEX

and

u= xa11 · · ·xakk , v = x

b11 · · ·xbkk xk+1, (NT.Ib)

wherea1, . . . , ak > 0, b1, . . . , bk 0 and rank( a1 ··· akb1 ··· bk

)= 2;

(2) 2 k n: u= 0 is a local equation ofDS , x1 · · ·xk = 0 is a local equation ofEX and

u = xa11 · · ·xakk , v = x

b11 · · ·xbkk , (NT.II)

wherea1, . . . , ak > 0, b1, . . . , bk 0 and rank( a1 ··· akb1 ··· bk

)= 2.

Suppose thatp ∈ EX is a 1 point such thatq = Φ(p) is a 1 point onDS . Let E bethe component ofEX containingp. Suppose that(u, v) are strongly prepared parameteat q such thatu = 0 is a local equation ofDS and(x1, . . . , xn) are strongly permissiblparameters atp with

u= xa1, v = xc1(xk+1 + α), α ∈ k.

By Lemma 3.7,a = νE(u) andc = νE(v) are independent of the choice of parameterp andq , so we can define an invariantI (Φ,p) = c − a. Moreover, following the proof oLemma 3.9, we see thatνE(u) andνE(v) evaluated atp are equal toνE(u) andνE(v),respectively, evaluated at any 1 pointp′ ∈ E. Therefore,I (Φ,E) = I (Φ,p) is a welldefined notion.

In the above notations,p is a toroidal point if and only ifc = 0. Thus either all 1 pointonE are toroidal or all of them are nontoroidal. DefineE to be a toroidal component if aleast one 1 point onE is toroidal, defineE to be nontoroidal otherwise. Set

I (Φ) = maxI (Φ,E) |E is a nontoroidal component ofEX

.


d

ents

ters

ters

e

t

Theorem 4.5. Suppose thatΦ :X → S is strongly prepared withZ(Φ)= ∅. Then the locusof nontoroidal points onX is a Zariski closed set of pure codimension1, consisting of allnontoroidal components ofEX.

Proof. Let Y be the union of all nontoroidal components ofEX . We will show that anynontoroidal point ofEX lies onY and there is no toroidal point lying onY.

Suppose thatp is k point, q = Φ(p). Suppose that(u, v) are strongly prepareparameters atq and(x1, . . . , xn) are strongly permissible parameters atp. LetE1, . . . ,Ek

be the components ofEX containing p with local equationsx1 = 0, . . . , xk = 0,respectively, and letU be a neighborhood ofp where (x1, . . . , xn) are uniformizingparameters.

We will assume thatp is a toroidal point forΦ and verify thatE1 is toroidal, thatis, contains a toroidal 1 point. By the symmetry, this will imply that all componE1, . . . ,Ek containingp are toroidal andp /∈ Y.

Suppose first that (T.Iz) holds atp; then a 1 pointp′ ∈ E1 ∩ V (xk+1) ∩ U istoroidal sinceΦ(p′) = Φ(p) is a 1 point and there exist strongly permissible parame(x1, . . . , xk, xk+1, xk+2, . . . , xn) atp′ such that

u= xa11 , v = xk+1.

Suppose that (T.In) holds atp; then a 1 pointp′ ∈ E1 ∩ V (xk+1) ∩ U is toroidalsince Φ(p′) = Φ(p) is a 2 point and there exist strongly permissible parame(x1, . . . , xk, xk+1, xk+2, . . . , xn) atp′ such that

u = xa1m1 , v = αx

a1t1 + x

a1t1 xk+1.

Suppose that (T.II) holds atp. After possibly interchangingu andv we can assumthat a1 > 0 andb1 0. Furthermore, since rank

( a1 ··· akb1 ··· bk

) = 2 there existi ∈ 1, . . . , ksuch thatbi − b1ai/a1 = 0. Thus after possibly permutingx2, . . . , xk we can assume thab2 − b1a2/a1 = 0.

Let p′ ∈ E1 ∩ U be a 1 point, then there exist nonzeroα2, . . . , αk ∈ k and regularparameters(x1, . . . , xn) atp′ such that

u= xa11 , v = x

b11 (x2 + α2)

b2−b1a2/a1 · · · (xk + αk)bk−b1ak/a1.

Set

α = αb2−b1a2/a12 · · ·αbk−b1ak/a1

k and

x2 = (x2 + α2)b2−b1a2/a1 · · · (xk + αk)

bk−b1ak/a1 − α

to get∗-permissible parameters(x1, x2, x3, . . . , xk, xk+1, . . . , xn) atp′ such that

u= xa1, v = x

b1(x2 + α).
1 1


ble

e

e

If b1 > 0 thenΦ(p′) = Φ(p) is a 2 point andp′ is toroidal since (T.In) holds atp′. Ifb1 = 0 then(u, v = v − α) are strongly prepared parameters atΦ(p′) andp′ is a toroidalpoint since either (T.Iz) or (T.II) holds atp′.

Assume now thatp is a nontoroidal point forΦ and find a componentEj containingpwhich is not toroidal, that is, contains a nontoroidal 1 point. This will implyp ∈ Y .

Suppose first that (NT.Ia) holds atp; then a 1 pointp′ ∈ E1 ∩ V (xk+1) ∩ U isnontoroidal sinceΦ(p′) = Φ(p) is a 1 point and there exist strongly permissiparameters(x1, . . . , xk, xk+1, xk+2, . . . , xn) atp′ such that

u= xa1m1 , v = αx

a1t1 + xa1t xk+1.

Suppose that (NT.Ib) holds atp. After possibly interchangingx1, . . . , xk, we can assumthatb1 > 0. Then a 1 pointp′ ∈ E1 ∩V (xk+1)∩U is nontoroidal sinceΦ(p′)=Φ(p) is a1 point and there exist strongly permissible parameters(x1, . . . , xn) atp′ such that (NT.Ia)holds:

u= xa11 , v = x

b11 xk+1.

Suppose that (NT. II) holds atp. After possibly interchangingx1, . . . , xk we can assumthatb1 > 0 andb2 − b1a2/a1 = 0.

Let p′ ∈ E1 ∩ U be a 1 point, then there exist nonzeroα2, . . . , αk ∈ k and regularparameters(x1, . . . , xn) atp′ such that

u= xa11 , v = x

b11 (x2 + α2)

b2−b1a2/a1 · · · (xk + αk)bk−b1ak/a1.

Set

α = αb2−b1a2/a12 · · ·αbk−b1ak/a1

k and

x2 = (x2 + α2)b2−b1a2/a1 · · · (xk + αk)

bk−b1ak/a1 − α

to get strongly permissible parameters(x1, x2, x3, . . . , xk, xk+1, . . . , xn) atp′ such that

u = xa11 , v = x

b11 (x2 + α). (∗∗∗)

ThusΦ(p′)=Φ(p) is a 1 point andp′ is nontoroidal since (NT.Ia) holds atp′. Remark 4.6. Suppose thatp ∈ EX is a nontoroidalk point in the form of (NT.II). In thenotations of Theorem 4.5, we have

u = xa11 · · ·xb1

k , v = xb11 · · ·xbkk

atp. Then formula(∗∗∗) shows thatI (Φ,E1)= b1 − a1.AnalogouslyI (Φ,Ei) = bi − ai for all i ∈ 1, . . . , k such thatbi > 0.


ly

ers

4

Theorem 4.7. Suppose thatΦ :X → S is strongly prepared andZ(Φ) = ∅, p ∈ X is a 1point such thatq =Φ(p) is a1 point. Suppose thatπ :S1 → S is the blow up ofq and therational mapΦ1 :X → S1 is a morphism in a neighborhood ofp.

If I (Φ,p) > 0 thenI (Φ1,p) < I (Φ,p). If I (Φ,p) 0 thenΦ1 is toroidal atp.

Proof. Let (u, v) be strongly prepared parameters atq and (x1, . . . , xn) be stronglypermissible parameters atp. Sinceq is a 1 point andmqOX,p is invertible atp, (NT.Ia)holds atp:

u= xa1, v = αxc1 + xc1x2,

and eitherc a or c < a and α = 0. Notice also that by the definition of strongpermissible parametersα = 0 if c = a.

Assume thatI (Φ,p) = c−a 0, then there exist strongly prepared parameters(u1, v1)

atΦ1(p) such that

u= u1, v = u1v1.

In case whenc > a we have

u1 = xa1, v1 = αxc−a1 + xc−a

1 x2,

andI (Φ1,p) = c − 2a < c − a = I (Φ,p).If c = a thenu1 = xa1 andv1 = x2 atp. Thusp is a toroidal point forΦ1.Assume thatI (Φ,p) < 0 and, therefore,α = 0. Then there exist prepared paramet

(u1, v1) atΦ1(p) and strongly permissible parameters(x1, x2, x3, . . . , xn) atp such that

u= u1v1, v = v1

and

u1 = xa−c1 , v1 = αa/(a−c)xc1 + xc1x2.

ThusΦ1(p) is a 2 point andp is toroidal forΦ1. Lemma 4.8. Suppose thatΦ :X → S is strongly prepared,Z(Φ) = ∅ and q ∈ S is a 1point. Suppose that a varietyC of codimension2 is a component ofNq(Φ) andC is not a2-variety.

Letπ :X1 → X be the blowup ofC, E = π−1(C) andΦ1 =Φ π . ThenΦ1 is stronglyprepared,Z(Φ1)= ∅ andI (Φ1,E) 0.

Proof. Theorem 3.24 implies thatΦ1 is strongly prepared andZ(Φ1)= ∅.Suppose thatp is a point onC, (u, v) are strongly prepared parameters atq and

(x1, . . . , xn) are strongly prepared parameters atp. Analyzing the proof of Theorem 3.2we see that ifπ−1(p) contains a 1 point, eitherp is a 1 point satisfying (N.Ia) orp is a


e

,

2 point satisfying (N.Ib). Since a generic point onC is a 1 point andI (Φ1,E) = I (Φ1, s)

for any 1 points ∈ E, it suffices to verify thatI (Φ1, s) 0 if s ∈ π−1(p) is a 1 point andp is 1 point satisfying (N.Ia).

In this case

u= xm1 , v = xt1x2 with t < m

andC is defined byx1 = x2 = 0 in the neighborhood ofp. Thens has strongly permissiblparameters(x1, x2, x3, . . . , , xn), wherex2 = x1(x2 + α) for somea ∈ k, and

u= xm1 , v = αxt+11 + xt+1

1 x2

with α = α if t + 1 =m andα = 0 if t + 1 =m.ThusIΦ1, s)= t + 1−m 0.

Lemma 4.9. Suppose thatΦ :X → S is strongly prepared,Z(Φ) = ∅ and q ∈ S is a 1point. Suppose that a2-varietyC is a component ofNq(Φ).

Letπ :X1 → X be the blowup ofC, E = π−1(C) andΦ1 =Φ π . ThenΦ1 is stronglyprepared,Z(Φ1)= ∅ andI (Φ1,E) < I (Φ).

Proof. Theorem 3.24 implies thatΦ1 is strongly prepared andZ(Φ1)= ∅.Suppose thatp is a point onC, (u, v) are strongly prepared parameters atq and

(x1, . . . , xn) are strongly prepared parameters atp. Analyzing the proof of Theorem 3.24we see that ifπ−1(p) contains a 1 point,p is a 2 point satisfying (N.IIa):

u = (xa11 x

a22

)m, v = x

b11 x

b22 with

b1

a1<m<

b2

a2,

and C is defined byx1 = x2 = 0 in the neighborhood ofp. Then OX1,s has regularparameters(x1, x2, x3, . . . , xn), wherex2 = x1(x2 + α) for some nonzeroa ∈ k, and

u = x(a1+a2)m1 (x2 + α)a2m, v = x

b1+b21 (x2 + α)b2.

Setf = a1b2−a2b1, x1 = x1(x2+α)a2/(a1+a2) andx2 = (x2+α)f/(a1+a2)−αf/(a1+a2).Then(x1, x2, x3, x4, . . . , xn) are∗-permissible parameters atp satisfying the equalities

u= x(a1+a2)m1 , v = αf/(a1+a2)x

b1+b21 + x

b1+b21 x2.

Thus

I (Φ1,E)= I (Φ, s) = (b1 + b2)− (a1 + a2)m= (b2 − a2m)+ (b1 − a1m) < b2 − a2m.

Denote byE2 the component ofEX with local equationx2 = 0; then by Remark 4.6

I (Φ1,E) < b2 − a2m = I (Φ,E2) I (Φ).


n

sd

.13,

a

draticion-2

s, bynsion-

Remark 4.10. It follows from the above proof that ifNq(Φ) contains a 2-variety theI (Φ) > 0.

Lemma 4.11. Suppose thatΦ :X → S is strongly prepared andZ(Φ) = ∅. Then thereexists a finite sequence of quadratic transformsπ :S1 → S and monoidal transformcentered at nonsingular varieties of codimension2 π2 :X1 → X such that the inducemapΦ :X1 → S1 is strongly prepared,Z(Φ) = ∅ andI (Φ) 0.

Proof. Suppose thatI (Φ) > 0 andE is a component ofEX such thatI (Φ,E) = I (Φ).ThenΦ(E) is a 1 pointq ∈DS .

Let π1 :S1 → S be the blowup ofq . Then, by Lemmas 3.27, 4.8, 3.30, 4.9, and 3there exists a sequence of blowups of nonsingular codimension-2 varietiesπ2 :X1 → X

such thatΦ1 :X1 → S is strongly prepared withZ(Φ1) = ∅, I (Φ1,E) < I (Φ1) = I (Φ)

if E is the exceptional divisor forΦ1 and the induced morphismΦ2 :X1 → S1 is stronglyprepared withZ(Φ2) = ∅.

Thus, if E is the strict transform ofE on X1, by Theorem 4.7I (Φ2, E) < I (Φ1) =I (Φ).

By induction on the number of componentsE of EX with I (Φ,E) = I (Φ) andinduction onI (Φ), we achieve the conclusion of the theorem.Theorem 4.12. Suppose thatΦ :X → S is a strongly prepared morphism fromnonsingularn-foldX to a nonsingular surfaceS.

Then there exists a finite sequence of quadratic transformsπ1 :S1 → S and monoidaltransforms, centered at nonsingular varieties of codimension2, π2 :X1 → X such that theinduced morphismΦ :X1 → S1 is toroidal.

Proof. From Theorem 3.31 and Lemma 4.11, we obtain a finite sequence of quatransformsS1 → S and monoidal transforms centered at nonsingular codimensvarietiesX1 →X such that the induced mapΦ1 :X1 → S1 is strongly prepared,Z(Φ1)= ∅andI (Φ1) 0.

If Φ1 is not toroidal, consider the set

T (Φ1)= q ∈ DS | q =Φ1(E) whereE is a nontoroidal component ofEX

.

T (Φ1) is a finite set, containing only 1 points.Let q ∈ T (Φ1) andπ1 :S2 → S1 be the blowup ofq . Then by Remark 4.10,Nq(Φ1)

contains only nonsingular codimension-2 varieties which are not 2-varieties. ThuLemmas 3.27 and 4.8, there exists a sequence of blowups of nonsingular codime2 varietiesπ2 :X2 →X1 such thatΦ2 :X2 → S1 is strongly prepared withZ(Φ2)= ∅ andI (Φ2) 0.

By Lemma 3.13 and Theorem 4.7, the induced morphismΦ3 :X2 → S2 is stronglyprepared,Z(Φ3)= 0 and all points inΦ−1

1 (q) are toroidal. Therefore,

T (Φ3) = T (Φ1)− q.


e

aps,

rlag,

ath.,

bra 28

ol. II,

By induction on the number of points inT (Φ1), we achieve the conclusion of ththeorem.

References

[1] S. Abhyankar, Good points of a hypersurface, Adv. Math. 68 (1988) 87–256.[2] A. Abramovich, K. Karu, K. Matsuki, J. Wlodarczyk, Torification and factorization of birational m

J. Amer. Math. Soc. 15 (2002) 531–572.[3] S. Akbulut, H. King, Topology of Real Algebraic Set, in: Math. Sci. Res. Inst. Publ., vol. 25, Springer-Ve

New York, 1992.[4] S.D. Cutkosky, Monomialization of Morphisms from 3 Folds to Surfaces, in: Lecture Notes in M

vol. 1786, Springer-Verlag, Heidelberg, 2002.[5] S.D. Cutkosky, O. Piltant, Monomial resolutions of morphisms of algebraic surfaces, in: Comm. Alge

(Hartshorne Volume) (2000) 5935–5960.[6] S.D. Cutkosky, O. Piltant, Ramification of valuations, Adv. Math., in press.[7] B. Teissier, Valuations, deformations and toric geometry, in: Valuation Theory and its Applications, v

Fields Inst. Commun., AMS, 2003.

monomialization of strongly prepared morphisms from nonsingular n-folds to surfaces

Documents