1
THE STRUCTURE AND SLOPE OF BIELLIPTIC
FIBRATIONS
Miguel A. BARJA
1
Depa amen de Ma ema ica Aplicada I
ETSEIB. Uni e si a Poli ecnica de Ca alunya
Diagonal 647, 08028 Ba celona, Spain
e-mail: b[email p o ec ed] c.es
1991 Ma hema ics Sub jec Classica ion: P ima y 14H10, Secunda y 14J29
A la memo ia de Fe nando
0 In o duc ion
Le
:
S
;!
B
be a
b a ion
, i.e. a su jec i e mo phism wi h connec ed b es, om
a smo o h su ace
S
on o a smo o h cu e
B
. A b a ion is said o b e
ela i ely minimal
when i has no e ical (
;
1)-cu e. Le
g
deno e he genus o a gene al b e and
b
he
genus o
B
.
Le
!
S=B
=
!
S
(
!
;
1
B
) b e he ela i e canonical bundle and le (
):= deg
(
!
S=B
).
I is known ha (
)
0 and ha (
) = 0 i and only i
is lo cally i ial. Assume
is no lo cally i ial. Then we dene he
slop e
o
as
(
):=
!
2
S=B
.
(
)
(see 19]). The e a e se e al esul s on he lowe slop e o ela i ely minimal b a ions o
genus
g
2. Fi s o all weha e
4
;
4
g
(see 8], 12], 13], 18] o he hyp e ellip ic
case and 19] o he gene al case) and equali y holds only in he hyp e ellip ic case (9]).
The e a e imp o emen s in he non-hyp e ellip ic case o
g
5 (see 4], 7], 9], 11], 14])
bu he p esen ly known echniques seem o ha e some limi a ions o ex end hese esul s
o highe genus.
Recen ly Konno is ying o nd go o d b ounds dep ending on some ex a nume ical
in a ian s o he gene al b e, such as he Clio d index. In 10], Konno nds b e e
b ounds o igonal and plane quin ic b a ions (so Clio d index 1), al hough hey do
no seem o b e sha p. Also in 11] he ge s gene al b ounds dep ending on he Clio d index
in some cases.
1
Pa ially supp o ed by CICYT PS93-0790 and HCM p o jec n.ERBCHRXCT-940557
2
MIGUEL A. BARJA
In his pap e we deal wi h he case o biellip ic b a ions (i.e., when he gene al b e
has a 2- o-1 map on o an ellip ic cu e). In he s udy o gene al b a ions i u ns ou
o b e ele an he p op e ies o he canonical emb eding o he gene al b e wi h esp ec
o quad ics. A s example is he igonal case whe e Konno uses he ac ha sucha
cu e lies in a a ional no mal sc oll which is he in e sec ion o he quad ics con aining
i . In ou case he ole o he quad ics h ough he canonical biellip ic cu e can no b e
used di ec ly. A die en app oach using ela i e B ill-No e he lo ci shows
Theo em 2.4
Le
:
S
;!
B
be a biel lip ic b a ion o genus
g
6
. Then
S
is,
bi a ional ly, a double co e o an el lip ic smoo h su ace
V
o e
B
.
In chap e 3 we p o e ha he conclusions o he ab o e heo em hold o genus 5
biellip ic b a ions
a e a base change
and wegi e an example whe e a non i ial base
change is needed.
As a by-p o duc we ob ain ha e e y smo o h biellip ic b a ion o genus
g
5is
iso i ial (see p op osi ion 2.6 and co olla y 3.3).
Finally, using heo em 2.4 and canonical esolu ion o singula i ies o double co e s
we ge he ollowing sha p b ound o he slop e o biellip ic b a ions
Theo em 4.1
Le
:
S
;!
B
bea ela i ely minimal biel lip ic b a ion o genus
g
6
.
Le
V
be he ela i e minimal model o he el lip ic b a ion ob ainedin heo em 2.4. Then
(a)
(
)
4+
2(
g
;
5)
XO
V
(
)
4
.
(b)
(
)= 4
i and only i
S
is he minimal desingula iza ion o a double co e
S
0
;!
V
o asmoo h el lip ic su ace such ha
All he b es o he el lip ic b a ion
:
V
;!
B
a esmoo h and isomo -
phic.
The b anch di iso o he double co e has only
negligeable
singula i ies.
In pa icula , he bound is sha p.
The au ho wan o hank, among o he s, p o esso Juan Ca los Na anjo o his en-
cou agemen and in e es ing commen s chap e 3 g ew om ui ul con e sa ions wi h
him.
Du ing he nal e ision o his pap e he ad iso o he au ho , p o esso Fe nando
Se ano, passed away.Iwould like o hank him hea ully o his supp o and con inuous
help, no only du ing he p epa a ion o his wo k bu also du ing he las yea s when I
enjoyed his eachings and iendship.
No a ions and con en ions
All h oughou his pap e wewo k o e he eld o complex numb e s
C
.
I
X
is a scheme and
G
is a shea o g aded algeb as and
E
is a lo cally ee cohe en
shea on
X
we dene P o j
G
and
P
(
E
) as in 5],I I.7.
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS
3
1 Some gene ali ies on b ed su aces
We ecall some basic ac s ab ou b ed su aces (see 9]). The ollowing a e some easy
bu use ul esul s.
Lemma 1.1
Le
X
be a smoo h a ie y and
'
:
X
;!
B
a mo phism on o a cu e. Le
X
be he b eo
'
o e
2
B
.
Fo any cohe en shea
F
on
X
and o any
a
2
Pic
(
B
)
le
F
(
a
)=
F
'
(
a
)
.
Suppose ha
a
is ample enough and ha
F
sa ises he ol lowing echnical condi ion:
o gene al
2
B
he sequence
0
;!F
(
;
)
;!F ;!F
j
X
;!
0
is exac .
Then, he na u al mo phism
H
0
(
X
F
(
a
))
;!
H
0
(
X
F
j
X
)
is su jec i e o gene al
2
B
.
P o o .
Conside he exac sequence
0
;!F
(
a
;
)
;!F
(
a
)
;!F
j
X
;!
0
o gene al
2
B
.Taking cohomology we ge
0
;!
H
0
(
X
F
(
a
;
))
;!
H
0
(
X
F
(
a
))
m
;!
H
0
(
X
F
j
X
)
I
a
is ample enough hen
h
1
(
B
(
'
F
)
(
a
;
)) = 0 and hen
h
0
(
X
F
(
a
;
)) =
h
0
(
B
(
'
F
)
(
a
;
)) do es no dep end on
by he Hi zeb uch-Riemann-Ro ch heo em
o cohe en shea es on
B
.
Fu he mo e
dim Im (
m
) =
h
0
(
X
F
(
a
))
;
h
0
(
X
F
(
a
;
)) =
=
h
0
(
B
(
'
F
)
a
)
;
h
0
(
B
(
'
F
)
(
a
;
)) =
=
d
+
(
a
+1
;
b
)
;
(
d
+
(
a
;
1+1
;
b
)) =
whe e
d
=deg
'
F
a
=deg
a
= ank (
'
F
)=
h
0
(
X
F
j
X
) o
2
B
gene al
2
Lemma 1.2
Le
:
S
;!
B
beab a ion. Le
L2
Pic
S
. I
a
2
Pic
B
is ample
enough, hen he na u al map
h
:
L
(
a
)
;!L
(
a
)
is an epimo phism jus excep a he base poin s o he linea sys em
jL
(
a
)
j
. Mo eo e , i
o a gene al b e
F
o
, he linea sys em
jL
j
F
j
is base-poin ee, hen such base poin s
a econcen a ed on a ni e numbe o b es.
4
MIGUEL A. BARJA
P o o .
F om he sequence o maps
S
;!
B
;!
Sp ec
C
we can conside he ollowing
na u al commu a i e diag am
e
L
(
a
)
k
=(
)
(
)
L
(
a
)=
H
0
(
S
L
(
a
))
C
O
S
/
/
L
(
a
)
s
s
h
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
L
(
a
)
Since
k
is su jec i e o
a
ample enough, i ollows ha su jec i i yo
h
is equi alen
o su jec i i yo
e
. This ails o b e an epimo phism p ecisely a he base p oin s o
jL
(
a
)
j
.
Finally, using lemma 1.1 one has ha
jL
(
a
)
j
has no base p oin s on a gene al b e
F
o
.
2
Then we can cons uc he ela i e canonical mo phism o
S
. Conside
L
=
!
S=B
=
!
S
!
;
1
B
hen
L
j
F
=
!
F
which has no base p oin s o smo o h
F
i
g
2.
L
(
a
)has
only base p oin s on singula b es o
, o
a
ample enough by lemma 1.2 and he na u al
map
h
:
L
(
a
)
;!L
(
a
)(1)
is an epimo phism away om such a base p oin s.
In ac weha e
e
h
:(
L
(
a
))
L
(
;
a
)
;!O
S
(2)
wi h Im
e
h
=
I
Z
=
O
S
(
;
R
)
I
;
b eing he ideal shea o such base p oin s, endowed wi h
some scheme s c u e, whe e
O
S
(
;
R
) is he ideal shea o i s di iso ial pa and
I
;
is he
ideal shea o i s disc e e pa (see 14]).
Conside a sequence o blow-ups
:
e
S
;!
S
such ha he base lo cus o
j
L
(
a
)
j
is o
pu e co dimension 1. Then, in pa icula weha e an epimo phism
I
;
;!
;
1
I
;
O
e
S
=
O
e
S
(
;
E
)
whe e
E
is a di iso on
e
S
excep ional wi h esp ec o
.
Thus om
L
(
a
)
;!L
(
a
)
O
S
(
;
R
)
I
;
(epimo phism) wege
(
L
(
a
))
;!
L
(
a
)
O
S
(
;
R
)
I
;
;!
L
(
a
)
O
S
(
;
R
)
O
e
S
(
;
E
)(3)
which is also an epimo phism.
We shall call
O
e
S
(
M
(
a
)) =
L
(
a
)
O
S
(
;
R
)
O
e
S
(
;
E
) he
mo ing pa
o
L
(
a
)
O
e
S
(
Z
(
a
)) =
O
S
(
R
)
O
e
S
(
E
) he
xed pa
o
L
(
a
)
(as one can easily see (
)
(
)
O
e
S
(
M
(
a
))
;!O
e
S
(
M
(
a
)) is an epimo phism so, in
iew o lemma 1.2,
j
M
(
a
)
j
hasnobasepoin s).
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS
5
Then (3) leads o
e
S
=
=
=
=
=
=
=
=
S
/
/
___
P
/
/
=
P
w
w
'
p
p
p
p
p
p
p
p
p
p
p
p
p
p
B
whe e
P
=
P
B
(
L
(
a
))
P
=
P
B
(
L
)
is he na u al isomo phism
such ha
O
P
(1) =
O
e
S
(
M
(
a
)). I wecall
=
wege
(
O
P
(1)
'
(
a
)) =
O
e
S
(
M
(
a
))
:
We shall call he image o
e
S
by
. In ac is he closu e in
P
o he image o he
bi a ional map induced on
S
by
h
in (1), so do esn' dep end on
a
.Thenwecan change
a
2
Pic
B
i needed. We shall call
he ela i e canonical image o
S
and
he ela i e
canonical map
.
We ema k ha o
F
S
a smo o h b e o
,
F
=
(
F
)
=
(
(
F
)), so we deno e
i always by
F
i no con usion a ises.
Fo
2
B
wecall
P
=
'
;
1
(
)
=
P
g
;
1
.
We shall call
O
P
(
T
) he au ological line bundle on
P
.
Rema k 1.3
Conside he shea
F
=
I
P
O
P
(2
T
)
'
(2
a
), whe e
I
P
is he ideal shea
o in
P
.Fo
2
B
weha e
F
j
P
=
I
F
P
(2). We claim ha
F
e ies he hyp o hesis o
lemma 1.1 and so ha o
a
ample enough weha e an epimo phism
H
0
(
P
I
P
O
P
(2
T
)
'
(2
a
))
;!
H
0
(
P
g
;
1
I
F
P
(2))
:
Indeed, jus conside
0 0 0
?
?
?
?
?
y
?
?
?
?
?
y
?
?
?
?
?
y
0
;;;! I
P
'
(
;
)
;;;! O
P
(
'
(
;
))
;;;! O
(
'
(
;
))
;;;!
0
?
?
?
?
?
y
?
?
?
?
?
y
?
?
?
?
?
y
0
;;;! I
P
;;;! O
P
;;;! O
;;;!
0
?
?
?
?
?
y
?
?
?
?
?
y
?
?
?
?
?
y
0
;;;! I
F
P
;;;! O
P
;;;! O
F
;;;!
0
?
?
?
?
?
y
?
?
?
?
?
y
?
?
?
?
?
y
0 0 0
6
MIGUEL A. BARJA
wi h he h ee ows and he wo igh columns i ially exac . Then he snake lemma
makes he le hand side column exac . Now jus enso wi h
O
P
(2
T
), which is lo cally
ee.
2 Biellip ic b a ions o genus
g
6
F om nowon we conside biellip ic b a ions only, i.e. b a ions such ha he gene al
b e admi s a 2- o-1 mo phism on o an ellip ic cu e. I seems easonable o hink ha
all hose maps can b e glued oge he o yield a global 2- o-1 map om
S
on o an ellip ic
su ace o e
B
.
A simila case, e y much s udied (see 8], 12], 13], 18]) is ha o hyp e ellip ic b a-
ions, i.e. when b es a e hyp e ellip ic. He e he hyp e ellip ic in olu ions glue oge he o
yield a bi a ional double co e o a uled su ace in an easy way: jus conside he ela i e
canonical mo phism o
:
S
;!
B
. The case whe e he gene al b es a e igonal also
globalizes o a 3- o-1 map om
S
on o a uled su ace o e
B
( his ac is implici in 10]).
Wep o ein his chap e ha such a double co e ac ually exis s o any biellip ic
b a ion o genus a leas 6.
To see his we need, s o all, a canonical way o cons uc ing he double co e o an
ellip ic cu e om
F
. Recall (1] Ch.IV) ha o any smo o h cu e
F
weha ea a ie y
W
d
(
F
) inside Pic
d
(
F
) pa ame izing he comple e linea se ies on
F
o deg ee
d
and
dimension a leas
.
Gi en a biellip ic in olu ion
:
F
;!
E
he e is a na u al wayo embedding
E
in
W
1
4
(
F
). This map is gi en by he comp osi ion o
wi h he hyp e ellip ic in olu ions
coming om
E
.
On he o he hand
W
0
4
(
F
), which has dimension ou , is singula p ecisely along
W
1
4
(
F
)
i
g
5 (see 1] p.160). Take
L
2
W
1
4
(
F
) we deno e
T
L
=
P
T
L
W
0
4
(
F
) he p o jec i ized
angen cone o
W
0
4
(
F
)a
L
,which s canonically in
P
(
H
0
(
F !
F
)
)
=
P
g
;
1
.I
g
5
weha eby he Kemp 's Singula i y Theo em (1] p.241) ha
T
L
is a h ee old o deg ee
(
g
;
3) in
P
g
;
1
con aining
F
.
Fo any eec i e di iso
D
on
F
wedeno eby
<D>
he linea subspace o
P
(
H
0
(
F !
F
)
)
spanned by
D
, in he sense o 1] p.12.
Lemma 2.1
Le
:
F
;!
E
be a biel lip ic co e . Then
(a) Fo
g
4
he union o he lines
<
(
p
)
>
,wi h
p
2
E
, is an el lip ic no mal
cone
R
o deg ee
(
g
;
1)
,con aining
F
.
(b) Fo
g
6
he biel lip ic in olu ion
:
F
;!
E
is unique,
W
1
4
(
F
)
=
E
and i
L
1
L
2
2
W
1
4
(
F
)
a e wo dis inc poin s, we ha e
R
=
T
L
1
T
L
2
.
P o o .
I
q
1
:::q
i
2
E
a e dis inc gene al p oin s, Riemann-Ro chon
E
and
F
shows
ha
dim
<
(
q
1
)+
:::
+
(
q
i
)
>
=
i
o
i
+1
g:
(4)
Then, o
i
= 3 and
g
4, i ollows ha all he lines
<
(
p
)
>
, o
p
2
E
, mee a he
same p oin
q
2
P
g
;
1
. So p o jec ion om
q
gi es he ellip ic cu e
E
as an ellip ic no mal
cu eo
P
g
;
2
, whichp o es (a).
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS
7
Le
1
:
F
;!
E
b e a biellip ic in olu ion. Assume he e exis s a 4- o-1 map
2
:
F
;!
P
1
dis inc om hose p o duced by
1
. Then he map
:
F
;!
E
P
1
gi en by
(
p
)=(
1
(
p
)
2
(
p
)) is bi a ional. I we deno e
a
=
E
p
2
g
,
b
=
p
1
g
P
1
o
p
1
2
E
,
p
2
2
P
1
weha e
(
F
)
2
a
+4
b
and hence, byadjun ion in
E
P
1
,we ge
g
(
F
)
p
a
(
(
F
)) = 5. So i
g
6 he biellip ic in olu ion is unique and
W
1
4
(
F
)
=
E
.
Fo
g
6weha e om (4) ha i
L
j
=
j
(
q
j
1
+
q
j
2
)
j
q
i
k
2
E j
=1
2
a e wo die en p oin s in
W
1
4
(
F
) hen
dim
<
(
q
1
1
+
q
1
2
)
>
<
(
q
2
1
+
q
2
2
)
>
=0 o 1(5)
dep ending on whe he
q
1
1
q
1
2
g
q
2
1
q
2
2
g
is emp y o has one p oin , esp ec i ely.
Then, again by Kemp 's Singula i y Theo em, weha e he se - heo e ic equali y
T
L
i
=
D
2
L
i
<D >
and hus
T
L
1
T
L
2
=
D
2
L
1
D
0
2
L
2
(
<D >
<D
0
>
)=
p
2
E
<
(
p
)
>
=
R
using (5), i
L
1
6
=
L
2
.
2
Theo em 2.2
Le
:
S
;!
B
be a biel lip ic b a ion o genus
g
6
. Le
P
=
P
(
!
S=B
)
be he ela i e canonical image o
S
as in sec ion 1. Then he e exis s a h ee old
W
such ha
(a)
W
P
.
(b) Fo
2
B
such ha he b e
F
is smoo h we ha e
W
P
=
R
( he el lip ic
no mal cone con aining
F
).
P o o .
Wep o e he heo em in se e al s eps.
S ep 1. We can assume ha he b a ion has enough sec ions
.
Indeed, conside a e y ample line bundle
L
on
S
and le
B
b e a global smo o h sec ion
o
L
. Conside he diag amm
S
/
/
>
>
>
>
>
>
>
>
b
S
b
/
/
S
B
/
/
B
whe e
=
j
B
,
b
is gi en by a base change and
:
S
;!
b
S
is a minimal desingula iza-
ion o
b
S
.
In his si ua ion we ema k ha
b
, and hence
, has a sec ion. Indeed, his sec ion is
gi en by
(
b b
)
2
B
B
B
S
B
B
=
b
S
g
which is a comp onen o
B
B
B
.We can ep ea
his cons uc ion in o de o ge as many sec ions as we need.
8
MIGUEL A. BARJA
Then weha e
b
(
(
!
S
)) =
(
(
!
S
)) by a base change
0
;!
(
(
!
S
))
;!
!
S
by amica ion o mula
(
(
(
!
S
))) =
b
(
(
!
S
)) =
b
(
(
!
S
)
O
S
) by p o jec ion o mula
0
;!O
b
S
;!
O
S
dominan
and hen
0
;!
(
!
S
)
;!
(
!
S
)
:
Being b o h lo cally ee shea es o he same ank we ge a bi a ional map gi en bya
sequence o elemen a y ans o ma ion on ce ain b es
P
B
(
!
S
)
;;;
'
P
B
(
!
S
)
which is an isomo phism o a gene al b e.
So nally we ge a gene ically ni e a ional map
:
P
B
(
!
S=
B
)
;;; !
P
B
(
!
S=B
)
gi en as he comp osi e
P
B
(
!
S=
B
)
=
P
B
(
!
S
)
;;; !
'
P
B
(
!
S
)
;!
P
B
(
!
S
)
=
P
B
(
!
S=B
)
which is linea on b es and es ic s o he na u al map om he ela i e canonical image
o
:
S
;!
B
on o he ela i e canonical image o
:
S
;!
B
.
Then supp ose he e exis s
W
as in he heo em o he biellip ic b a ion
:
S
;!
B
.
We had ha o smo o h
F
,
W
P
=
R
.Now jus conside
W
=
(
W
)which e ies
he desi ed condi ions. Indeed, o
2
B
such ha
F
is smo o h weha e
W
P
=
(
0
)=
(
W
P
0
)=
(
0
)=
(
R
0
)=
R
b ecause
(
R
0
) is an ellip ic no mal cone con aining
F
=
(
F
0
), whichisuniqueby lemma
2.1.
S ep 2. We can assume ha
is smo o h and has enough disjoin sec ions
.
Once weha e enough sec ions we can es ic he b a ion o he nonemp y Za iski
op en se whe e
is smo o h and he sec ions do no mee (weonly ha e o a oid a ni e
numb e o b es). I he e exis s
W
U
P
U
(
i
(
!
S=B
)), whe e
i
:
U
;!
B
is he na u al
inclusion, e i ying he heo em, hen as
W
i is enough o ake he closu e o
W
U
inside
P
=
P
B
(
!
S=B
).
S ep 3. Exis ence o
W
.
F om he p e ious s eps weha e
:
S
;!
B
a smo o h b a ion o e a, p ossibly non
comple e, cu e wi h enough disjoin sec ions. Unde hese assump ions (see 16) he e
exis schemes
W
d
(
), Pic
d
(
)o e
B
such ha
(i) Fo e e y
2
B
,(
W
d
(
))
=
W
d
(
F
), (Pic
d
(
))
=Pic
d
(
F
).
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS
9
(ii) (Base change p op e y) I
:
B
;!
B
is a base change and
:
S
;!
B
sa ises
he same go o d p op e ies as
, hen
W
d
(
)=
W
d
(
) and Pic
d
(
)=Pic
d
(
) (whe e, as
always,
X
deno es he base change o
X
;!
B
).
Then we can conside
W
1
4
(
)
W
0
4
(
)
Pic
4
(
). Mo eo e we ema k ha
W
1
4
(
)
;!
B
is an ellip ic b a ion which, up o base change, we can assume has a
leas wo sec ions (by he base change p op e y hiswould co esp ond o a base change
o
ha , as p o en a s ep 1, can always b e done).
We also ema k ha , i we se
J
=Pic
4
(
)
;!
B
, hen he shea o ela i e die en-
ials o
is jus
1
J=B
=
(
!
S=B
) (see 17] p.2).
Then we can p o ceed as ollows. Conside
W
1
4
(
)
W
0
4
(
)
i
Pic
4
(
)=
J
B
s
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
whe e
s
is a sec ion o
j
W
1
4
(
)
(we a e assuming ha suchan
s
exis s). Le
e
B
b e he image
o
s
.I we call
I
1
,
I
2
he ideal shea es o
e
B
in
W
0
4
(
)and
J
esp ec i ely, he na u al
epimo phism
I
2
;!
i
I
1
induces he epimo phisms
S
I
2
.
I
2
2
;!S
i
I
1
.
(
i
I
1
)
2
=
S
I
1
.
I
2
1
;!
i
I
i
1
.
I
i
+1
1
(whe e
S
deno es he symme ic algeb a) and hen also he inclusions
Z
1
= P o j
i
I
i
1
.
I
i
+1
1
,
!
Z
2
=
P
e
B
I
1
.
I
2
1
,
!
Z
3
=
P
e
B
I
2
.
I
2
2
:
Since
W
0
4
(
) is singula along
W
1
4
(
), hence along
e
B
,weha e ha
Z
1
is he ela i e
p o jec i ized angen cone o
W
0
4
(
)along
e
B
which s canonically in o
Z
2
and
Z
3
, he
ela i e p o jec i ized Za iski angen spaces o
W
0
4
(
)and
J
, esp ec i ely, along
e
B
.
On he o he hand, since
e
B
J
is smo o h, weha e an exac sequence
0
;!
I
2
.
I
2
2
;!
1
J=B
O
e
B
;!
1
e
B=B
;!
0
k
0
ha leads o
I
2
.
I
2
2
=
1
J=B
O
e
B
.Then
Z
3
=
P
e
B
I
2
.
I
2
2
=
P
e
B
1
J=B
O
e
B
=
P
B
s
1
J=B
=
=
P
B
s
!
S=B
=
P
B
!
S=B
16
MIGUEL A. BARJA
Al hough his example shows ha heo em 2.4 is no ue o genus 5 biellip ic b a-
ions weha e
P op osi ion 3.2
Le
:
S
;!
B
be a biel lip ic b a ion o genus 5. Then, he e exis s
abase change
e
:
e
S
;!
e
B
o
such ha
e
S
is, bi a ional ly, a double co e o an el lip ic
b a ion
e
:
e
V
;!
e
B
.
I he gene al b eo
has only one biel lip ic in olu ion, hen we can ake
e
=
.
P o o .
As in s ep 3 o heo em 2.2 weha e a e a base change
W
1
4
(
b
)
;!
b
B
(a
leas o e a non-emp y op en subse o
b
B
). Le
b
L
b e an i educible comp onen o
W
1
4
(
b
)
such ha o gene al
2
b
B
b
L
has only ellip ic comp onen s. Le
L
b e a desingula iza ion
o
L
. A e a base change we can ge
e
L
/
/
L
e
B
/
/
b
B
such ha
e
L
;!
e
B
has i educible gene al b es and a leas wo sec ions. I
e
:
e
S
;!
e
B
is he induced b a ion weha e ha
e
L
is an i educible comp onen o
W
1
4
(
e
)
;!
e
B
.
Then s ep 3 in heo em 2.2 wo ks and wege
e
S
;;
>
e
W
P
e
B
(
e
!
e
S=
e
B
). The
exis ence o
W
is all wha we need o use he same p o o o heo em 2.4.
I he gene al b e o
has only one biellip ic in olu ion (hence he e exis s only one
ellip ic cone con aining
F
P
4
) hen om he exis ence o
e
W
we can deduce he
exis ence o
W
in he same way as s ep 1 in heo em 2.2 (uniqueness o ellip ic cones
con aining he gene al b e is all wha we need).
2
This esul is enough o ex end p op osi ion 2.6 o he genus 5 case.
Co olla y 3.3
Le
:
S
;!
B
be a genus 5 b a ion such ha al l b es a e smoo h and
biel lip ic. Then
is iso i ial.
P o o .
We can check iso i ialli y a e a base change. The esul ollows hen om
p op osi ion 3.2 and he p o o o p op osi ion 2.6.
2
4 Double co e s and he slop e o biellip ic b a ions
We ecall some basic ac s ab ou double co e s (see 6], 3]).
By a double co e we mean a ni e, deg ee womapbe ween su aces,
0
:
S
0
;!
V
0
.
This map is de e mined by a di iso
Z
0
on
V
0
( he b anch di iso ) and a line bundle
L
0
such ha
L
2
0
=
O
V
0
(
Z
0
). I
V
0
is smo o h,
S
0
is no mal ( esp ec i ely smo o h) i and only
i
Z
0
is educed ( esp ec i ely smo o h).
Conside a double co e as ab o ewi h
S
0
no mal and
V
0
smo o h. Then he e exis s a
canonical esolu ion o singula i ies
o
S
0
which consis s on a ni e sequence o maps
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS
17
S
k
k
;;;;;;;!
S
k
;
1
;;;;;;;!
:::
;;;;;;;!
S
1
1
;;;;;;;!
S
0
k
?
?
?
?
?
?
y
k
;
1
?
?
?
?
?
?
y
:::
?
?
?
?
?
?
y
1
?
?
?
?
?
?
y
0
V
k
;;;;;;;!
k
V
k
;
1
;;;;;;;!
:::
;;;;;;;!
V
1
;;;;;;;!
1
V
0
sa is ying:
(i)
j
is he blow-up o
V
j
;
1
a a singula p oin
p
j
;
1
o
Z
j
;
1
( he b anching di iso o
j
;
1
).
(ii)
j
is he double co e o
V
j
dened by
L
2
j
=
O
(
Z
j
), wi h
Z
j
=
j
(
Z
j
;
1
)
;
2
m
j
;
1
E
j
,
L
j
=
j
(
L
j
;
1
)
O
V
j
(
;
m
j
;
1
E
j
), whe e
E
j
is he excep ional di iso o
j
and
p
j
;
1
is a
singula p oin o
Z
j
;
1
o mul iplici y2
m
j
;
1
o 2
m
j
;
1
+1.
(iii)
j
is a bi a ional mo phism induced by he ca esian diag am o
j
and
j
;
1
.
(i )
Z
k
is smo o h and, hence,
S
k
is a smo o h su ace.
Nowwe can use his as ollows. Recall om sec ion 2 ha weha e ob ained
:
e
S
;!
V
a gene ically 2- o-1 mo phism (we can supp ose ha
is e e ywhe e dened up o blow-
ups) om a blow-up o
S
on o an ellip ic b a ion
V
o e
B
whichwe can supp ose
ela i ely minimal a e some blow-downs. Supp ose ha
is ela i ely minimal.
Now conside
e
S
+
+
u
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
S
=
S
k
{
{
x
x
x
x
x
x
x
x
x
x
x
k
/
/
:::
/
/
S
0
0
S
:::
V
=
V
k
/
/
:::
/
/
V
0
=
V
s
s
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
B
whe e:
=
0
u
is he S ein ac o iza ion o
,wi h
u
bi a ional,
0
ni e (so i is a double
co e ) and
S
0
no mal.
k
:
S
k
;!
V
k
is he canonical esolu ion o singula i ies o
0
:
S
0
;!
V
0
.
:
S
k
;!
S
is he bi a ional mo phism dened by he ela i eminimali yo
.
Theo em 4.1
Le
:
S
;!
B
bea ela i ely minimal biel lip ic b a ion o genus
g
6
.
Le
V
be he ela i e minimal model o he el lip ic b a ion ob ainedin heo em 2.4. Then
18
MIGUEL A. BARJA
(a)
!
2
S=B
;
4(
)
2(
g
;
5)
XO
V
.Inpa icula , i
is no local ly i ial
(
)
4+
2(
g
;
5)
XO
V
(
)
4
(b)
(
)= 4
i and only i
S
is he minimal desingula iza ion o a double co e
S
0
;!
V
o asmoo h el lip ic su ace such ha
All he b es o he el lip ic b a ion
:
V
;!
B
a esmoo h and isomo -
phic.
The b anch di iso o he double co e has only
negligeable
singula i ies
(i.e., al l he mul iplici ies
m
j
in he abo e p ocess a e
2
o
3
(see 13],
17])).
In pa icula , he bound is sha p.
P o o .
(a) Fi s o all weha e
!
2
S=B
;
4(
)= (
K
2
S
;
4
XO
S
)
;
4(
b
;
1)(
g
;
1)
(
K
2
S
;
4
XO
S
)
;
4(
b
;
1)(
g
;
1)
:
(6)
Fo smo o h double co e s
k
:
S
;!
V
weha e (see 3] p.183):
XO
S
= 2
XO
V
+
1
2
L
k
K
V
+
1
2
L
k
L
k
K
2
S
= 2
K
2
V
+4
L
k
K
V
+2
L
k
L
k
so weha e
K
2
S
;
4
XO
S
=2
K
2
V
k
;
4
XO
V
k
]+ 2
L
k
K
V
k
:
(7)
Mo eo e , in each blow-up
j
:
V
j
;!
V
j
;
1
we ge
XO
V
j
=
XO
V
j
;
1
K
V
j
=
j
K
V
j
;
1
+
E
j
L
j
=
j
L
j
;
1
;
m
j
;
1
E
j
:
Then
2
K
2
V
j
;
4
XO
V
j
]+ 2
L
j
K
V
j
=2
K
2
V
j
;
1
;
4
XO
V
j
;
1
]+
2
L
j
;
1
K
V
j
;
1
+2(
m
j
;
1
;
1)
2
K
2
V
j
;
1
;
4
XO
V
j
;
1
]+ 2
L
j
;
1
K
V
j
;
1
:
(8)
Finally as
:
V
;!
B
is an ellip ic minimal b a ion, nume ically weha e
K
V
h
2(
b
;
1) +
XO
V
+
P
i
(
n
i
;
1)
n
i
i
E
(3] p.162) whe e
E
deno es a smo o h b e o
and
n
i
g
a e he mul iplici ies o singula b es o
. In pa icula
K
2
V
0.
As
L
2
0
=
O
V
0
(
Z
0
)and
Z
0
is he b anch di iso o
0
wege
L
0
E
=(
g
;
1) by Hu wi z
o mula. So
2
K
2
V
0
;
4
XO
V
0
]+ 2
L
0
K
V
0
=
;
8
XO
V
0
+(9)
+2
L
0
E
"
2(
b
;
1) +
XO
V
0
+
X
i
(
n
i
;
1)
n
i
#
4(
b
;
1)(
g
;
1) + 2(
g
;
5)
XO
V
:
THE STRUCTURE AND SLOPE OF BIELLIPTIC FIBRATIONS
19
Then (a) ollows om (6), (7), (8) and (9) and om he ac ha
XO
V
0 o ellip ic
b a ions.
(b) Lo oking a he p o o o (a) we see ha
=4 i
XO
V
= 0 and equali y holds in
(6), (7), (8) and (9). So weha e
=4 i
S
is he minimal desingula iza ion o a double
co e o an ellip ic, ela i ely minimal, b a ion
:
V
;!
B
such ha :
has no mul iple b es (
8
i n
i
= 1).
XO
V
=0.
The b anch di iso
Z
0
o he double co e has only
negligeable singula i ies
(see
13], 17]), i.e. all he mul iplici es o he singula i ies o he b anch di iso s in
he p o cess o canonical esolu ion a e 2 o 3.
Bu he s wo condi ions a e equi alen o he ac ha
is smo o h and iso i ial
(see 15 hms. 6,7 Ch.IV). This allows us o cons uc examples wi h
(
)= 4 which a e
esen ially he same as in 19] example 4.3. So he b ound is sha p.
2
Rema k 4.2
Al hough we canno use double co e s o he case o biellip ic b a ions o
genus 5 we al eady know ha
4 also holds o such b a ions (see 9] hm.5.1, 11]).
Re e ences.
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Th ee olds wi h nega i e Kodai a dimension and posi i e
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91
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3. W. Ba h, C. Pe e s, A. Van de Ven.
Compac complex su aces
. E gebnisse de
Ma hema ik und ih e G enzgebie e
4
. Sp inge -Ve lag (1984).
4. Z. Chen.
On he bound o he slope o a non-hype el lip ic b a ion o genus 4
.In e n.
J. Ma h., ol
4
, No.3 (1993), 367{378.
5. R. Ha sho ne.
Algeb aic Geome y
. G adua e Tex s in Ma hema ics
52
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Ve lag (1977).
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On de o ma ion o Quin ic Su aces
.In en iones ma h.
31
(1975),
43{85.
7. E. Ho ikawa.
No es on canonical su aces
.Tohoku Ma h. J. I
43
(1991), 141{148.
8. E. Ho ikawa.
Algeb aic su aces o gene al ype wi h smal l
c
2
1
,V
.J.Fac. Sci. Uni .
Tokyo. Sec . A
28
(1981), 745{755.
9. K. Konno.
Non-hype el lip ic b a ions o smal l genus and ce ain i egula canonical
su aces
. Ann. Sc. No m. Pisa Se . IV ol XX (1993), 575{595.
20
MIGUEL A. BARJA
10. K. Konno.
A lowe bound o he slope o igonal b a ions
.In e n. J. Ma h., ol
7
, No.1 (1996), 19{27.
11. K. Konno.
Clio d index and he slope o b edsu aces
. P ep in . Oc ob e 1996.
12. S. Ma susaka.
Some nume ical in a ian s o hype el lip ic b a ions
.J.Ma h.Kyo o
Uni .
30
-1 (1990), 33{57.
13. U. Pe sson.
Che n in a ian s o su aces o gene al ype
. Comp osi io Ma h.
43
(1982), 3{58.
14. M. Reid.
P oblems on pencils o smal l genus
. P ep in .
15. I.R. Sha a e ich, e al.
Algeb aic Su aces
. P o c. S eklo Ins . Ma h.
75
(1965),
AMS T ansla ions, P o idence R.I. (1967).
16. M. Teixido .
On ansla ion in a iance o
W
d
. J. eine angew. Ma h.
385
(1988),
10{23.
17. G. Xiao.
Su aces b ees en cou bes de gen e deux
. Lec . No es Ma h.
1137
.
Sp inge -Ve lag 1985.
18. G. Xiao.
The b a ions o algeb aic su aces
. Shangai Scien ic & Technical Pub-
lishe s 1992 (Chinese).
19. G. Xiao.
Fibe ed algeb aic su aces wi h low slope
. Ma h. Ann.
276
(1987), 449{
466.
