ZARISKI DECOMPOSITIONS ON ARITHMETIC SURFACES

77 Pages·2010·0.44 MB·English

ZARISKI DECOMPOSITIONS ON ARITHMETIC SURFACES
ATSUSHIMORIWAKI
ABSTRACT. Inthispaper,weestablishtheZariskidecompositionsofarithmeticR-divisors
ofcontinuoustypeonarithmeticsurfacesandinvestigateseveralproperties. Wealsode-
velopthegeneraltheoryofarithmeticR-divisorsonarithmeticvarieties.
CONTENTS
Introduction 1
1. Zariskidecompositionsinvectorspaces 8
2. GreenfunctionsforR-divisors 12
3. Gromov’sinequalityanddistorsionfunctionsforR-divisors 21
4. Plurisubharmonicupperenvelopes 26
5. ArithmeticR-divisors 35
6. PositivityofarithmeticR-divisors 43
7. LimitofnefarithmeticR-divisorsonarithmeticsurfaces 56
8. (cid:27)-decompositionsonarithmeticsurfaces 60
9. Zariskidecompositionsandtheirpropertiesonarithmeticsurfaces 65
References 77
INTRODUCTION
Let S be a non-singular projective surface over an algebraically closed field and let
Div(S) be the group of divisors on S. An element of Div(S)⊗R is called an R-divisor
onS. Inaddition,itissaidtobeeffectiveifitisalinearcombinationofcurveswithnon-
negativerealnumbers. TheproblemoftheZariskidecompositionforaneffectiveR-divisor
D istofindadecompositionD = P +N withthefollowingproperties:
(1) P;N ∈ Div(S)⊗R.
(2) P isnef,thatis,(P ·C) ≥ 0forallreducedandirreduciblecurvesC onS.
(3) N iseffective.
(4) Assuming N 6= 0, let N = c C + ··· + c C be the decomposition such that
1 1 l l
c ;:::;c ∈ R and C ;:::;C are reduced and irreducible curves on S. Then
1 l >0 1 l
thefollowing(4.1)and(4.2)hold:
(4.1) (P ·C ) = 0foralli.
i
(4.2) Thel×lmatrixgivenby((Ci·Cj))1(cid:20)i(cid:20)l isnegativedefinite.
1(cid:20)j(cid:20)l
In 1962, Zariski [19] established the decomposition in the case where D ∈ Div(S). By
the recent work due to Bauer [1] (see also Section 1), P is characterized by the greatest
elementin
{M ∈ Div(S)⊗R | D−M iseffectiveandM isnef}:
Date:21/June/2010,15:00(Kyoto),(Version1.80).
1991MathematicsSubjectClassification. Primary14G40;Secondary11G50.
1
2 ATSUSHIMORIWAKI
Inthispaper,wewouldliketoconsideranarithmeticanalogueoftheaboveproblemonan
arithmeticsurface. Inordertomakethemaintheoremclear,weneedtointroducealotof
conceptsandterminology.
•GreenfunctionsforR-divisors. LetV beanequidimensionalsmoothprojectivevariety
over C. An element of Div(V)R := Div(V)⊗R is called an R-divisor on V. For an R-
divisor D on V, we would like to introduce several types of Green functions for D. We
set D = a D +···+a D , where a ;:::;a ∈ R and D ’s are reduced and irreducible
1 1 l l 1 l i
divisors on V. Let g : V → R∪{±∞} be a locally integrable function on V. We say g
isaD-GreenfunctionofC1-type(respaD-GreenfunctionofC0-type)onV if,foreach
point x ∈ V, there are a small open neighborhood U of x, local equations f ;:::;f of
x 1 l
1
D ;:::;D over U respectively and a C -function (resp. continuous function) u over
1 l x x
U suchthat
x
∑l
g = u + (−a )log|f |2 (a:e:)
x i i
i=1
1
holds on U . These definitions are counterparts of C -metrics and continuous metrics.
x
Besides them, it is necessary to introduce a degenerated version of semipositive metrics.
We say g is a D-Green function of PSHR-type on V if the above ux is taken as a real
valuedplurisubharmonicfunctiononU (i.e.,u isaplurisubharmonicfunctiononU and
x x x
u (y) ∈ R for all y ∈ U ). To say more generally, let L1 be the sheaf consisting of
x x loc
locallyintegrablefunctions,thatis,
L1 (U) = {g : U → R∪{±∞} | g islocallyintegrable}
loc
foranopensetU ofV,andletusfixasubsheafT ofL1 satisfyingthefollowingcondi-
loc
tions(inthefollowing(1),(2)and(3),U isanarbitraryopensetofV):
(1) Ifu;v ∈ T (U)anda ∈ R(cid:21)0,thenu+v ∈ T (U)andau ∈ T (U).
(2) Ifu;v ∈ T (U)andu ≤ v almosteverywhere,thenu ≤ v.
(3) If (cid:30) ∈ O(cid:2)(U) (i.e., (cid:30) is a nowhere vanishing holomorphic function on U), then
V
log|(cid:30)|2 ∈ T (U).
This subsheaf T is called a type for Green functions on V. Moreover, T is said to be
real valued if u(x) ∈ R for any open set U, u ∈ T (U) and x ∈ U. Using T , we say g
is a D-Green function of T -type on V if the above u is an element of T (U ) for each
x x
x ∈ V. The set of all D-Green functions of T -type on V is denoted by GT(V;D). If
x 6∈ Supp(D),then,byusing(2)and(3)inthepropertiesofT ,wecanseethatthevalue
∑l
u (x)+ (−a )log|f (x)|2
x i i
i=1
doesnotdependonthechoiceofthelocalexpression
∑l
g = u + (−a )log|f |2 (a:e:)
x i i
i=1
∑
of g, so that u (x)+ l (−a )log|f (x)|2 is called the canonical value of g at x and
x i=1 i i
it is denoted by g (x). Note that g ∈ T (V \ Supp(D)) and g = g (a:e:) on
can can can
V \Supp(D). Further,ifT isrealvalued,theng (x) ∈ R.
can
ZARISKIDECOMPOSITIONSONARITHMETICSURFACES 3
?H0(V;D)foranR-divisorD anditsnormarisingfromaGreenfunction. LetD bean
R-divisor. IfV isconnected,thenH0(V;D)isdefinedby
{ (cid:12) }
(cid:12)
H0(V;D) := (cid:30) (cid:12)(cid:12) (cid:30)onisVawnoitnh-z((cid:30)er)o+raDtio≥nal0function ∪{0}:
Ingeneral,letV = V ∪···∪V bethedecompositionintoconnectedcomponentsofV.
1 r
Then
⊕r
H0(V;D) := H0(V ; D| ):
i Vi
i=1
Let g be a D-Green function of C0-type on V. For (cid:30) ∈ H0(V;D), it is easy to see that
|(cid:30)| := exp(−g=2)|(cid:30)| coincides with a continuous function almost everywhere on V, so
g
thatthesupremumnormk(cid:30)k of(cid:30)withrespecttog isdefinedby
g
k(cid:30)k := esssup{|(cid:30)| (x) | x ∈ V}:
g g
•ArithmeticR-divisors. LetX bead-dimensionalgenericallysmoothnormalprojective
arithmetic variety. Let Div(X) be the group of Cartier divisors on X. As before, an
elementofDiv(X)R := Div(X)⊗RiscalledanR-divisoronX. Itissaidtobeeffective
ifitisalinearcombinationofprimedivisorswithnon-negativerealnumbers. Inaddition,
forD;E ∈ Div(X)R,ifD−E iseffective,thenitisdenotedbyD ≥ E orE ≤ D.
Let D be an R-divisor on X and let g be a locally integrable function on X(C). A
pair D = (D;g) is called an arithmetic R-divisor if F1(cid:3) (g) = g (a:e:), where F1 is
the complex conjugation map on X(C). Moreover, D is said to be of C1-type (resp.
of C0-type, of PSHR-type) if g is a D-Green function of C1-type (resp. of C0-type, of
PSHR-type). More generally, for a fixed type T for Green functions, D is said to be of
T -type if g is a D-Green function of T -type. For arithmetic R-divisors D = (D ;g )
1 1 1
andD = (D ;g ),wedefineD = D andD ≤ D asfollows:
2 2 2 1 2 1 2
{
D = D ⇐de⇒f D = D andg = g (a:e:);
1 2 1 2 1 2
D ≤ D ⇐de⇒f D ≤ D andg ≤ g (a:e:):
1 2 1 2 1 2
IfD ≥ (0;0),thenD issaidtobeeffective. Further,theset
{M | M isanarithmeticR-divisoronX andM ≤ D}
isdenotedby(−∞;D].
?VolumeofarithmeticR-divisorsofC0-type. LetDdivC0(X)R bethegroupofarithmetic
R-divisors of C0-type on X. For D ∈ DdivC0(X)R, we define H0(X;D), H^0(X;D),
h^0(X;D)andvcol(D)asfollows:
 { (cid:12) }
(cid:12)

H0(X;D) := (cid:12)(cid:12) isanon-zerorationalfunction ∪{0};
 (cid:12) onX with( )+D ≥ 0


H^0(X;D) := { ∈ H0(X;D) | k k ≤ 1};
g
h^0(X;D) := log#(H^0(X;D));


c h^0(X;nD)
vol(D) := limsup :
n!1 nd=d!
Notethat
{ (cid:12) }
(cid:12)
H^0(X;D) = (cid:12)(cid:12) isanon-zderorationalfunction ∪{0}:
onX with( )+D ≥ (0;0)
4 ATSUSHIMORIWAKI
Thecontinuityof
vcol : Pcic(X)Q → R
is proved in [13], where Pcic(X)Q := Pcic(X) ⊗ Q. Moreover, in [14], we introduce
PcicC0(X)R asanaturalextensionofPcic(X)Q andprovethatvcol : Pcic(X)Q → Rhasthe
continuousextension
vcol : PcicC0(X)R → R:
Theorem5.2.2showsthatthereisanaturalsurjectivehomomorphism
OR : DdivC0(X)R → PcicC0(X)R
suchthatvcol(D) = vcol(OR(D))forallD ∈ DdivC0(X)R. Inparticular,byusingresultsin
[5],[6],[13],[14],[15]and[17],wehavethefollowingpropertiesofvcol : DdivC0(X)R →
R(cf. Theorem5.2.2andTheorem6.6.1):
(1) vcol : DdivC0(X)R → Rispositivelyhomogeneousofdegreed,thatis,vcol(aD) =
advcol(D)foralla ∈ R(cid:21)0 andD ∈ DdivC0(X)R.
(2) vcol : DdivC0(X)R → R is continuous in the following sense: for arithmetic R-
divisors D1;:::;Dr, A1;:::;Ar0 of C0-type, there is a positive constant C de-
pendingonlyonX andD1;:::;Dr;A1;:::;Ar0 suchthat
(cid:12)   (cid:12)
(cid:12) ( )(cid:12)
(cid:12) ∑r ∑r0 ∑r (cid:12)
(cid:12)(cid:12)vcol aiDi+ (cid:14)jAj +(0;(cid:30))−vcol aiDi (cid:12)(cid:12)
(cid:12) (cid:12)
i=1 j=1 i=1
   
d(cid:0)1
∑r ∑r0 ∑r0
≤ C |a |+ |(cid:14) | k(cid:30)k + |(cid:14) |
i j sup j
i=1 j=1 j=1
foralla1;:::;ar;(cid:14)1;:::;(cid:14)r0 ∈ Rand(cid:30) ∈ C0(X).
c
(3) vol(D)isgivenby“lim”,thatis,
c h^0(tD)
vol(D) = lim ;
t!1 td=d!
whereD ∈ DdivC0(X)R andt ∈ R>0.
(4) vcol(−)1=d is concave, that is, for arithmetic R-divisors D ;D of C0-type, if D
1 2 1
and D are pseudo-effective (for the definition of pseudo-effectivity, see SubSec-
2
tion6.1),then
vcol(D +D )1=d ≥ vcol(D )1=d+vcol(D )1=d:
1 2 1 2
(5) (Fujita’sapproximationtheoremforR-divisors)IfD isanarithmeticR-divisorof
c
C0-type and vol(D) > 0, then, for any positive number (cid:15), there are a birational
morphism (cid:22) : Y → X of generically smooth and normal projective arithmetic
varieties and an ample arithmetic Q-divisor A of C1-type on Y (cf. Section 6)
suchthatA ≤ (cid:22)(cid:3)(D)andvcol(A) ≥ vcol(D)−(cid:15).
(6) (The generalized Hodge index theorem for R-divisors) If D is an arithmetic R-
divisor of (C0 ∩PSH)-type and D is nef on every fiber of X → Spec(Z), then
vcol(D) ≥ ddeg(Dd)(seedescriptionsin“PositivityofarithmeticR-divisors”below
d d
orProposition6.4.2forthedefinitionofdeg(D )).
ZARISKIDECOMPOSITIONSONARITHMETICSURFACES 5
?IntersectionnumberofanarithmeticR-divisorwitha1-dimensionalsubscheme. LetT
be a real valued type for Green functions such that C0 ⊆ T and −u ∈ T whenever
u ∈ T . LetD = (D;g)beanarithmeticR-divisorofT -type. LetC bea1-dimensional
closed integral subscheme of X. Let D = a D +···+a D be a decomposition such
1 1 l l
that a1;:::;al ∈ R and Di’sare Cartier divisors. Forsimplicity, weassume that Di(cid:12)’sare
effective, C 6⊆ Supp(D ) for all i and that C is flat over Z. In this case, ddeg(D(cid:12) ) is
i C
definedby
(cid:12) ∑l ∑
ddeg(D(cid:12) ) := a log#(O (D )=O )+ 1 g (x):
C i C i C 2 can
i=1 x2C(C)
In general, see Section 5.3. Let Z be a 1-cycle on X with coefficients in R, that is, there
are a ;:::;a ∈ R and 1-dimensional closed integral subschemes C ;:::;C on X such
1 l ( ) 1 l
thatZ = a C +···+a C . Thenddeg D | Z isdefinedby
1 1 l l
( ) ∑l ( (cid:12) )
ddeg D | Z := a ddeg D(cid:12) :
i Ci
i=1
? Positivity of arithmetic R-divisors. An arithmetic R-divisor D is called nef if D is of
PSHR-type and ddeg(D| ) ≥ 0 for all 1-dimensional closed integral subschemes C of
C
X. The cone of all nef arithmetic R-divisors on X is denoted by Ndef(X)R. Moreover,
the cone of all nef arithmetic R-divisors of C1-type (resp. C0-type) on X is denoted by
d d c
NefC1(X)R (resp. NefC0(X)R). Further,wesayD isbig ifvol(D) > 0.
dNef d d
LetDivC0 (X)RbethevectorsubspaceofDivC0(X)RgeneratedbyNefC0(X)R. Then,
byProposition6.4.2,
DdivC1(X)R+DdivC0\PSH(X)R ⊆ DdivNCe0f(X)R
andthesymmetricmulti-linearmap
DdivC1(X)R×···×DdivC1(X)R → R
givenby (D ;:::;D ) 7→ ddeg(D ···D )(cf. Proposition-Definition6.4.1) extendstoa
1 d 1 d
uniquesymmetricmulti-linearmap
DdivNCe0f(X)R×···×DdivNCe0f(X)R → R
suchthat(D;:::;D) 7→ vcol(D)forD ∈ NdefC0(X)R.
• Zariski decompositions on arithmetic surfaces. Let X be a regular projective arith-
meticsurface. Themaintheoremofthispaperisthefollowing:
TheoremA(cf. Theorem9.2.1andTheorem9.3.4). LetD beanarithmeticR-divisorof
C0-typeonX suchthattheset
(−∞;D]∩Ndef(X)R = {M | M isanefarithmeticR-divisoronX andM ≤ D}
is not empty. Then there is a nef arithmetic R-divisor P of C0-type such that P gives the
greatestelementof(−∞;D]∩Ndef(X)R,thatis,P ∈ (−∞;D]∩Ndef(X)R andM ≤ P
for all M ∈ (−∞;D]∩Ndef(X)R. Moreover, if we set N = D −P, then the following
propertieshold:
c c d 2
(1) vol(D) = vol(P) = deg(P ).
6 ATSUSHIMORIWAKI
(cid:12)
(2) ddeg(P(cid:12) ) = 0 for all 1-dimensional closed integral subschemes C with C ⊆
C
Supp(N).
(3) If L is(cid:12)an arithmetic R-divisor of PSHR-type on X such that 0 ≤ L ≤ N and
deg(L(cid:12) ) ≥ 0 for all 1-dimensional closed integral subschemes C with C ⊆
C
Supp(N),thenL = 0.
TheabovedecompositionD = P +N iscalledtheZariskidecompositionofDandwe
say P (resp. N) is the positive part (resp. the negative part) of the decomposition. For
example, let P1 = Proj(Z[x;y]), C = {x = 0}, z = x=y and (cid:11);(cid:12) ∈ R with (cid:11) > 1
Z 0 >0
and(cid:12) < 1. Thenthepositivepartofanarithmeticdivisor
(C ;−log|z|2+logmax{(cid:11)2|z|2;(cid:12)2})
0
of(C0∩PSH)-typeonP1 is
Z
((cid:18)C ;−(cid:18)log|z|2+logmax{(cid:11)2|z|2(cid:18);1});
0
where (cid:18) = log(cid:11)=(log(cid:11) − log(cid:12)) (cf. Subsection 9.4). This example shows that an R-
divisor is necessary for the arithmetic Zariski decomposition. In addition, an example in
Remark9.4.3showsthattheArakelovChowgroupconsistingofadmissiblemetricsdueto
FaltingsisinsufficienttogettheZariskidecomposition.
We assume that N 6= 0. Let N = c C +···+c C be the decomposition of N such
1 1 l l
that c ;:::;c ∈ R and C ’s are 1-dimensional closed integral subschemes on X. Let
1 l >0 i
(C1;g1);:::;(Cl;gl)beeffectivearithmeticdivisorsofPSHR-typesuchthat
c (C ;g )+···+c (C ;g ) ≤ N;
1 1 1 l l l
whichispossiblebyProposition2.4.2andLemma9.1.3. Then,byusingLemma1.2.3,the
above(3)yieldsaninequality
( ( ))
(−1)ldet ddeg (C ;g )| > 0:
i i Cj
This is a counterpart of the property (4.2) of the Zariski decomposition on an algebraic
surface. Ontheotherhand,ourZariskidecompositionisarefinementofFujita’sapproxi-
mationtheoremduetoChen[6]andYuan[17]onanarithmeticsurface. ActuallyFujita’s
approximationtheoremonanarithmeticsurfaceisaconsequenceoftheabovetheorem(cf.
Proposition9.3.6).
Let D be an effective arithmetic R-divisor of C0-type. For each n ≥ 1, we set F (D)
n
andM (D)asfollows:
n

∑ { }
 1
F (D) = min mult (((cid:30))+nD) | (cid:30) ∈ H^0(X;nD)\{0} C;
n C
n
 C

M (D) = D−F (D):
n n
LetV(nD)beacomplexvectorspacegeneratedbyH^0(X;nD). Itiseasytoseethat
1
g := g+ logdist(V(nD);ng)
Mn(D) n
1
is an M (D)-Green function of C -type (for the definition of distorsion functions, see
n
Subsection3.2). Thenwehavethefollowing:
TheoremB(Asymptoticorthogonality). IfD isbig,then
(( )(cid:12) )
(cid:12)
d
lim deg M (D);g (cid:12) F (D) = 0:
n!1 n Mn(D) n
ZARISKIDECOMPOSITIONSONARITHMETICSURFACES 7
• Technical results for the proof of the arithmetic Zariski decomposition. In order to
get the greatest element of (−∞;D]∩Ndef(X)R, we need to consider the nefness of the
limitofaconvergentsequenceofnefarithmeticR-divisors. Thefollowingtheoremisour
solutionforthisproblem:
Theorem C (cf. Theorem 7.1). Let X be a regular projective arithmetic surface. Let
{M = (M ;h )}1 beasequenceofnefarithmeticR-divisorsonX withthefollowing
n n n n=0
properties:
(a) There is an arithmetic divisor D = (D;g) of C0-type such that M ≤ D for all
n
n ≥ 1.
(b) ThereisaproperclosedsubsetEofX suchthatSupp(D) ⊆ EandSupp(M ) ⊆
n
E foralln ≥ 1.
(c) limn!1multC(Mn)existsforall1-dimensionalclosedintegralsubschemesC on
X.
(d) limsup (h ) (x)existsinRforallx ∈ X(C)\E(C).
n!1 n can
ThenthereisanefarithmeticR-divisorM = (M;h)onX suchthatM ≤ D,
∑( )
M = lim mult (M ) C
C n
n!1
C
and that h | is the upper semicontinuous regularization of the function given
can X(C)nE(C)
byx 7→ limsup (h ) (x)overX(C)\E(C).
n!1 n can
c c
Moreover, forthefirstpropertyvol(P) = vol(D)ofthearithmeticZariskidecomposi-
tion, it is necessary to observe the following behavior of distorsion functions, which is a
consequenceofGromov’sinequalityforanR-divisor(cf. Proposition3.1.1).
Theorem D (cf. Theorem 3.2.3). Let V be an equidim⊕ensional smooth projective variety
over C and let D be an R-divisor on V. Let R = R be a graded subring of
⊕ n(cid:21)0 n
H0(V;nD). IfgisaD-GreenfunctionofC1-type,thenthereisapositiveconstant
n(cid:21)0
C withthefollowingproperties:
(1) dist(R ;ng) ≤ C(n+1)3dimV foralln ≥ 0.
n
dist(R ;ng) dist(R ;mg) dist(R ;(n+m)g)
(2) n · m ≤ n+m foralln;m ≥ 0.
C(n+1)3dimV C(m+1)3dimV C(n+m+1)3dimV
ThemostdifficultpointfortheproofofthearithmeticZariskidecompositionistocheck
the continuous property of the positive part. For this purpose, the following theorem is
needed:
Theorem E (cf. Theorem 4.6). Let V be an equidimensional smooth projective variety
over C. Let A and B be R-divisors on V with A ≤ B. If there is an A-Green function
h of C1-type such that ddc([h])+(cid:14) is represented by either a positive C1-form or the
A
zero form, then, for a B-Green function g of C0-type, there is an A-Green function g of
B
(C0∩PSH)-typesuchthatg isthegreatestelementoftheset
GPSHR(V;A)(cid:20)gB := {u ∈ GPSHR(V;A) | u ≤ gB (a:e:)}
modulo null functions, that is, g ∈ GPSHR(V;A)(cid:20)gB and u ≤ g (a:e:) for all u ∈
GPSHR(V;A)(cid:20)gB.
For the proof, we actually use a recent regularity result due to Berman-Demailly [3].
1
EvenstartingfromanarithmeticdivisorD ofC -type,itisnotexpectedthatthepositive
partP isofC1-typeagain(cf[16]). ItcouldbethatP isofC1;1-type.
8 ATSUSHIMORIWAKI
Acknowledgement. I would like to thank Prof. Bauer, Prof. Caiba˘r and Prof. Kennedy
forsendingmetheirwonderfulpaperconcerningZariskidecompositionsinvectorspaces,
which were done independently. I also express my hearty thanks to Prof. Yuan for his
questionsandcomments.
1. ZARISKI DECOMPOSITIONS IN VECTOR SPACES
LogicallythecontextsofthissectionarenotnecessaryexceptLemma1.2.3. Theyhow-
ever give an elementary case for our considerations and provide a good overview of our
paperwithoutanymaterials.
1.1. In the paper [1], Bauer presents a simple proof of the existence of Zariski decompo-
sitionsonanalgebraicsurface. Unfortunately,heuseslinerseriesonthealgebraicsurface
toshowthenegativedefinitenessofthenegativepartoftheZariskidecomposition. Inthis
section,wewouldliketogivealinearalgebraicproofwithoutusinganymaterialsofalge-
braic geometry. The technical main result for our purpose is Lemma 1.2.3. After writing
the first draft of this paper, Bauer, Caiba˘r and Kennedy kindly informed me that, in the
paper [2], they had independently obtained several results similar to the contexts of this
section. Theirpaperiswrittenforageneralreader.
LetV beavectorspaceoverR. Leteee = {e(cid:21)}(cid:21)2(cid:3) beabasisofV andlet(cid:30)(cid:30)(cid:30) = {(cid:30)(cid:21)}(cid:21)2(cid:3)
beafamilyofelementsofHomR(V;R)suchthat(cid:30)(cid:21)(e(cid:22)) ≥ 0for(cid:21) 6= (cid:22). Thispair(eee;(cid:30)(cid:30)(cid:30))
ofeeeand(cid:30)(cid:30)(cid:30)iscalledasystemofZariskidecompositionsinV.
Let us fix several notations which work only in this section. For (cid:21) ∈ (cid:3), the coefficient
of x at e(cid:21) in∑the linear combination of x with respect to the basiseee is denoted by x((cid:21);eee),
thatis,x = x((cid:21);eee)e . Let≤ beanorderrelationofV givenby
(cid:21) (cid:21) eee
x ≤ y ⇐de⇒f x((cid:21);eee) ≤ y((cid:21);eee)forall(cid:21) ∈ (cid:3):
eee
We often use y ≥ x instead of x ≤ y. Supp(x;eee), [x;y] , (−∞;x] , [x;∞) , Nef((cid:30)(cid:30)(cid:30))
eee eee eee eee eee
andNum((cid:30)(cid:30)(cid:30))aredefinedasfollows:

Supp(x;eee) := {(cid:21) ∈ (cid:3) | x((cid:21);eee) 6= 0};


[x;y]eee := {v ∈ V | x ≤eee v ≤eee y};

(−∞;x] := {v ∈ V | v ≤ x};
eee eee
[x;∞) := {v ∈ V | v ≥ x};
 eee eee

Nef((cid:30)(cid:30)(cid:30)) := {v ∈ V | (cid:30) (v) ≥ 0forall(cid:21) ∈ (cid:3)};
 (cid:21)

Num((cid:30)(cid:30)(cid:30)) := {v ∈ V | (cid:30) (v) = 0forall(cid:21) ∈ (cid:3)}:
(cid:21)
ForanelementxofV,adecompositionx = y+z iscalledaZariskidecompositionof
xwithrespectto(eee;(cid:30)(cid:30)(cid:30))ifthefollowingconditionsaresatisfied:
(1) y ∈ Nef((cid:30)(cid:30)(cid:30))andz ≥ 0.
eee
(2) (cid:30){(cid:21)(y)∑= 0forall(cid:21) ∈ Supp(z;eee). }
(3) x ∈ (cid:21)2Supp(z;eee)R(cid:21)0e(cid:21) | (cid:30)(cid:21)(x) ≥ 0forall(cid:21) ∈ Supp(z;eee) = {0}.
Wecally (resp. z)thepositivepartofx(resp. negativepartofx).
Thepurposeofthissectionistogivetheproofofthefollowingproposition.
Proposition1.1.1. ForanelementxofV,wehavethefollowing:
(1) Thefollowingareequivalent:
(1.1) AZariskidecompositionofxwithrespectto(eee;(cid:30)(cid:30)(cid:30))exists.
(1.2) (−∞;x] ∩Nef((cid:30)(cid:30)(cid:30)) 6= ∅.
eee
(2) IfaZariskidecompositionexists,thenitisuniquelydetermined.
ZARISKIDECOMPOSITIONSONARITHMETICSURFACES 9
(3) IfaZariskidecompositionofxwithrespectto(eee;(cid:30)(cid:30)(cid:30))existsandthenegativepartz
ofxisnon-zero,thenz hasthefollowingproperties:
(3.1) LetQbethematrixgivenby((cid:30)(cid:21)(e(cid:22)))(cid:21);(cid:22)2Supp(z;eee). Then
(−1)#(Supp(z;eee))detQ > 0:
Moreover,ifQissymmetric,thenQisnegativedefinite.
(3.2) {e(cid:21)}(cid:21)2Supp(z;eee) islinearlyindependentonV=Num((cid:30)(cid:30)(cid:30)).
1.2. Proofs. HereletusgivetheproofofProposition1.1.1.
Forx ;:::;x ∈ V,max {x ;:::;x } ∈ V isgivenby
1 r eee 1 r
∑
max {x ;:::;x } := max{x ((cid:21);eee);:::;x ((cid:21);eee)}e :
eee 1 r 1 r (cid:21)
(cid:21)2(cid:3)
Letusbeginwiththefollowinglemma.
Lemma1.2.1. Ifx ;:::;x ∈ Nef((cid:30)(cid:30)(cid:30)),thenmax{x ;:::;x } ∈ Nef((cid:30)(cid:30)(cid:30)).
1 r 1 r
Proof. Itis sufficientto seethatif (cid:30) (x ) ≥ 0forall i, then (cid:30) (max {x ;:::;x }) ≥ 0.
(cid:21) i (cid:21) eee 1 r
We set z = max {x ;:::;x }. Note that Supp(z −x ;eee)∩···∩Supp(z −x ;eee) = ∅.
eee 1 r 1 r
Thusthereisiwith(cid:21) 6∈ Supp(z−x ;eee). Then(cid:30) (z−x ) ≥ 0,andhence
i (cid:21) i
(cid:30) (z) = (cid:30) (z−x )+(cid:30) (x ) ≥ 0:
(cid:21) (cid:21) i (cid:21) i
(cid:3)
Lemma1.2.2. LetxbeanelementofV suchthat(−∞;x] ∩Nef((cid:30)(cid:30)(cid:30)) 6= ∅. Thenthereis
eee
thegreatestelementy in(−∞;x] ∩Nef((cid:30)(cid:30)(cid:30)),thatis,y ∈ Nef((cid:30)(cid:30)(cid:30))∩(−∞;x] andy ≥ v
eee eee eee
forallv ∈ Nef((cid:30)(cid:30)(cid:30))∩(−∞;x] . Thisgreatestelementy isdenotedby
eee
max(Nef((cid:30)(cid:30)(cid:30))∩(−∞;x] ):
eee
Further,y andz := x−y satisfythefollowingproperties:
(a) y ∈ Nef((cid:30)(cid:30)(cid:30)),z ≥ 0andx = y+z.
eee
(b) (cid:30){(cid:21)(y)∑= 0forall(cid:21) ∈ Supp(z;eee). }
(c) v ∈ (cid:21)2Supp(z;eee)R(cid:21)0e(cid:21) | (cid:30)(cid:21)(v) ≥ 0forall(cid:21) ∈ Supp(z;eee) = {0}.
Proof. Wechoosex0 ∈ (−∞;x] ∩Nef((cid:30)(cid:30)(cid:30)). Letusseethefollowingclaim.
eee
Claim1.2.2.1. Thereisthegreatestelementy ofNef((cid:30)(cid:30)(cid:30))∩[x0;x] .
eee
Proof. Notethat[x0;x] = x0+[0;x−x0] . Moreover,itiseasytoseethat
eee eee
Nef((cid:30)(cid:30)(cid:30))∩[x0;x]
eee{ }
= x0+ v ∈ [0;x−x0] | (cid:30) (v) ≥ −(cid:30) (x0)forall(cid:21) ∈ Supp(x−x0;eee) :
eee (cid:21) (cid:21)
Therefore, Nef((cid:30)(cid:30)(cid:30)) ∩ [x0;x]eee is⊕a translation of a bounded convex polyhedral set in a fi-
nite dimensional vector space Re . Hence Nef((cid:30)(cid:30)(cid:30))∩[x0;x] is a convex
(cid:21)2Supp(x(cid:0)x0;eee) (cid:21) eee
polytope, that is, there are (cid:13) ;:::;(cid:13) ∈ Nef((cid:30)(cid:30)(cid:30)) ∩ [x0;x] such that Nef((cid:30)(cid:30)(cid:30)) ∩ [x0;x] =
1 l eee eee
Conv{(cid:13) ;:::;(cid:13) } (cf. [18, Theorem 3.2.5 or Finite basis theorem]). If we set y =
1 l
max{(cid:13) ;:::;(cid:13) }, then, by Lemma 1.2.1, y ∈ Nef((cid:30)(cid:30)(cid:30)) ∩ [x0;x] . Moreover, for v =
1 l eee
a1(cid:13)1 + ··· + al(cid:13)l ∈ Nef((cid:30)(cid:30)(cid:30)) ∩ [x0;x]eee (a1;:::;al ∈ R(cid:21)0 and a1 + ··· + al = 1), we
have
y = a y+···+a y ≥ a (cid:13) +···+a (cid:13) = v:
1 l eee 1 1 l l
(cid:3)
10 ATSUSHIMORIWAKI
Thisyisactuallythegreatestelementof(−∞;x] ∩Nef((cid:30)(cid:30)(cid:30)). Indeed,ifv ∈ (−∞;x] ∩
eee eee
Nef((cid:30)(cid:30)(cid:30)),thenmax{v;y} ∈ [x0;x] ∩Nef((cid:30)(cid:30)(cid:30))bylemma1.2.1,andhence
eee
v ≤ max{v;y} ≤ y:
Letuschecktheproperties(a),(b)and(c). Firstofall,(a)isobvious. Inordertosee(b)
and(c),wemayassumethatz 6= 0.
(b)Weassumethat(cid:30) (y) > 0for(cid:21) ∈ Supp(z;eee). Let(cid:15)beasufficientlysmallpositive
(cid:21)
number. Theny+(cid:15)e ≤ xand
(cid:21) eee
(cid:30) (y+(cid:15)e ) = (cid:30) (y)+(cid:15)(cid:30) (e ) ≥ 0
(cid:22) (cid:21) (cid:22) (cid:22) (cid:21)
for all (cid:22) ∈ (cid:3) because 0 < (cid:15) (cid:28) 1. Thus y + (cid:15)e ∈ Nef((cid:30)(cid:30)(cid:30)), which contradicts to the
(cid:21)
maximalityofy. Therefore,(cid:30)(cid:21)(y) = 0(fo∑r(cid:21) ∈ Supp(z;eee). )
(c)Nextweassumethatthereisv ∈ (cid:21)2Supp(z;eee)R(cid:21)0e(cid:21) \{0}suchthat(cid:30)(cid:21)(v) ≥ 0
for all (cid:21) ∈ Supp(z;eee). Then there is a sufficiently small positive number (cid:15)0 such that
y + (cid:15)0v ≤ x. Note that (cid:30) (y + (cid:15)0v) ≥ 0 for all (cid:22), which yields a contradiction, as
eee (cid:22)
before. (cid:3)
Lemma 1.2.3. Let W be a vector space over R. Let e ;:::;e ∈ W and (cid:30) ;:::;(cid:30) ∈
1 n 1 n
HomR(W;R)withthefollowingproperties:
(a) {(a ;:::;a ) ∈ Rn | a e +···+a e = 0} = {(0;:::;0)}.
1 n (cid:21)0 1 1 n n
(b) (cid:30) (e ) ≥ 0foralli 6= j.
i j
(c) {x ∈ R(cid:21)0e1+···+R(cid:21)0en | (cid:30)i(x) ≥ 0foralli} = {0}.
Thenwehavethefollowing:
(1) LetQbethe(n×n)-matrixgivenby((cid:30) (e )). Thenthereare(n×n)-matricesA
i j
andB withthefollowingproperties:
(1.1) A(resp. B)isalower(resp. upper)trianglematrixconsistingofnon-negative
numbers.
(1.2) detA > 0,detB > 0and
 
−1 ··· 0
 
AQB =  ... ... ... :
0 ··· −1
(1.3) IfQissymmetric,thenB = tA.
(2) Thevectorse ;:::;e arelinearlyindependentin
1 n
W={x ∈ W | (cid:30) (x) = ··· = (cid:30) (x) = 0}:
1 n
Proof. (1)Letusbeginwiththefollowingclaim.
Claim1.2.3.1. (cid:30) (e ) < 0foralli.
i i
Proof. If(cid:30)i(ei) ≥ 0,thenei ∈ {x ∈ R(cid:21)0e1+···+R(cid:21)0en | (cid:30)j(x) ≥ 0forallj}. Thisis
acontradictionbecausee 6= 0. (cid:3)
i
Theaboveclaimproves(1)inthecasewheren = 1. Hereweset
(cid:30)0 = −(cid:30) (e )(cid:30) +(cid:30) (e )(cid:30) (i ≥ 2); e0 = −(cid:30) (e )e +(cid:30) (e )e (j ≥ 2):
i 1 1 i i 1 1 j 1 1 j 1 j 1
Weclaimthefollowing:
Claim1.2.3.2. (i) (cid:30) (e ) = 0and(cid:30) (e0) = 0foralli ≥ 2andj ≥ 2.
i 1 1 j
(ii) e0;:::;e0 and(cid:30)0;:::;(cid:30)0 satisfyallassumptions(a)∼(c)ofthelemma.
2 n 2 n