arXiv:1304.5949v1 [math.AG] 22 Apr 2013
JUNGKAI ALFRED CHEN
Abstract. We show that 3-fold terminal flips and divisorial con-tractions may be factored into a sequence of flops, blow-downs to a smooth curve in a smooth 3-fold or divisorial contractions to points with minimal discrepancies.
1. Introduction
In birational geometry, one of the main task is to find a good model inside a birational equivalence class and study the geometry of models. This goal can be achieved by minimal model program. The minimal model conjecture asserts that for any given nonsingular or mildly sin-gular projective variety, there exists a minimal model or a Mori fiber space after a sequence of flips and divisorial contractions. Moreover, different minimal models are connected by a sequence of flops. There-fore divisorial contractions, flips and flops are the elementary birational maps of the minimal model program.
Together with some recent advances on geometry of 3-folds, for ex-ample, m-th canonical maps is birational for m ≥ 73 and the canonical volume ≥ 26601 (cf. [2, 3]), one might hope to build up an explicit clas-sification theory for 3-folds similar to the theory of surfaces by using the minimal model program explicitly. To this end, it is thus natu-ral to ask how explicit do we know about birational maps in three-dimensional minimal model program. Even though the minimal model program for 3-folds was ”proved” in more than twenty years ago by Mori and others, the more detailed and explicit description of bira-tional maps in 3-dimensional minimal model program was available only quite recently and not completely satisfactory. To give a quick tour of known results: Mori and then Cutkosky classified birational maps from a nonsingular and Gorenstein 3-fold respectively [18, 6], and Tziolas has a series of works on divisorial contractions to curves passing through Gorenstein singularities (cf. [22, 23, 24]). Divisorial contrac-tions to points are probably most well-understood mainly thanks to the work of Kawamata, Hayakawa, Markushevich and Kawakita (cf.
The author was partially supported by NCTS/TPE and National Science Council of Taiwan. We are indebted to Cascini, Hacon, Hayakawa, Kawakita, Kawamata, Koll´ar and Mori for many useful discussion. Some of this work was done during visits of the author to RIMS and Imperial College London. The author would like to thank both institutes for their hospitality.
[15, 7, 8, 9, 20, 10, 11, 12, 13, 14]). Also, the structure of flops are studied in Koll´ar’s article [16]. Flips are still quite mysterious except for some examples in [17, 1] and toric flips [21].
Instead of classifying birational maps completely, we work on the problem to factorize birational maps into a composition of simplest ones. Such factorization can be very useful for comparing various in-variants between birational models. It is also useful in classifying bi-rational maps. In the previous joint work with Christopher Hacon [4], we are able to factorize flips and divisorial contractions to curves. Our previous work [5] factorizes divisorial contractions to a point of index r > 1 with non-minimal discrepancy a
r > 1
r. The purpose of this note is
to show that one can factor threefold birational maps in minimal model program into some simple and explicit ones by combing previous work [4, 5] and considering divisorial contraction to a point of index r = 1.
Definition 1.1. A birational map f : X 99K Y is factorizable if it admits a factorization into a sequence of birational maps:
X = X0 99K X1 99K . . . 99K Xn= Y,
such that each map Xi−1 99K Xi is one of the following
(1) a divisorial contraction (or its inverse) to a point Pi ∈ Xi of
index ri ≥ 1 with minimal discrepancy;
(2) a blowup along a smooth curve in a smooth neighborhood; (3) a flop.
Theorem 1.2 (=Main Theorem). A three dimensional divisorial
con-traction f : X → W (resp. flip φ : X 99K X+) is factorizable.
Remark 1.3. Given a divisorial contraction to a point f : X → W ∋ P with exceptional divisor E. Then we can write KX = f∗KW+ aE. We
say that the contraction f has discrepancy a.
Given P a terminal singularity of index r, then the minimal discrep-ancy among all divisorial contractions to P is 1
r by [20] and [15]. If
P ∈ W is a nonsingular point, then the minimal discrepancy among all contractions to P is 2 by [10].
The key observation is that for any complicated divisorial contraction X → W (resp. flip X 99K X+), there exists singular points of index
r > 1 on X. By choosing Q ∈ X a point of higher index and choosing a divisorial contraction Y → X to the point Q ∈ X with discrepancy
1
Y ✲ Y♯ X g ❄ X♯ g♯ ❄ ❅ ❅ ❅ f ❘ ✠ f♯ W ‡
where Y 99K Y♯consists of a sequence of flips and flops, g♯is a divisorial
contraction, and f♯ is also a divisorial contraction (resp. f♯ is the
flipped map). We thus call that ‡ is a factoring diagram for X → W
(resp. X 99K X+).
If f is a weighted blowup, then the factoring diagram can be con-structed by using toric geometry and a few computation. This was the approach in [5]. In the remaining divisorial contractions which are not known to be weighted blowups, usually there is a unique non-Gorenstein singularity P ∈ X of pretty high index. By choosing a divisorial contraction g : Y → X with minimal discrepancy, one can verify that there is only a little change in the intersections. Compu-tation shows that −KY /W is nef and one can thus play the so-called
2-ray game to obtain the factoring diagram.
Moreover, by considering depth (cf. [4]) and discrepancy, one sees that Y, Y♯, X♯ has milder singularities in some sense. Our result then
follows by induction using the factoring diagram.
2. notations and preliminary
We always work on complex threefolds with Q-factorial singulari-ties (unless the image of flipping contraction). Recall that threefold terminal singularities of index 1 are isolated cDV points and terminal singularities of index r > 1 are classified by Mori (cf. [19]).
This work can be considered as a continuation of our previous work [4, 5]. We usually adapt the constructions and notations there.
Given a threefold terminal singularity P ∈ X of index r > 1, by [7, 8], there exists a partial resolution
Xn→ . . . → X1 → X0 = X †
such that Xn has Gorenstein singularities and each Xi+1→ Xi is a
di-visorial contraction to a point Pi ∈ Xi of index ri > 1 with discrepancy 1
ri. The definition of depth was introduced in [4].
dep(P ∈ X) := min{n|Xn→ X ∋ P is a partial resolution as above}.
The following properties for depth are useful.
(1) Let φ : X 99K X+ be a flip (resp. flop), then dep(X) >
dep(X+) (resp. dep(X) = dep(X+)).
(2) Let f : X → W be a divisorial contraction to a curve, then dep(X) ≥ dep(W ). Equality holds if and only if dep(X) = dep(W ) = 0.
(3) Let f : X → W be a divisorial contraction to a point, then dep(X) + 1 ≥ dep(W ).
Proof. All the statements were proved in [4, Proposition 2.15, 3.5, 3.6]
except the strict inequality for divisorial contractions to curves when dep(X) > 0. Recall that by [4], there is a factoring diagram
Y ✲ Y♯ X g ❄ X♯ g♯ ❄ ❅ ❅ ❅ f ❘ ✠ f♯ W
such that Y → X is a divisorial contraction to a higher index point Q ∈ X with r(Q ∈ X) > 1 and discrepancy 1
r, and dep(Y ) = dep(X) − 1.
Moreover, Y♯ → X♯ is a divisorial contraction to a curve and X♯ → W
is a divisorial contraction to a point.
If dep(X) = 1 and suppose that dep(W ) > 0, then dep(W ) = 1 for dep(X) ≥ dep(W ) by [4, Proposition 3.6]. Then by definition of depth, it is easy to see tat W has only one quotient singularity of type 1
2(1, 1, 1). It follows that X → W is the weighted blowup with
weights v = 12(1, 1, 1) by [15], which is absurd. We thus conclude that dep(W ) = 0 < dep(X).
In general dep(X) = d > 1, then dep(Y♯) ≤ dep(Y ) = d − 1. By
induction hypothesis, one has dep(X♯) < dep(Y♯) ≤ d − 1. It follows
that dep(W ) ≤ dep(X♯) + 1 < d by [4, Proposition 2.15].
3. divisorial contractions to curves
The purpose of this section is to factorize threefold divisorial con-traction to curves. Let f : X → W be a divisorial concon-traction to a curve Γ ⊂ W such that X has at worst terminal Gorenstein singulari-ties. By [18, 6], it is known that W is smooth near Γ and Γ ⊂ W is a lci curve. Moreover, f is the blowup along Γ.
If Γ is a nonsingular curve, then f : X → W is nothing but the blowups along Γ. If the curve Γ is singular at o, then one can factorize the divisorial contraction f : X → W by the following diagram. Proposition 3.1. Keep the notation as above. Then there is a factor-ing diagram as ‡ of birational maps such that
(1) Y 99K Y♯ consists of a sequence of flops;
(2) f♯ is the blowup along o ∈ W ;
(3) g♯ is the blowup of X♯ along Γ♯, where Γ♯ is the proper transform
of Γ in X♯;
(4) the induced map Γ♯ → Γ is isomorphic to the blowup of Γ over
o;
(5) g is a divisorial contraction to a singular point Q ∈ X of type cA with discrepancy 1.
Proof. Recall that a weighted blowup for a toric variety can be obtained
by subdivision along a primitive vector v and the exceptional divisor is the divisor corresponding to the vector v. Also a weighted blowup for a complete intersection in a toric variety is considered to be the induced map from its proper transform. For detailed description, please see [5] for example.
By shrinking W , we may assume that X is an open subset in C3,
Γ = (x3 = h(x1, x2) = 0) ⊂ W ⊂ C3 and o ∈ Γ is the only singular
point of Γ. Let τ := multoh(x1, x2) ≥ 2.
We consider towers of weighted blowups X2 πg
→ X1 πf
→ X0, where
X0 = C4, πg (resp πf) are weighted blowup along the vector v2 =
(1, 1, τ − 1, τ ) (resp v1 = (0, 0, 1, 1)). More explicitly, πf is the blowup
of X0 along Σ := (x3 = x4 = 0) and X1 is covered by two affine pieces
U3 ∪ U4. One sees also that πg is the weighted blowup over the origin
of U3 with weights (1, 1, τ − 1, 1).
We may consider an embedding W ֒→ C4 that W = (x
4−h(x1, x2) =
0). Now Γ = W ∩ Σ and the given divisorial contraction f : X → W coincides with the induced map πf |
X. On X, there is a unique
singularity Q3 of cA type locally given by x3x4−h(x1x2) = 0. Moreover,
let Y be the proper transform of X in X2. The induced map g : Y → X,
which is the weighted blowup with weights (1, 1, τ − 1, 1) over Q3, is
clearly a divisorial contraction to Q3 with discrepancy 1.
Let l := f−1(o) ∼= P1, l
Y be the proper transform of l in Y . It is easy
to see that l · KX = −1 and lY · KY = 0. We remark that there is only
one singularity on Y , which is a quotient singularity of index τ − 1 and does not contained in lY. By the same argument in [4, Theorem 3.3],
one has a factoring diagram as ‡ and a tower of divisorial contractions Y♯→ X♯ → W .
On the other hand, we may consider Y′ → X′ → W by weighted
blowup with vector v2 = (1, 1, τ − 1, τ ) and then v1 = (0, 0, 1, 1). By
the same argument as in [5, Theorem 2.7], the tower Y′ → X′ → W is
isomorphic to Y♯→ X♯→ W .
Let Γ′ be the proper transform of Γ in X′. Computation shows that
both X′ → W and Γ′ → Γ are isomorphic to the blowup over o ∈ W
and o ∈ Γ. Moreover, Y′ → X′ is the blowup along Γ′. Since the only
singularity on X′ is a quotient singularity Q′
not contains Q′
3. Therefore dep(X′) = dep(Y′) = τ − 1 = dep(Y ). It
follows that Y 99K Y′consists of a sequence of flops only by Proposition
2.1. This completes the proof.
By the above diagram successive over the singular points of Γ, one get the following consequence immediately.
Corollary 3.2. Let f : X → W be a divisorial contraction to a curve
Γ ⊂ W such that X has at worst terminal Gorenstein singularities.
The f : X → W is factorizable.
4. divisorial contractions to points
Divisorial contractions to points was intensively studied by Kawa-mata, Hayakawa, and Kawakita [15, 7, 8, 10, 11, 12, 13, 14]. We give a brief summary of the known classification.
• If f : X → W ∋ P is a divisorial contraction to a point P ∈ W of index r > 1 with discrepancy a
r ≥ 1
r, then f is completely
classified. Any of these can be realized as a weighted blowup explicitly (cf. [15, 7, 8, 13, 14]).
• If f : X → W ∋ P is a divisorial contraction to a point P ∈ W of index r = 1 with discrepancy 1.
• If f : X → W ∋ P is a divisorial contraction to a point P ∈ W of index r = 1 with discrepancy a > 1, then f is one of following cases in Table A.
Table A.
type P ∈ W discrepancy w. blowup reference
Ia nonsingular a + b Yes [10, Theorem 1.1]
Ib cA a ≥ 1 Yes [13, Theorem 1.2.i]
Ic cD a > 1, odd Yes [13, Theorem 1.2.ii.a]
Id cD a > 1 Yes [13, Theorem 1.2.ii.b]
IIa cA1 4 Yes [11, Theorem 2.5]
IIb cE7,8 2 ? [13, Table 3, e9]
IIc cE7 2 ? [13, Table 3, e5]
IId cA2, cD, cE6 3 ? [13, Table 3, e3]
IIe cD, cE6,7 2 ? [13, Table 3, e2]
IIf cD 2 ? [13, Table 3, e1]
IIg cD 4 ? [13, Table 3, e1]
The purpose of this section is to construct a factoring diagram ‡ for divisorial contraction with non-minimal discrepancy a > 1 as listed in Table A. Given a divisorial contraction with non-minimal discrepancy f : X → W ∋ P . Let E be its exceptional divisor. By the classification of [18],[6], X can not be Gorenstein. We will pick a point Q ∈ X of index p > 1.
For any divisor D on X passing through Q, we set DW = f∗D,
DY = g∗−1D to be the proper transform of D on W, Y respectively. Let
EY denotes the proper transform of E on Y . We have
f∗DW = D + c0 nE, g ∗D = D Y + q0 pF, g ∗E = E Y + q pF for some c0, q0, q ∈ Z>0.
Proposition 4.1. [5, Proposition 2.4] Let f : X → W be a divisorial
contraction to a point P ∈ W of index n with discrepancy an and E the
exceptional divisor of f . Let g : Y → X be a divisorial contraction to
a point Q ∈ E of index p with discrepancy b
p. Suppose that there is a
divisor D on X such that D ∩ E is irreducible. Then −KY /W is nef if
the following inequalities holds:
T (f, g, D) := −ac0 n2 E3+ q0qb p3 F3 ≤ 0; bc0− aq0 ≤ 0. †
In [14, Theorem 1.5], Kawakita give an affirmative answer to the General Elephant Conjecture. In particular, let f : X → W be a divisorial contraction, then a general element SX ∈ | − KX| is normal
and has only Du Val singularities.
Proposition 4.2. [5, Proposition 2.5] Let f : X → W be a divisorial
contraction to a point with exceptional divisors E and g : Y → X be a divisorial contraction to a point Q ∈ E ⊂ X of index p with discrepancy
1
p. Let F be the exceptional divisor of g. Suppose that −KY /W is nef
and there is an irreducible curve l ⊂ SX ∩ E such that lY · KY < 0,
then we have the factoring diagram ‡ such that
(1) φ : Y 99K Y♯ is a sequence of flips and flops (or just the identity
map);
(2) g♯ is a divisorial contraction contracting E Z♯;
(3) f♯ is a divisorial contraction contracting F
Y♯ to the point P ∈
W .
We will need the following variant. The proof is almost the same as [5, Corolary 2.6].
Corollary 4.3. Let f : X → W be a divisorial contraction to a point with exceptional divisors E and g : Y → X be a divisorial contraction
to a point Q ∈ E ⊂ X of index p with discrepancy 1
p. Let F be the
exceptional divisor of g. Suppose that lY · KY ≤ 0 for any irreducible
curve l ⊂ SX ∩ E and T (f, g) := −a
2
n2 E3 +
q
p3F3 < 0. Then we have a
factoring diagram ‡ as in Proposition 4.2.
An immediate but useful consequence is the following:
Corollary 4.4. Keep the notation as in Corollary 4.3. Suppose that
Suppose furthermore that q
p3F3 < 1p. Then there exists a factoring
diagram ‡ as in Proposition 4.2.
Proof. Suppose that [SX ∩ E] = [P cili] as 1-cycle for some ci ∈ Z>0.
Note that li,Y · KY ≥ li· KX for all i. Hence for all i,
li,Y · KY = li· KX + (li,Y · KY − li· KX)
≤ li· KX +Pi(li,Y · KY − li· KX)
≤ −1p + q
p3F3 < 0.
.
By Corollary 4.3, there exists a factoring diagram. We remark that once there is a factoring diagram, then the induced map f♯ : X♯ → W is a divisorial contraction to P ∈ W with exceptional
divisor FX♯ and discrepancy a := aq+n
p ∈ Z>0.
We now study the divisorial contraction to a Gorenstein point with non-minimal discrepancies case by case (cf. Table A).
Case Ia. Suppose that P ∈ W is nonsingular.
By [10], f is the weighted blowup of weight (1, m, n) with (m, n) = 1, 1 < m < n, and the discrepancy is a = m + n.
On X, the highest index point, say Q, is a terminal quotient singu-larity of type 1
n(1, m, −1). Let g : Y → X be the Kawamata blowup,
which is the weighted blowup of weights n1(t, 1, n − t), where t is the minimal positive integer satisfying mt = ns + 1. Clearly t < n, s < m.
Pick D = f−1
∗ div(x2). Then l = D ∩ E is clearly irreducible. Since
c0 = m, q0 = 1 and q = n − t, one has
T (f, g, D) = −m + n
n +
1 nt < 0.
Hence we have the factoring diagram by Proposition 4.2. By Theorem 2.7 of [5], one sees that both f♯, g♯are weighted blowups. The factoring
diagram indeed fits into the following diagram. Y −−−→99K Y♯ 1 n ywt=w2 s+t ywt=w ′ 2 Q3 ∈ X X♯∋ Q♯1 m+n ywt=w1 m+n−s−t ywt=w ′ 1 W −−−→= W where w1 = (1, m, n), w′1 = (1, m − s, n − t), w2 = n1(t, 1, n − t), w′2 = (1, s, t).
Case Ib. This contraction is described in [10, Theorem 1.2.i]. In fact, the factoring diagram is described in [5, Subsection 3.5] with n = 1. We give a brief review for reader’s convenience. The equation of P ∈ W is given by
The map f is given by weighted blowup with weight v1 = (r1, r2, a, 1).
We may write r1 + r2 = da for some d > 0 with the term xd3 ∈ ϕ.
Moreover, (a, r1) = (a, r2) = 1. Hence, there exist 0 < s∗i < ri and
0 < ai < a so that
1 + a1r1 = s∗1a;
1 + a2r2 = s∗2a.
Note that as∗
2 = 1 + a2r2 = 1 + a2(ad − r1). Therefore, a(s∗2− a2d) =
1 − a2r1. By (a, r1) = 1 and comparing it with as∗1 = 1 + a1r1, we have
a1 = −a2+ ta for some t ∈ Z. Since 0 < a1+ a2 < 2a, it follows that
a1+ a2 = a.
Suppose that r1 > 1. We have the following factoring diagram.
Y −−−→99K Y♯ 1 r1 ywt=w2 a1 ywt=w ′ 2 Q1 ∈ X X♯∋ Q♯4 a ywt=w1 a 2 ywt=w ′ 1 W −−−→= W where w1 = (r1, r2, a, 1), w1′ = (r1− s∗1, r2− a1d + s∗1, a2, 1) w2 = r11(r1− s∗1, d, 1, s∗1), w2′ = (s∗1, a1d − s∗1, a1, 1).
Suppose that r2 > 1. We have the following factoring diagram.
Y −−−→99K Y♯ 1 r2 ywt=w2 a2 ywt=w ′ 2 Q2 ∈ X X♯∋ Q♯4 a ywt=w1 a 1 ywt=w ′ 1 W −−−→= W where w1 = (r1, r2, a, 1), w1′ = (r1+ s∗2− a2d, r2− s∗2, a1, 1) w2 = r12(d, r2− s∗2, 1, s∗2), w2′ = (a2d − s∗2, s∗2, a2, 1).
Case Ic. This contraction is described in [10, Theorem 1.2.ii.a] and the discussion is parallel to the that in [5, Subsection 3.2]. The local equation of P ∈ W is given by
(ϕ : x21+ x22x4+ x1q(x23, x4) + λx2x23+ µx33+ p(x2, x3, x4) = 0) ⊂ C4,
f is the weighted blowup with weights v1 = (r + 1, r, a, 1), 2r + 1 = ad
and both a, d are odd. Notice that wtv1(ϕ) = 2r + 1 and we have that
xd
There are two quotient singularities Q1, Q2 of index r + 1, r
re-spectively. We take g : Y → X the weighted blowup with weights w2 = 1r(d, r − d, 1, d) over Q2. Then
E3 = 2r + 1 ar(r + 1), F
3 = r2
d(r − d), q= r − d, a= a − 2. In this case, we pick S = f∗−1div(x3) ∈ | − KX|, then S ∩ E is
irreducible. Now T (f, g) = 1 r(− a(2r + 1) r + 1 + 1 d) < 0.
Therefore there exists a factoring diagram by Proposition 4.2.
Y −−−→99K Y♯ 1 r ywt=w2 2 ywt=w ′ 2 Q2 ∈ X X♯∋ Q♯4 a ywt=w1 a−2 ywt=w ′ 1 W −−−→= W where w1 = v1 = (r + 1, r, a, 1), w′1 = v2 = (r + 1 − d, r − d, a − 2, 1), w2 = 1r(d, r − d, 1, d), w′2 = (d, d, 2, 1).
Case Id. In the case (1.2.ii.b), the local equation of P ∈ W is given by (P ∈ W ) ∼= o ∈ ϕ1 : x 2 1 + x2x5+ p(x2, x3, x4) = 0 ϕ2 : x2x4+ xd3 + q(x3, x4)x4+ x5 = 0 ⊂ C5,
f is a weighted blowup with weights v1 = (r + 1, r, a, 1, r + 2), and
r + 1 = ad.
There are quotient singularities Q2, Q5 of index r, r + 2 respectively.
We take g : Y → X the weighted blowup with weights w2 = r+21 (d, 2d, 1, r−
d + 2, d) over Q5. Then E3 = 2r + 2 ar(r + 2), F 3 = (r + 2)2 d(r − d + 2), q= d, a= 1. We pick D = f−1
∗ div(x2). It is easy to check that E ∩D is irreducible
but non-reduced. We have c0 = r, q0 = 2d, hence c0 − aq0 < 0 and
moreover
T (f, g, D) = 1
r + 2(−(2r + 2) + 2d
r − d + 2) < 0. Therefore there exists a factoring diagram by Proposition 4.2.
Y −−−→99K Y♯ 1 r+2 ywt=w2 a−1 ywt=w ′ 2 Q5 ∈ X X♯∋ Q♯4 a ywt=w1 1 ywt=w ′ 1 X −−−→= X where w1 = v1 = (r + 1, r, a, 1, r + 2), w2 = r+21 (d, 2d, 1, r − d + 2, d), w′ 1 = v2 = (d, d, 1, 1, d), w′ 2 = (r − d + 1, r − d, 2, a − 1, 1, r − d + 2).
Case IIa. This contraction is described in [11, Theorem 1.1.(2)]. The local equation of P ∈ W is given by
(ϕ : x1x2+ x23+ x34 = 0) ⊂ C4,
and f is the weighted blowup with weights v1 = (1, 5, 3, 2).
There is a unique singularity Q2 on E, which is a quotient
singular-ities of index 5. We take g : Y → X the weighted blowup with weights w2 = 15(4, 1, 2, 3) over Q2. Thus q = 1, a = 1 and 5q3F3 =
1 30 <
1 5.
Therefore there exists a factoring diagram by Corollary 4.4. Y −−−→99K Y♯ 1 5 ywt=w2 3 ywt=w ′ 2 Q2 ∈ X X♯∋ Q♯1 4 ywt=w1 1 ywt=w ′ 1 W −−−→= W where w1 = v1 = (1, 5, 3, 2), w′1 = v2 = (1, 1, 1, 1), w2 = 15(4, 1, 2, 3), w′2 = (1, 4, 2, 1).
Case IIb. f is of type e9 with discrepancy 2. This case was studied in [12]. We summarize some results in [12]. There are two singularities Q1, Q2 of type 15(1, 1, −1) and 13(1, 1, −1) respectively. Pick any general
elephant S ∈ | − KX|, then [S ∩ E] = 2[l], where l ∼= P1 and l passes
through both Q1, Q2 [12, Lemma 5.1]. We may assume that, near Q1,
S = div(x), E = div(y2) (after coordinate change) and l = (x = y = 0).
Now E3 = 1
15 and l · E = −1 15.
Let g : Y → X be the Kawamata blowup over Q1 with weights 1
5(1, 1, 4). One sees that q = 2, a = 1. Notice that
2lY · KY = 2l · KX + 2 53F 3 = −2 15 + 2 20 < 0.
By Proposition 4.2, there exists a factoring diagram. Y ✲ Y♯ X g 1 5 ❄ X♯ g♯ ❄ ❅ ❅ ❅a=2 f ❘ ✠ 1 f♯ W
where f♯ is a divisorial contraction with exceptional divisor F
X♯ and
discrepancy a = 1.
Case IIc. f is of type e5 with discrepancy 2.
There is only one singularity Q ∈ X, which is of type 17(1, 1, 6). Let g : Y → W be the weighted blowup of weights 1
6(1, 1, 6) over Q and let
µ : Z → Y → X ∋ Q be the economic resolution by further weighted blowups. Clearly, ( KZ = µ∗KX +P6j=1 j 7Fj; µ∗E = E Z+P6j=1 qj 7Fj,
for some qj, where F1 = F is the exceptional divisor of g. Hence
KZ = µ∗f∗KW + 2EZ+ 6 X j=1 ajFj,Z with aj = 2qj7+j ∈ Z.
Suppose that E is given by (φ : P cαβγxαyβzγ = 0) ⊂ C3/17(1, 1, 6)
locally around Q. Then
qj := min{αj + βj + γ(7 − j)|xαyβzγ ∈ φ} ≥ min{j, 7 − j}.
By [20], there must exists a exceptional divisor with discrepancy 1 centering at P ∈ W . Since Z → W is a Gorenstein partial resolution, the exceptional with discrepancy 1 must appear in Z, that is, among {Fj,Z}j=1,...,6. One can verify that F1 is the only exceptional divisor
with discrepancy 1 and q = q1 = 3. Hence pq3F3 =
1 14 <
1 7. By
Corollary 4.4, we have a factoring diagram so that f♯ : X♯ → W is a
divisorial contraction contracting FX♯ with discrepancy 1.
Case IId. f is of type e3 with discrepancy 3.
There is only one singularity Q ∈ X, which is of type cAx/4 with axial weight 2. More precisely, Q ∈ X is given by
(ϕ : x2+ y2+ f (z, u) = 0) ⊂ C4/1
4(1, 3, 1, 2), such that u3 ∈ ϕ and wt1
4(1,2)f (z, u) =
6
4. By [7, Theorem 7.4], there
is a unique divisorial contraction g : Y → X over Q with discrepancy
1
4, which is the weighted blowup of weights 1
resolution ν : Z → Y over the unique higher index point, which is a quotient singularity of index 5, and let µ := g ◦ ν : Z → X. Then we ends up with
(
KZ = µ∗KX +14F +P4j=1b4jFj;
µ∗E = E
Z+4qF +P4j=1q4jFj,
where Fj are ν-exceptional divisors and (b1, b2, b3, b4) = (2, 2, 3, 4).
Hence KZ = (f ◦ µ)∗KW + aF + 4 X j=1 ajFj,
where a = 1+3q4 and aj = bj+3q4 j. Since aj := bj+3q4 j > 1 for all j, it
follows that F is the only exceptional divisor with discrepancy 1 over W and hence q = 1 and a = 1. Thus q
p3F3 = 201 < 14. By Corollary
4.4, we have a factoring diagram such that f♯ : X♯→ W is a divisorial
contraction with exceptional divisor FX♯ and discrepancy 1.
Case IIe. f is of type e2 with discrepancy 2.
There is a unique higher index point Q ∈ X of type cA/r or cD/3 with axial weight 2.
Subcase 1. Q is of type cD/3.
Let µ : Z → X be a common resolutions of Q dominating all divisorial contractions with minimal discrepancies over Q. We have
KZ = µ∗KX + N X j=1 1 3Fj + Xcl 3Gl,
where {Fj}j=1,...,N is the set all all exceptional divisors with discrepancy 1
3 over Q and cl ≥ 2. Suppose that µ
∗E = E Z+Pq3jFj+Pt3lGl, then KX = µ∗f∗KW + 2EZ+ N X j=1 ajFj + X blGl,
where aj = 2qj3+1 and bl = 2tl3+cl > 1. Since there exists an exceptional
divisor with discrepancy 1 over P ∈ W , we may assume that a1 = 1.
By [9, Section 9], a cD/3 point can be classified as cD/3-1, cD/3-2 and cD/3-3. Unless Q ∈ X is of type cD/3-3 and Equation ∗ holds (cf. [9, p.549]), we know that any exceptional divisor with minimal discrepancy 13 over a cD/3 point is obtained by a divisorial contraction. Hence there is a divisorial contraction g : Y → X with exceptional divisor F = F1 and discrepancy 13. We thus have q = 1 and a = 1.
It is also straightforward to check that q 33F3 =
1
12 for any such
di-visorial contraction with discrepancy 1
3. By Corollary 4.4, we have a
factoring diagram such that f♯ : X♯ → W is a divisorial contraction
In the remaining situation that Q ∈ X is of type cD/3-3 and Equa-tion ∗ holds (cf. [9, p.549]), then there is only one divisorial contracEqua-tion g : Y → X, which is a weighted blowup with weights v2 = 13(5, 4, 1, 6).
There is another valuation with discrepancy 13 given by the weighted blowup with weights v1 = 14(2, 4, 1, 3). We write KZ = µ∗KX + 13F1 +
1 3F2+ Pcl 3Gl, and KZ = µ∗f∗KW + 2EZ + a1F1+ a2F2+ X blGl,
where Fi corresponds to the valuation with weights vi for i = 1, 2.
Let (φ = 0) ⊂ C3/1
3(2, 1, 1, 0) be the local equation of E near Q.
Since a1 = 1, then q1 = 1 and q31 = wtv1(φ) =
1
3. One sees that φ
contains z. It follows that q2
3 = wtv2(φ) =
1
3 and hence q = 1 and a = 1
holds.
Now we have q 33F3 =
1
10. By Corollary 4.4 again, we have a
factor-ing diagram such that f♯ : X♯ → W is a divisorial contraction with
exceptional divisor FX♯ and discrepancy 1.
Subcase 2. Q is of type cA/r.
After coordinate changes, we may assume that local equation near Q is given by (ϕ : xy + ztr + u2 = 0) ⊂ C4/1
r(1, −1, 2, r) for some t ≥ 2.
Set r = 2k + 1. Let Y → X be the weighted blowup with weights v1 := 2k+11 (k + 1, 3k + 1, 1, 2k + 1) with exceptional divisor F . There
are quotient singularities R1, R2 of index k + 1, 3k + 1. Let Z → Y be
the economic resolution of R1, R2. Then we have
KZ = µ∗KX +2k+11 F +Pkj=1 2k+12j Fj+ Pk i=1(2k+12i+1G0i+ 2i 2k+1G1i+ 2i−1 2k+1G2i).
More explicitly, the resolution over R1 is obtained by weighted blowups
of weights k+11 (j, 2k + 2 − 2j, j, k + 1 − j) for 1 ≤ j ≤ k. Over Q these weights corresponds to vectors 2k+11 (j, 4k + 2 − j, 2j, 2k + 1). Similarly, the resolution over R2 is obtained by weighted blowups of
weights 3k+11 (2i, 3k + 1 − i, 3i, i),3k+11 (2k + 2i, 2k + 1 − i, 3i − 1, k + i),
and 1
3k+1(4k + 2i, k + 1 − i, 3i − 2, 2k + i) for 1 ≤ i ≤ k. Over Q, these
weights corresponds to vectors 1 2k+1(k + 1 + i, 3k + 1 − i, 2i + 1, 2k + 1), 1 2k+1(2k + 1 + i, 2k + 1 − i, 2i, 2k + 1), 1 2k+1(3k + 1 + i, k + 1 − i, 2i − 1, 2k + 1). for 1 ≤ i ≤ k respectively.
Suppose that E is given by (φ :P cαβγδxαyβzγuδ = 0) ⊂ C4/1r(1, −1, 2, r)
locally around Q. We write µ∗E = E
Z + 2k+1q F + Pkj=1 qj 2k+1Fj + Pk i=1(2k+1t0i G0i+ t1i 2k+1G1i+ t2i 2k+1G2i) and hence KZ = µ∗f∗KW + 2EZ+ aF + k X j=1 ajFj+ k X i=1 (b0iG0i+ b1iG1i+ b2iG2i),
with a := 2k+12q+1, aj := 2q2k+1j+2j , b0i := 2t0i2k+1+2i+1, b1i := 2t2k+11i+2i, b2i := 2t2i+2i−1
2k+1 . There exists an exceptional divisor with discrepancy 1. Hence
either a, b0i or b2i = 1 for some i because aj and b1i are even.
Claim. a= 1.
Suppose that b0i= 1 for some i. Then t0i = k − i. Since
t01= min{α(k+1+i)+β(3k+1−i)+γ(2i+1)+δ(2k+1)|xαyβzγuδ∈ φ}.
It follows that φ contains zγ with γ(2i + 1) = k − i. Hence
q 2k + 1 = wtv1φ ≤ k − i 2k + 1 ≤ k − 1 2k + 1 and a < 1, a contradiction.
Suppose that b2i = 1 for some i. Then similarly, one sees that
φ contains zγ with γ(2i − 1) = k − i + 1. This leads to the same
contradiction unless b21 = 1 and φ contains zk. It follows that q = k
and a = 1 in this situation.
Now q
(2k+1)3F3 = (k+1)(3k+1)(2k+1)2k < 2k+11 . By Corollary 4.4, there
is a factoring diagram such that f♯ is a divisorial contraction with
discrepancy a = 1.
Case IIf. f is of type e1 with discrepancy 2.
In this case, there is a unique higher point Q of type 1r(1, −1, 4). Subcase 1. r = 4k + 3.
Let Y → X be the Kawamata blowup along Q with weights 4k+31 (k + 1, 3k + 2, 1). Suppose that the local equation of E near Q is given by (φ : P cαβγxαyβzγ = 0). Let µ : Z → X be the economic resolution
over Q, which factors through Y . Then we have ( KZ = µ∗KX +P4k+2j=1 j 4k+3Fj; µ∗E = E Z+P4k+2j=1 qj 4k+3Fj, where F1 = F and qj := min{α(k + 1)j + β(3k + 2)j + γj|xαyβzγ ∈ φ}.
We have KZ = g∗f∗KW+ 2EZ+Pj=14k+2ajFj with aj = 2q4k+3j+j ∈ Z. Note
that aj ≡ j (mod 2) and aj = 1 for some j.
Claim. a1 ≤ 3.
Suppose on the contrary that a1 ≥ 5. For all monomial xαyβzγ ∈ φ,
we have
q1 = α(k + 1) + β(3k + 2) + γ ≥ 10k + 7. †
If aj = 1 for some j, then
qj = 2k − 2s + 1 = (k + s + 1)α + (3k − s + 2)β + (4s + 1)γ, if j = 4s + 1;2k − 2s = (3k + s + 3)α + (k − s)β + (4s + 3)γ, if j = 4s + 3,
for some xαyβzγ∈ φ, which is a contradiction to †. This completes the
Notice that if a1 = 3, i.e. q1 = 6k + 4, then y2 ∈ φ and aj = 1 if
and only if j = 4s + 3 with s < k. In this case, there are exactly k − 1 exceptional divisors with discrepancy 1. Hence k ≥ 2 in this situation. Also, if a1 = 1, then q1 = 2k + 1. Thus in any event,
q (4k + 3)3F 3 = 2q (k + 1)(3k + 2)(4k + 3) ≤ 4 3(4k + 3). For any l ⊂ S ∩ E, one has l · E ≥ 1
4k+3 and hence l · KX ≤ −2 4k+3.
Therefore, lY · KY < 0 for all i. Hence there exists a factoring diagram
by Corollary 4.3. The resulting divisorial contraction f♯ : X♯ → W is
a divisorial contraction with discrepancy 1 or 3. Subcase 2. r = 4k + 1.
Similarly, let Y → X be the Kawamata blowup along Q with weights
1
4k+1(3k + 1, k, 1) and µ : Z → X be the economic resolution over Q,
which factors through Y .
Thus we have KZ = g∗f∗KW+ 2EZ+Pj=14k ajFj with aj = 2q4k+1j+j ∈ Z
and
qj := min{α(3k + 1)j + βkj + γj|xαyβzγ ∈ φ}.
Note that aj ≡ j (mod 2) and aj = 1 for some j.
Claim. a1 = 1.
Suppose on the other hand that a1 ≥ 3. For all monomial xαyβzγ ∈ φ,
we have
q1 = α(3k + 1) + βk + γ ≥ 6k + 1. †
Suppose that aj = 1, it is straightforward to see that
qj = 2k − 2s + 1 = (k + s)α + (3k − s + 1)β + (4s − 1)γ, if j = 4s − 1;2k − 2s = (3k + s + 1)α + (k − s)β + (4s + 1)γ, if j = 4s + 1,
for some xαyβzγ ∈ φ, which is a contradiction to †. The Claim now
follows.
Now a = a1 = 1, q = 2k and thus
q (4k + 1)3F 3 = 4 (3k + 1)(4k + 1) ≤ 1 4k + 1.
For any l ⊂ S ∩ E, one has l · E ≥ 4k+11 and hence l · KX ≤ 4k+1−2 .
Therefore, lY · KY < 0 for all i. Hence there exists a factoring diagram
by Corollary 4.3. The resulting map f♯ : X♯ → W is a divisorial
contraction with discrepancy 1.
Case IIg. f is of type e1 with discrepancy 4.
In this case, there is a unique higher index point Q of type 1r(1, −1, 8). One can work out this case similar to Case IIf.
Subcase 1. r = 8k + 7.
Let Y → X be the Kawamata blowup along Q with weights 1 8k+7(k +
1, 7k + 6, 1) and µ : Z → X be the economic resolution over Q, which factors through Y . Suppose that the local equation of E near Q is
given by (φ : P cαβγxαyβzγ = 0). Thus we have KZ = µ∗f∗KW +
4EZ+P8k+6j=1 ajFj with aj = 4q8k+7j+j ∈ Z and
qj := min{α(k + 1)j + β(7k + 6)j + γj|xαyβzγ ∈ φ}.
Note that aj ≡ −j (mod 4) and aj = 1 for some j.
Claim. a1 = 3 or 7. 1
Suppose on the contrary that a1 ≥ 11. For all monomial xαyβzγ ∈ φ,
we have
q1 ≥ α(k + 1) + β(7k + 6) + γ ≥ 22k + 19. †
Suppose that aj = 1, it is straightforward to see that
qj = 2k − 2s + 1 = (3k + s + 3)α + (5k − s + 4)β + (8s + 3)γ, if j = 8s + 3;2k − 2s = (7k + s + 1)α + (k − s)β + (8s + 7)γ, if j = 8s + 7,
for some xαyβzγ ∈ φ, which is a contradiction to †. The Claim now
follows.
Now q ≤ 14k + 12 and thus q (8k + 7)3F 3 = 2q (k + 1)(7k + 6)(8k + 7) ≤ 4 (k + 1)(8k + 7). For any li ⊂ S ∩ E, one has li · E ≥ 8k+71 and hence li · KX ≤ 8k+7−4 .
Therefore, li,Y · KY ≤ 0 for all i and strictly < 0 for some i. Hence
there exists a factoring diagram by Proposition 4.3. The resulting map f♯ : X♯→ W is a divisorial contraction with discrepancy 3 or 7.
Subcase 2. r = 8k + 5.2
Similar argument shows that a1 = 1 or 5 ( since a1 ≡ 1 (mod 4)) and
there exists a factoring diagram by Corollary 4.3. The resulting map f♯ : X♯→ W is a divisorial contraction with discrepancy 1 or 5.
Subcase 3. r = 8k + 3.
Similar argument shows that a1 = 3 ( since a1 ≡ −1 (mod 4)) and
there exists a factoring diagram by Proposition 4.3. The resulting map f♯ : X♯→ W is a divisorial contraction with discrepancy 3.
Subcase 4. r = 8k + 1.
Similar argument shows that a1 = 1 ( since a1 ≡ 1 (mod 4)) and
there exists a factoring diagram by Proposition 4.3. The resulting map f♯ : X♯→ W is a divisorial contraction with discrepancy 3.
5. proof of the main theorem
Proof. We prove by induction on depth and discrepancies.
1if a
1= 7, then y2∈ φ and aj = 1 if and only if j = 8s + 3 with s < k. In this
case, there are exactly k − 1 exceptional divisors with discrepancy 1.
2if a
1= 5, then y2∈ φ and aj = 1 if and only if j = 8s + 5 with s < k. In this
1. Suppose first that dep(X) = 0, that is, X has at worst Gorenstein terminal singularities. By the classification of Mori and Cutkosky [18, 6], f can not be a flipping contraction.
If f : X → W is a divisorial contraction to a point then f is a divisorial contraction with minimal discrepancy (cf. [18, 6]).
If f : X → W be a divisorial contraction to a curve, then f is a blowup along a lci curve in a smooth neighborhood by the classification of Mori and Cutkosky again. By Proposition 3.1, f is factorizable.
2. Let f : X → W be a divisorial contraction to a curve Γ with dep(X) = d > 0. By [4], there is a factoring diagram
Y ✲ Y♯ X g ❄ X♯ g♯ ❄ ❅ ❅ ❅ f ❘ ✠ f♯ W satisfying:
(1) Y → X is a divisorial contraction to a highest index point of index r > 1 with discrepancy 1
r;
(2) Y → Y♯ is a sequence of flips and flops;
(3) g♯ : Y♯ → X♯ is divisorial contraction to the proper transform
of Γ;
(4) f♯ is a divisorial contraction to a point.
Note that dep(Y ) = d −1, and dep(Y♯) ≤ dep(Y ) = d −1. Therefore
by Proposition 2.1,
dep(X♯) ≤ min(0, dep(Y♯) − 1) < d.
It follows that X → W can be factored into X 99K Y 99K Y♯ → X♯ → W so that each map is factorizable by induction on depth.
3. Let f : X → W be a flipping contraction. By [4], there is a factoring diagram as above so that f♯ : X♯ = X+ → W is the flipped
contraction. Similarly, each map of
X 99K Y 99K Y♯→ X♯ = X+ is factorizable by induction on depth.
4. Let f : X → W be a divisorial contraction to a point P ∈ W of index r with dep(X) = d and discrepancy 1r. Nothing to do.
5. Let f : X → W be a divisorial contraction to a point P ∈ W of index r > 1 with dep(X) = d and discrepancy ar > 1r. By [5], there is a factoring diagram satisfying:
(1) Y → X is a divisorial contraction to a highest index point of index r > 1 with discrepancy 1
r;
(2) Y → Y♯ is a sequence of flips and flops;
(3) f♯ is a divisorial contraction with discrepancy a′
r < a r;
(4) g♯ is divisorial contraction to a point Q of index r with
dis-crepancy ar′′ < ar and a′′ + a′ = a if P ∈ W is not of type
cE/2;
(5) g♯ is divisorial contraction to a point Q of index 3 with
discrep-ancy 13 if P ∈ W is of type cE/2.
Notice that dep(Y♯) ≤ dep(Y ) = d − 1 and dep(X♯) ≤ dep(Y♯) +
1 ≤ d. By induction on depth, both Y 99K Y♯ and Y♯ → X♯ are
factorizable. If dep(X♯) < dep(X), then we are done by induction.
If dep(X♯) = dep(X), then we may proceed by induction on a which
measures the discrepancy.
6. Let f : X → W be a divisorial contraction to a point P ∈ W of index 1 with dep(X) = d and discrepancy a > 1.
6.1 If P ∈ W is a non-singular point, then by the study of Case Ia, f is factorizable by induction on a.
6.2 If P ∈ W is of type cA. By the studies in Case Ib, IIa, and IId, there exists a factoring diagram such that f♯ : X♯ → W has
discrepancy a1 < a (Case Ib) or 1 (Case IIa, IId). Moreover dep(X♯) ≤
d. Therefore, f♯ is factorizable by induction on discrepancy a hence so
is f : X → W because Y 99K Y♯ → X♯ having dep < d.
6.3 If P ∈ W is of type cD or cE and the discrepancy a is odd. This could be Case Ic, Id, IId. There exists a factoring diagram such that f♯ : X♯→ W has discrepancy a
2 < a (Case Ic) or 1 (Case Id, IId).
Similarly f is factorizable by induction on a and on depth.
6.4If P ∈ W is of type cD or cE and the discrepancy a is even. This could be Case Id, IIb, IIc, IIe, IIf, and IIg. There exists a factoring diagram such that f♯: X♯ → W has odd discrepancy a
1 (Case IIf, IIg)
or 1 (other cases). Therefore, f is factorizable by 6.3 and induction on depth.
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National Center for Theoretical Sciences (NCTS/TPE) and Department of Math-ematics, National Taiwan University, Taipei, 106, Taiwan
