Log del Pezzo surfaces with large volumes

0.24 MB·

by  Kento Fujita

LOG DEL PEZZO SURFACES WITH LARGE
VOLUMES
4
KENTO FUJITA
1
0
2

n
a Abstract. WeclassifyallofthelogdelPezzosurfacesS ofindex
J a such that the volume (−K2) is larger than or equal to 2a.
S
8

] 1. Introduction
G
A The purpose of this paper is to classify all of the log del Pezzo sur-
. faces of fixed index with large anti-canonical volumes. A normal pro-
h
t jective variety S is called a log Fano variety if S is log-terminal and the
a
m anti-canonical divisor −KS is ample. We call 2-dimensional log Fano
varieties as log del Pezzo surfaces. For a log Fano variety S, the index
[

of S is defined as the smallest positive integer a such that −aK is
1 S
v Cartier. Log del Pezzo surfaces S with fixed index a have been treated
8 by many authors in various viewpoint.
8
We recall the viewpoint in terms of the classification problem of log
5
1 del Pezzo surfaces. If a = 1 then all such S are classified by several
.
1 authors, see [Bre80, Dem80, HW81]; if a = 2 then all such S are
0 classified in [AN88, AN89, AN06, Nak07]; and if a = 3 then all such
4
S are classified in [FY14]. However, it is not realistic to classify all of
1
: the log del Pezzo surfaces of (fixed) index a ≥ 4 since even for the case
v
i a = 3 there are 300 types of such log del Pezzo surfaces (see [FY14]).
X
On the other hand, the problem of the boundedness for log Fano va-
r
a rieties of fixed index a is also important in the area of minimal model
program. The problem is equivalent to bound the anti-canonical vol-
ume (−Kn) for such S if the characteristic of the base field is zero.
S
Such problem is called the Batyrev conjecture. See [Bor01, HMX12].
We remark that the authors in [HMX12] showed that the Baryrev
conjecture is true (in any dimension, if the characteristic of the base
field is zero). Nowadays, a generalized version of the Batyrev conjec-
ture, called the BAB conjecture, is considered by many authors, see
[BB92, Ale94, AM04, Lai12, Jia13]. The BAB conjecture is true for
Date: January 9, 2014.
2010 Mathematics Subject Classification. Primary 14J26;Secondary 14E30.
Key words and phrases. del Pezzo surface, rational surface, extremal ray.
1
2 KENTO FUJITA
log del Pezzo surfaces but is open in higher-dimensional case. Note
that, as a corollary of [Jia13], any log del Pezzo surface S of index
a ≥ 2 satisfies that (−K2) ≤ (2a2+4a+2)/a and equality holds if and
S
only if S is isomorphic to the weighted projective plane P(1,1,2a).
In this paper, we classify the log del Pezzo surfaces S of index a ≥ 4
such that (−K2) ≥ 2a. The motivation is related to both the classi-
S
fication problem and the boundedness problem. We note that log del
Pezzo surfaces S of index a ≤ 3 are completely classified.
Theorem 1.1. Let S be a log del Pezzo surface of index a ≥ 4. Then
(−K2) ≥ 2a holds if and only if S is isomorphic to the log del Pezzo
S
surface associated to the a-fundamental multiplet of length b with b =
⌊(a+1)/2⌋ whose type is one of the following:
(1) hOi ((−K2) = (2a2 +4a+2)/a),
a S
(2) hIi ((−K2) = (2a2 +3a+2)/a),
a S
(3) hIIi ((−K2) = (2a2 +2a+2)/a),
a S
(4) hIIIi ((−K2) = (2a2 +a+2)/a),
a S
(5) hAi (if a = 5) ((−K2) = 54/5),
5 S
(6) hIVi ((−K2) = (2a2 +2)/a),
a S
(7) hBi (if a = 4) ((−K2) = 8),
4 S
(8) hCi (if a = 4) ((−K2) = 8).
4 S
We describe the above types in Section 4.1.
By Theorem 1.1 and the arguments in Section 4, we have the follow-
ing result. We note that we do not use the results in [Jia13] in order
to prove Corollary 1.2.
Corollary 1.2. Let S be a log del Pezzo surface of index a ≥ 2. Then
(−K2) > 2(a + 1) holds if and only if S is isomorphic to one of the
S
following (We describe the following varieties in Section 4.2.) :
(1) P(1,1,2a) ((−K2) = (2a2 +4a+2)/a),
S
(2) S ((−K2) = (2a2 +3a+2)/a),
I,a S
(3) S ((−K2) = (2a2 +2a+2)/a),
II1,a S
(4) S ((−K2) = (2a2 +2a+2)/a),
II2,a S
(5) P(1,1,3) (if a = 3) ((−K2) = 25/3).
S
We remark that all of them are toric varieties.
The strategy for the classification is essentially same as the strat-
egy in [FY14] based on the earlier work in [Nak07]. First, we reduce
the classification problem of log del Pezzo surfaces of index a to the
classification problem of the pair of its minimal resolution and (−a)
times the discriminant divisor. We call such pair an a-basic pair (see
Section 3.1). Next, by contracting (−1)-curves very carefully, we get
LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 3
an a-fundamental multiplet from an a-basic pair (see Section 3.2). See
also the flowcharts in [FY14, §1]. From the assumption (−K2) ≥ 2a,
S
thestructureoftheassociateda-fundamentalmultiplet hasveryspecial
structure. Thus we can get the multiplets in Section 4.1.
Acknowledgments. The author is partially supported by a JSPS Fel-
lowship for Young Scientists.
Notation and terminology. We work in the category of algebraic
(separated and of finite type) scheme over a fixed algebraically closed
field k of arbitrary characteristic. A variety means a reduced and irre-
ducible algebraic scheme. A surface means a two-dimensional variety.
For a normal variety X, we say that D is a Q-divisor (resp. divisor
or Z-divisor) if D is a finite sum D = a D where D are prime
P i i i
divisors and a ∈ Q (resp. a ∈ Z). For a Q-divisor D = a D ,
i i P i i
the value a is denoted by coeff D and set coeffD := {a } . For an
i Di i i
effective Q-divisor or a scheme D on X, let |D| be the support of D. A
normal variety X is called log-terminal if the canonical divisor K is
X
Q-Cartier and the discrepancy discrep(X) of X is bigger than −1 (see
[KM98, §2.3]). For a proper birational morphism f: Y → X between
normal varieties such that both K and K are Q-Cartier, we set
X Y
KY/X := X a(E0,X)E0,
E0⊂Y f-exceptional
where a(E ,X)is thediscrepancy ofE with respects toX (see[KM98,
0 0
§2.3]). (We note that if aK and aK are Cartier for a ∈ Z , then
X Y >0
aK is a Z-divisor.)
Y/X
ForanonsingularsurfaceS andaprojectivecurveC whichisaclosed
subvariety of S, the curve C is called a (−n)-curve if C is a nonsingular
rational curve and (C2) = −n. For a birational map M 99K S between
normal surfaces and for a curve C ⊂ S, the strict transform of C on
M is denoted by CM. If the birational map is of the form M 99K M ,
i j
then the strict transform of C ⊂ M on M is denoted by Ci.
j i
Let S be a nonsingular surface and let D = a D be an effective
P j j
divisor on S (a > 0) such that |D| is simple normal crossing and D ,
j i
D are intersected at most one point. (It is sufficient in our situation.)
j
The dual graph of D is defined as follows. A vertex corresponds to a
component D . Let v be the vertex corresponds to D . The v and
j j j i
v are joined by a (simple) line if and only if D , D are intersected.
j i j
In the dual graphs of divisors, a vertex corresponding to (−n)-curve is
expressed as (cid:13)n . On the other hand, an arbitrary irreducible curve is
expressed by the symbol ⊘ when it is not necessary a (−n)-curve.
4 KENTO FUJITA
Let F → P1 be a Hirzebruch surface P (O⊕O(n)) of degree n with
n P1
the P1-fibration. A section σ ⊂ F with (σ2) = −n is called a minimal
n
section. If n > 0, then such σ is unique. We usually denote a fiber of
F → P1 by l.
n
For a real number t, let ⌊t⌋ be the greatest integer not grater than t.
2. Elimination of subschemes
In this section, we recall the results in [Nak07, §2] (see also [Fuj14,
§2]). Let X be a nonsingular surface and ∆ be a zero-dimensional
subscheme of X. The defining ideal sheaf of ∆ is denoted by I .
∆
Definition 2.1. Let P be a point of ∆.
(1) Let ν (∆) := max{ν ∈ Z |I ⊂ mν}, where m is the maxi-
P >0 ∆ P P
mal ideal sheaf in O defining P. If ν (∆) = 1 for any P ∈ ∆,
X P
then we say that ∆ satisfies the (ν1)-condition.
(2) The multiplicity mult ∆ of ∆ at P is given by the length of
P
the Artinian local ring O .
∆,P
(3) The degree deg∆ of ∆ is given by mult ∆.
PP∈∆ P
Definition 2.2. Assume that ∆ satisfies the (ν1)-condition. Let V →
X be the blowing up along ∆. The elimination of ∆ is the birational
projective morphism ψ: Y → X which is defined as the composition
of the minimal resolution Y → V of V and the morphism V → X.
For any divisor E on X and for any positive integer s, we set E∆,s :=
ψ∗E −sK .
Y/X
Proposition 2.3 ([Nak07, Proposition 2.9]). (1) Assume that the
subscheme ∆ satisfies the (ν1)-condition and let ψ: Y → X be
the elimination of ∆. Then the anti-canonical divisor −K
Y
is ψ-nef. More precisely, for any P ∈ ∆ with mult ∆ =
P
k, the set-theoretic inverse image ψ−1(P) is the straight chain
k Γ of nonsingular rational curves and the dual graph of
Pj=1 P,j
ψ−1(P) is the following:
Γ Γ Γ Γ
P,1 P,2 P,k−1 P,k
(cid:13)2 (cid:13)2 (cid:13)2 (cid:13)1
(2) Conversely, for a proper birational morphism ψ: Y → X be-
tween nonsingular surfaces such that −K is ψ-nef, the mor-
Y
phismψ is the eliminationof ∆whichsatisfiesthe (ν1)-condition
defined by the ideal I := ψ O (−K ).
∆ ∗ Y Y/X
LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 5
Definition 2.4. Under the assumption of Proposition 2.3 (1), we al-
ways denote the exceptional curves of ψ over P by Γ ,...,Γ . The
P,1 P,k
order is determined as Proposition 2.3 (1).
Now we see some examples of the dual graphs of E∆,s.
Example 2.5 ([Fuj14, Example 2.5]). Assume that ∆ satisfies the
(ν1)-condition such that |∆| = {P}. Let E = eC be a divisor on X
such that P ∈ C and C is nonsingular. Let m := deg∆ and k :=
mult (∆∩C). Then we have
P
k m
E∆,s = eCY +Xi(e−s)ΓP,i + X (ek −si)ΓP,i,
i=1 i=k+1
where ψ: Y → X is the elimination of ∆. Moreover, the dual graph of
ψ−1(E) is the following:
Γ Γ Γ Γ
P,1 P,k P,m−1 P,m
(cid:13)2 (cid:13)2 (cid:13)2 (cid:13)1
⊘
CY
Example 2.6. [Fuj14, Example 2.6] Assume that ∆ satisfies the (ν1)-
condition such that |∆| = {P}. Let E = e C + e C be a non-zero
1 1 2 2
effective divisor such that C and C are nonsingular and intersect
1 2
transversally at a unique point P = C ∩ C . Let m := deg∆ and
1 2
k := mult (∆∩C ). By [Nak07, Lemma 2.12], we may assume that
j P j
k = 1. Then we have
1
k2 m
E∆,s = e1C1Y +e2C2Y +X(i(e2−s)+e1)ΓP,i+ X (e1+k2e2−is)ΓP,i,
i=1 i=k2+1
where ψ: Y → X is the elimination of ∆. Moreover, the dual graph of
ψ−1(E) is the following:
CY Γ Γ Γ Γ
1 P,1 P,k2 P,m−1 P,m
⊘ (cid:13)2 (cid:13)2 (cid:13)2 (cid:13)1
⊘ CY
2
3. Log del Pezzo surfaces
We define the notion of log del Pezzo surfaces, a-basic pairs, and
a-fundamental multiplets, and we see the correspondence among them.
6 KENTO FUJITA
3.1. Log del Pezzo surfaces and a-basic pairs.
Definition 3.1. (1) A normal projective surface S is called a log
del Pezzo surface if S is log-terminal and the anti-canonical
divisor −K is an ample Q-Cartier divisor.
S
(2) Let S be a log del Pezzo surface. The index of S is defined as
min{a ∈ Z |−aK is Cartier}.
>0 S
Remark 3.2. AnylogdelPezzo surfaceisarationalsurfaceby[Nak07,
Proposition3.6]. Inparticular, thePicardgroupPic(S)ofS isafinitely
generated and torsion-free Abelian group.
Definition 3.3 ([FY14, Definition 3.3]). Fix a ≥ 2. A pair (M,E )
M
is called an a-basic pair if the following conditions are satisfied:
(C1) M is a nonsingular projective rational surface.
(C2) E is a nonzero effective divisor on M such that coeffE ⊂
M M
{1,...,a−1} and |E | is simple normal crossing.
M
(C3) A divisor L ∼ −aK −E (called the fundamental divisor of
M M M
(M,E ))satisfiesthatK +L isnefand(K +L ·L ) > 0.
M M M M M M
(C4) For any irreducible component C ≤ E , (L ·C) = 0 holds.
M M
We see the correspondence between log del Pezzo surfaces and a-
basic pairs.
Proposition 3.4 ([FY14, Proposition 3.4]). Fix a ≥ 2.
(1) Let S be a non-Gorenstein log del Pezzo surface such that −aK
S
is Cartier. Let α: M → S be the minimal resolution of S and
let E := −aK . Then (M,E ) is an a-basic pair and the
M M/S M
divisor α∗(−aK ) is the fundamental divisor of (M,E ).
S M
(2) Let (M,E ) be an a-basic pair and L be the fundamental
M M
divisorof (M,E ). Thenthere exists aprojective andbirational
M
morphism α: M → S such that S is a non-Gorenstein log del
Pezzo surface with −aK Cartier and L ∼ α∗(−aK ) holds.
S M S
Moreover, the morphism α is the minimal resolution of S.
In particular, under the correspondence between S and (M,E ), we
M
have a(−K2) = (1/a)(L2 ), where L is the fundamental divisor.
S M M
Definition 3.5. Let a ≥ 2.
(1) For a log del Pezzo surface S of index a, the corresponding
a-basic pair (M,E ) which is given in Proposition 3.4 (1) is
M
called the associated a-basic pair of S.
(2) For an a-basic pair (M,E ), the corresponding log del Pezzo
M
surface S which is given in Proposition 3.4 (2) is called the
associated log del Pezzo surface of (M,E ). We note that a is
M
divisible by the index of S.
LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 7
We discuss that when the log del Pezzo surface S associated to an
a-basic pair (M,E ) is of index a.
M
Lemma 3.6. Let S be a log del Pezzo surface of index a ≥ 2 and
(M,E ) be the associated a-basic pair. Pick any irreducible component
M
C ≤ E and set e := coeff E and d := −(C2). Then 2 ≤ d ≤
M C M
2a/(a−e). In particular, d ≤ 2a.
Proof. By Proposition 3.4, d ≥ 2. We know that −ed ≤ (E · C) =
M
(−aK ·C) = a(2−d) since (L ·C) = 0, where L is thefundamental
M M M
divisor. Thus d ≤ 2a/(a−e). Since e ≤ a−1, we have d ≤ 2a. (cid:3)
Corollary 3.7. Let (M,E ) be an a-basic pair with a ≥ 2. Assume
M
that there exists an irreducible component C ≤ E such that either
M
(C2) < −a, or coeff E and a are coprime. Then the associated log
C M
del Pezzo surface S is of index a.
Proof. Let a′ be the index of S. Then a′ > 1 and a is divisible by a′.
Moreover, coeff E is divisible by a/a′. Thus the assertion follows
C M
(cid:3)
from Lemma 3.6.
3.2. a-fundamental multiplets. In order to determine a-basic pairs,
we define the notion of a-fundamental multiplets given in [FY14, §1].
Definition 3.8. Fix a ≥ 2 and 1 ≤ i ≤ a − 1. We define the fol-
lowing notion inductively. A multiplet (M ,E ;∆ ,...,∆ ) is called an
i i 1 i
a-pseudo-fundamental multiplet of length i if the following conditions
are satisfied:
(F1) M is a nonsingular projective rational surface and E is a
i i
nonzero effective divisor on M .
i
(F2) A divisor L ∼ −aK − E (called the fundamental divisor)
i Mi i
satisfies that iK +L is nef and ((i+1)K +L ·γ) ≥ 0 for
Mi i Mi i
any (−1)-curve γ ⊂ M .
i
(F3) ∆ ⊂ M is a zero-dimensional subscheme which satisfies the
i i
(ν1)-condition.
(F4) Let π : M → M be the elimination of ∆ and E :=
i i−1 i i i−1
E∆i,a−i. Then the following holds:
i
• If i = 1, then (M ,E ) is an a-basic pair.
0 0
• If i ≥ 2, then (M ,E ;∆ ,...,∆ ) is an a-pseudo-
i−1 i−1 1 i−1
fundamental multiplet of length i−1.
Moreover, if (i+1)K +L is not nef, then we call the multiplet an
Mi i
a-fundamental multiplet of length i.
Definition 3.9. (1) Let(M ,E ;∆ ,...,∆ )beana-pseudo-funda-
i i 1 i
mentalmultiplet. Wecallthea-basicpair(M ,E )(constructed
0 0
from the multiplet inductively) as the associated a-basic pair.
8 KENTO FUJITA
(2) We sometimes call a-basic pairs as a-pseudo-fundamental mul-
tiplets of length zero for convenience.
The following proposition is important.
Proposition3.10. Fix a ≥ 2and 0 ≤ i ≤ a−1. Let(M ,E ;∆ ,...,∆ )
i i 1 i
be an a-pseudo-fundamental multiplet of length i, π : M → M be
j j−1 j
the elimination of ∆ , E := E∆j,a−j and L be the fundamental
j j−1 j j
divisor of (M ,E ;∆ ,...,∆ ) for any 0 ≤ j ≤ i.
j j 1 j
(1) Assume that (i+1)K +L is nef. Then i ≤ a−2 and iK +L
Mi i Mi i
is nef and big. Moreover, there exists a projective and birational
morphism π : M → M between nonsingular surfaces such
i+1 i i+1
that the following conditions are satisfied:
• There exists a zero-dimensional subscheme ∆ ⊂ M
i+1 i+1
which satisfies the (ν1)-condition such that π is the elim-
i+1
ination of ∆ .
i+1
• Set E := (π ) E and L := (π ) L . Then E =
i+1 i+1 ∗ i i+1 i+1 ∗ i i
E∆i+1,a−i−1 and L = L∆i+1,i+1.
i+1 i i+1
• (M ,E ;∆ ,...,∆ ) is ana-pseudo-fundamentalmul-
i+1 i+1 1 i+1
tiplet of length i+1 and L is the fundamental divisor.
i+1
(2) Assume that (i + 1)K + L is not nef, that is, the multiplet
Mi i
(M ,E ;∆ ,...,∆ ) is an a-fundamental multiplet. Then M is
i i 1 i i
isomorphic to either P2 or F , and ((i + 1)K + L · l) < 0,
n Mi i
where l is a line (if M ≃ P2); a fiber (if M ≃ F ).
i i n
(3) L is nef and big. Moreover, we have the following:
i
• (L ·E ) = i j(a−j)deg∆ .
i i Pj=1 j
• (K +L ·L )−(K +L ·L ) = i j(j −1)deg∆ .
Mi i i M0 0 0 Pj=1 j
• (L ·C) = i jdeg(∆ ∩Cj) for any nonsingular com-
i Pj=1 j
ponent C ≤ E .
i
• (1/a)(L2) = (−K ·L )− i jdeg∆ .
0 Mi i Pj=1 j
Proof. We can assume by induction on i that L is nef and big and
i−1
L = L∆i,i if i ≥ 1. Thus L is nef and big.
i−1 i i
(1) Assume that (i+1)K +L is nef. Since a((i+1)K +L ) ∼
Mi i Mi i
−(i+1)E +(a−(i+1))L , we must have a−(i+1) > 0. Moreover,
i i
since (i+1)(iK +L ) = i((i+1)K +L )+L , iK +L is nef and
Mi i Mi i i Mi i
big. From now on, we run ((i+2)K +L )-minimal model program
Mi i
and let π : M → M be the composition of the morphisms in
i+1 i i+1
the program. More precisely, we obtain the morphism π which is
i+1
the composition of monoidal transforms such that each exceptional
(−1)-curve intersects the strict transform of (i+2)K +L negatively.
Mi i
Furthermore, ((i + 2)K + L · γ) ≥ 0 holds for any (−1)-curve
Mi+1 i+1
LOG DEL PEZZO SURFACES WITH LARGE VOLUMES 9
γ ⊂ M . Since (i+1)K +L is nef, any (−1)-curve in each step of
i+1 Mi i
themonoidaltransformintersects thestrict transformof(i+1)K +L
Mi i
trivially. Thus(i+1)K +L = π∗ ((i+1)K +L ). Inparticular,
Mi i i+1 Mi+1 i+1
−K is π -nef. By Proposition 2.3, there exists a zero-dimensional
Mi i+1
subscheme ∆ ⊂ M which satisfies the (ν1)-condition such that
i+1 i+1
π is the elimination of ∆ . Furthermore, we have L = L∆i+1,i+1
i+1 i+1 i i+1
and E = E∆i+1,a−i−1. Since each step of the ((i+2)K +L )-minimal
i i+1 Mi i
model programcan be seen as a step of (−E )-minimal model program,
i
E is nonzero effective. Thus the assertion (1) follows.
i+1
(2) Follows from [Mor82, Theorem 2.1].
(3) We know that (−K ·L ) = (−K ·L )−i(K2 ) =
Mi i Mi−1 i−1 Mi−1/KMi
(−K · L ) +ideg∆ . Thus we have (1/a)(L2) = (−K ·L ) −
Mi−1 i−1 i 0 Mi i
i jdeg∆ by induction on i. Other assertions follow similarly. (cid:3)
Pj=1 j
As a direct corollary of Proposition 3.10 (1), we have the following.
Corollary 3.11. Fix a ≥ 2. Let (M,E ) be an a-basic pair. Then
M
there exists a positive integer 1 ≤ b ≤ a − 1 and an a-fundamental
multiplet (M ,E ;∆ ,...,∆ ) of length b such that the associated a-
b b 1 b
basic pair is isomorphic to (M,E ).
M
From the next proposition, we can replace an a-fundamental multi-
pletwithanother one. Theproofissameasthatof[Nak07, Proposition
4.4]. See also [Fuj14, Proposition 3.14] and [FY14, Theorem 3.12].
Proposition 3.12. Fix a ≥ 2 and 1 ≤ b < a. Let (M ,E ;∆ ,...,∆ )
b b 1 b
be an a-fundamental multiplet of length b and L be the fundamental
b
divisor. Assume that M = F and bK +L is not big and not trivial.
b n Mb b
Thenthere existsan a-fundamentalmultiplet(M′,E′;∆ ,...,∆ ,∆′)
b b 1 b−1 b
such that the associated a-pseudo-fundamental multiplets of length b−1
are same, Mb′ = Fn′ for some n′ ≥ 0 and ∆′b ∩σ′ = ∅, where σ′ ⊂ Fn′
is a section with (σ′2) = −n′.
The next proposition is useful to know when a given multiplet is an
a-pseudo-fundamental multiplet.
Proposition 3.13. Fix 1 ≤ i < a. Let X be a nonsingular projective
surface, E be a divisor on X, L ∼ −aK − E be a divisor such that
X
iK +L is nef, ∆ ⊂ X be a zero-dimensional subscheme which satisfies
X
the (ν1)-condition, ψ: Y → X be the elimination of ∆, L := L∆,i and
Y
E := E∆,a−i. If E is effective and (L ·C) ≥ 0 for any irreducible
Y Y Y
component C ≤ E , then jK +L is nef for any 0 ≤ j ≤ i.
Y Y Y
Proof. Assume that there exists an integer 0 ≤ j ≤ i such that jK +
Y
L is non-nef. Since iK + L = ψ∗(iK + L), we have j < i. Pick
Y Y Y X
10 KENTO FUJITA
an irreducible curve B ⊂ Y such that (jK +L ·B) < 0. Then
Y Y
0 > (a−i)(jK +L ·B)
Y Y
= (a−j)(iK +L ·B)+(i−j)(E ·B) ≥ (i−j)(E ·B).
Y Y Y Y
Thus B ≤ E . Hence (L ·B) ≥ 0. However,
Y Y
0 > i(jK +L ·B) = j(iK +L ·B)+(i−j)(L ·B) ≥ 0,
Y Y Y Y Y
(cid:3)
which leads to a contradiction.
3.3. Local properties. In this section, we fix the following notation:
Let 1 ≤ i < a, (M ,E ;∆ ,...,∆ ) be an a-pseudo-fundamental multi-
i i 1 i
plet of length i, L be its fundamental divisor, π : M → M be the
i j j−1 j
elimination of ∆ , E := E∆j,a−j and L := L∆j,j for 1 ≤ j ≤ i.
j j−1 j j−1 j
Moreover, we fix a point P ∈ ∆ .
i
Lemma 3.14. (1) The multiplicity of E at P is bigger than or
i
equal to a−i.
(2) Pick any π -exceptional curve Γ ⊂ M . If (Γ2) = −1, then
i i−1
(L ·Γ) = i. If (Γ2) = −2, then (L ·Γ) = 0.
i−1 i−1
(3) Assume that E = eC around P, where P ∈ C and C is non-
i
singular and e ≤ a−i. Then e = a−i, ∆ ⊂ C around P and
i
E is the strict transform of E around over P.
i−1 i
(4) Assume that E = (a−1)C around P, where P ∈ C and C is
i
nonsingular. If2i ≤ a+1, mult (∆ ∩C) = 1 andmult ∆ ≥ 2,
P i P i
then 2i = a+1 and mult ∆ = 2.
P i
Proof. (1) The value coeff E must be nonnegative.
ΓP,1 i−1
(2) Follows from the fact (iK +L ·Γ) = 0.
Mi−1 i−1
(3) Set m := mult ∆ and k := mult (∆ ∩ C). By Example
P i P i
2.5, coeff E = e − (a − i). Thus e = a − i. If k < m, then
ΓP,1 i−1
coeff E = (a−i)(k −m) < 0 by Example 2.5, a contradiction.
ΓP,m i−1
(4) We know that coeff E = a−1−2(a−i) ≤ 0 by Example
ΓP,2 i−1
2.5. Thus 2i = a+1 and mult ∆ = 2. (cid:3)
P i
Lemma 3.15. Assume that a ≥ 4, i = 1, 1 ≤ e ≤ 2, E = (a−1)C +
1 1
eC around P, where C , C are nonsingular and intersect transver-
2 1 2
sally at P. Then e = 2 and mult ∆ = mult (∆ ∩ C ). Moreover,
P 1 P 1 2
(a,mult ∆ ) = (5,2) or (4,3).
P 1
Proof. Set m := mult ∆ and k := mult (∆ ∩C ). By Lemma 3.6,
P 1 j P 1 j
coeff E = 0. If k > 1, then k = 1 by Example 2.6. However, if
ΓP,m 0 1 2
m = k , then coeff E = e; if m > k , then coeff E = e−(a−
1 ΓP,m 0 1 ΓP,k1+1 0
1) < 0. This leads to a contradiction. Thus k = 1. If m > k , then
1 2
coeff E = a−1+ek −m(a−1) ≤ 2k −k (a−1) < 0 by Example
ΓP,m 0 2 2 2