(1,4)-polarized Abelian surfaes and their moduli
Pol Vanhaeke
Abstrat
In thispaper we disusssome algebrai-geometri aspets of a (family of) integrable quarti
potential(s)intwodegreesoffreedom. Itisaspeialaseoftheso-alledGarniersystem,whihwas
rst introdued by Garnier when studying isomonodromi deformations of dierential equations.
We show that the omplex invariant manifolds of this integrable system omplete into Abelian
surfaes of type (1;4) and use the spei geometry of these surfaes to prove that the system is
algebraiompletelyintegrable. The limitingaseofthepotential(q 2
1 +q
2
2 )
2
willalsobedisussed,
inpartiularaLaxpairforthislimitingpotentialwillbefoundfromtheLaxpairweonstrutfor
thegeneri ase.
We alsoshow thatevery Abeliansurfaeof type(1;4) oursasan invariantmanifoldforone
of these integrable potentials. This allows us(among other expliitthings) to omputeexpliitely
a anonial map between themodulispae of Abeliansurfaes of type (1;4) to the moduli spae
of Jaobiansof genustwo urves.
Address: Max-Plank-Institut furMathematik
Gottfried-Claren-Strae26
D-5300 Bonn3
Germany
E-mail: Vanhaekantigone.mpim-bonn.mpg.de
It is well-known that there is a rih interation between algebrai geometry and algebrai
ompletelyintegrablesystems(a..i.systems)bothinthenite-dimensionalase(e.g.Todalatties,
geodesi ows on Lie groups, lassial tops) and the innite-dimensionalase (e.g. KdV and KP
equations, non-linearShrodingerequation)(see [AvM1℄, [D℄,[M2℄, [Sh℄).
The main fat is that the generi integral urve of the Hamiltonian vetor eld of suh an
integrablesystemisdenseinanAbelianvariety,i.e.,inaomplexalgebraitorus(runwithomplex
time). The dierent Abelianvarieties whih orrespond to the dierent integral urvesllup the
phase spae and are alled the (omplex) invariant manifolds of the vetor eld. Equations for
(anaÆne part of) these invariantmanifolds aregiven bya maximalset ofindependentfuntions,
invariantforthevetor eld(often alledonstants of motion orrst integrals) one of whihisthe
Hamiltonian funtiondening thevetor eld. It follows that knowingthese onstants of motion
leads to expliit equations for aÆne parts of Abelian surfaes. On the one hand they yield by
diret methodssome interestingresults about thefamily of Abelianvarieties whihappear inthe
system, whih often desribethefullmoduliof Abelianvarieties of agiven type(at leastin small
dimensions). RememberthatAbelianvarieties(of dimensiong) aredesribedbymeansofaset of
disreteparameters(Æ
1
;:::;Æ
g
)givingthe(polarization)typeandbymeansofaRiemannmatrixZ
(i.e.,asymmetriggmatrixwithpositivedeniteimaginarypart). Ontheother handalgebrai
geometryanbeusedtostudytheintegrablesystem,forexampleto linearizetheowofthevetor
eldorto ndtransformationsbetweendierent systems (see[V1℄ andSetion 2.2below).
Thepresentpaperdealswithanintegrablesystemdenedbyaquartipotentialintwodegrees
offreedom,whosegeneriinvariantmanifoldsareAbeliansurfaesofpolarizationtype(1;4). Inone
diretion,thespeigeometry of these Abeliansurfaes willbe used to prove algebraiomplete
integrability of the potential and in the other diretion the expliit (aÆne) oordinates provided
bythesystem willbeused to prove some newresults and perform some expliitonstrutionsfor
Abelian surfaes of type (1;4). In this way we provideand exploitan essentially new ase of the
interationbetweenalgebraigeometry and a..i.systems (the presentpotentialis therst known
a..i. systemleading to Abeliansurfaesof type (1;4)).
The potentialisa quadratiperturbation
V
= q 2
1 +q
2
2
2
+q 2
1 +q
2
2
(1)
of thepotential
V
00
= q 2
1 +q
2
2
2
;
thelatterbeingobviouslyintegrablesineitisaentralpotential. However,althoughV
00
aswellas
V
areonlyLiouvilleintegrable(butnota..i.) theperturbationV
beomesa..i.for6=. V
an be interpreted as a potential whih desribes an anisotropi harmoni osillator in a entral
eld; remark that the entral eld V
00
is exeptional in the sense that an anisotropi harmoni
osillatorina generalentraleld isnotintegrable.
Newton'sequationsof motiontake thesymmetri form
q
1
= 2q
1 2q
2
1 +2q
2
2 +
;
q
2
= 2q
2 2q
2
1 +2q
2
2 +
;
and itis heked at onethat
F =( q
1 _ q
2 q
2 _ q
1 )
2
( ) q_ 2
1 +2q
4
1 +2q
2
1 q
2
2 +2q
2
1
isa onstant of motion,independentof theHamiltonian
H= 1
2 _ q 2
1 +q_
2
2
+ q 2
1 +q
2
2
2
+q 2
1 +q
2
2 :
It waspointedoutto mebyA. Perelomov that thispotentialwasrst studied by Garnierin
the beginning of this entury. In fat the Garnier system is a muh more general system whih
ontainsalot ofintegrablesystems;thederivation ofthepotentialsV
(andtheirgeneralizations
to higherdimensions)willbegiven intheAppendix(see [G℄, [P℄).
ToprovethatthepotentialsV
deneana..i.systemweusetheresultof[BLS℄(explainedin
Setion2.1)whihstatesthatthelinebundleLwhihdenesthepolarizationonageneriAbelian
surfaeof type (1;4) induesa birational map
L :T
2
!IP 3
, whoseimage is an oti of a ertain
type;an equation forthisotiis given withresptto well-hosenoordinatesforIP 3
by
2
0 y
2
0 y
2
1 y
2
2 y
2
3 +
2
1 (y
4
0 y
4
1 +y
4
2 y
4
3 )+
2
2 (y
4
1 y
4
3 +y
4
0 y
4
2 )+
2
3 (y
4
0 y
4
3 +y
4
1 y
4
2 )+
2
1
2 (y
2
0 y
2
1 +y
2
2 y
2
3 )(y
2
1 y
2
3 y
2
0 y
2
2 )+2
1
3 (y
2
0 y
2
3 y
2
1 y
2
2 )(y
2
0 y
2
1 y
2
2 y
2
3 )+
2
2
3 (y
2
1 y
2
2 +y
2
0 y
2
3 )(y
2
1 y
2
3 +y
2
0 y
2
2 )=0;
(2)
forsome(
0 :
1 :
2 :
3 )2IP
3
nSwhereSissomedivisorofIP 3
,whihwewilldetermine. Moreover
eah oti of this type ours inthat way. It willallow us to show that the invariant surfaes of
the Hamiltonian vetor eld assoiated to the potential V
;( 6= ), are Abelian surfaes, and
we show that the ow of this vetor eld is linear on the invariant tori. Combining these results
leads to the proof that the potentials V
dene an a..i. system for 6= and we derive a Lax
representationforit. Our proof ofalgebrai ompleteintegrabilityis unusualinthe sensethatwe
do notusetheLaurent solutionsto the dierentialequations(see [AvM3℄),northe Laxequations
(whihoftenonlyome up at theend)(see [Gr℄).
DotheAbeliansurfaesgeneratedbythepotentials(1)aountforallmoduliof(1;4)-polarized
Abelian surfaes? The answeris yes. In order to state preisely this answer (as given in Setion
4), we rst make adetailedstudy ofthe modulispae A
(1;4)
of Abeliansurfaes oftype(1,4) and
of some assoiatedmodulispaes(Setion 4). We usesome resultsfrom[BLS℄ to onstrut amap
from A
(1;4)
to an algebrai one M 3
of dimension 3, whih lives in weighted projetive spae
IP
(1;2;2;3;4)
. Themapisbijetiveonthedensesubset
~
A
(1;4)
ofAbeliansurfaesforwhihtheabove
map
L
isbirationalandtheimageisanaÆnevarietyM 3
nDwhereDissomedivisorinM 3
;the
two-dimensionalsubsetA
(1;4) n
~
A
(1;4)
whihonsistsofthose Abeliansurfaes(T 2
;L)forwhih
L
is2:1 howevermaps toa urve C (minustwopointsP;Q),whih itselfisadivisorinD. It follows
thatthe imageof themap :A
(1;4)
!IP
(1;2;2;3;4)
onsistsof theunion
I =(M 3
nD) [ (CnfP;Qg);
and theone M 3
an beonsidered asa ompatiation of A
(1;4)
. Equations forM 3
;D;C and
oordinatesforthe pointsP and Qwillbe expliitlyalulated. We prove that forevery pointin
the one M 3
(exept for its vertex) there is at least one invariant surfae of some potential V
orrespondingto itunder (Theorem3).
Wealsodeneamapfrom
~
A
(1;4)
ontothemodulispaeoftwo-dimensionalJaobians,orwhat
isthesame themodulispae of smoothurvesof genustwo. Namelywe show (Setion5) thatfor
every T 2
~
A
(1;4)
there exists exatlyone Jaobi surfaeJ =J(T ) (with urve = (T )) suh
that the map 2
J
(multipliationby 2 in J) fatorizes over T 2
(hene also over its dual
^
T 2
,) i.e.,
there isa ommutative diagram
J 4:1
!
^
T 2
?
?
y 4:1
2
J
&
?
?
y 4:1
T 2
4:1
! J
(3)
We all this Jaobian the anonial Jaobian (of T 2
); it will also appear naturally in Setion 3
whenlinearizingthevetorelddenedbythepotentialsV
. Onesees fromthediagramthatT 2
annot be reonstruted from J (or ); indeed T 2
indues a deomposition =
1
2
of any
lattie deningJ =C 2
= (and a partition W =W
1 [W
2
of the set of Weierstra pointsof )
and thisextradatum suÆesto reonstrut T 2
from J (or ). Thiswillbeshown inSetion5.
Theproblemarisesto alulatethismapexpliitely aswellastheextradata. We knowof no
diretalgebrai wayto do this. Insteadwe solve thisproblem(inSetion 6) by relyingheavilyon
the partiular oordinates provided by the potentials V
. Some geometrial investigations then
lead to thefollowingresult: theurve (T 2
) orresponding to T 2
isgiven by
y 2
=x(x 1) 4 2
2 x
3
( 2
0 +2
2
1 +6
2
2 +2
2
3 )x
2
+( 2
0 2
2
1 +2
2
2 +6
2
3
)x 4 2
3
;
when theoordinate x is hosen suh that it sends the points of W
2
to 0;1 and 1; W
1
ontains
theother 3Weierstra pointson this urve. We obtain thisresult intwo dierent ways: one way
uses the over J ! T 2
and the other uses the over T 2
! J. It would be nie to alulate this
mapina diret way,i.e., withoutusingtheV
.
In the nal setion (Setion 7) we study the degenerate ase V
as a limit of the generi
ase V
( 6= ). Sine the potentials V
are entral they are obviously integrated using polar
oordinates;theseoordinateswillbeobtainedasalimitof thelinearizingvariablesforthegeneri
ase (V
; 6=) as wellas theLax representation(with a spetralparameter). This shows that
thesystemati tehniquesdevelopped in[V1℄ to obtainlinearizingvariablesand Laxequationsfor
generitwo-dimensionala..i.systemsanleadtothesedataforintegrablesystemswhoseinvariant
manifolds are not Abelian varieties. We prove that in this degenerate ase the aÆne invariant
manifolds areC
-bundles over an ellipti urve, whih itself is the spetral urve going with the
Lax pair. Alsowe show that the invariant manifolds of all entral potentials V
orrespondsto
thespeial pointP 2M 3
at the boundaryof I.
1. Introdution
2. Preliminaries
2.1 Abeliansurfaes of type(1,4)
2.2 Two-dimensionala..i. systems
3. ThequartipotentialV
and itsintegrability
4. Somemodulispaes ofAbeliansurfaes oftype (1,4)
5. Thepreise relationwiththe anonial Jaobian
6. Therelationwith theanonial Jaobianmade expliit
6.1 Anembeddingof theAbeliansurfaesinIP 15
6.2 Abeliansurfaes of type(1;4) asquotientsof theiranonial Jaobians
6.3 Abeliansurfaes of type(1;4) asovers of theiranonial Jaobians
6.4 Theexeptional lousS IP 3
7. Theentralpotentials V
8. Appendix: The Shlesingersystem, theGarniersystemand thequartipotentials V
Inthis setion we reall some resultsaboutAbeliansurfaes of type (1,4) whih willbe used
in this paper (see [BLS℄, [GH℄, [LB℄), as well as the basi tehniques to study two-dimensional
(algebrai) ompletelyintegrable systems (see [V1℄).
2.1. Abelian surfaes of type (1,4)
Let be a rank 4 lattie inC 2
; and form the assoiated omplex torus T 2
=C 2
=. By a
theorem of Riemann,T 2
is an Abeliansurfae (i.e., an be embedded inprojetive spae) if and
onlyifthereexistsa omplexbasefe
1
;e
2 g forC
2
and an integer basef
1
;:::;
4
g forsuhthat
thelatter basean bewritten interms oftheformer as
=
Æ
1
0 a b
0 Æ
2 b
(i.e.,
1
=Æ
1 e
1
;::: )whereÆ
1 jÆ
2
2INand=
a b
b
>0:TheintegersÆ
1 andÆ
2
arenotinvariants
fortheAbeliansurfaeT 2
itself,butforT 2
equippedwithsome additional data: ifL isan ample
linebundleonT 2
(i.e.,alinebundleforwhihthesetionsofsomepowerofthelinebundleembeds
thesurfaeinprojetivespae) thenabase
1
;:::;
4
foranbehosensuhthattherstChern
lass
1
(L)is givenin termsof oordinatesx
1
;:::;x
4
;dual to
1
;:::;
4
;by
1
(L)=Æ
1 dx
1
^dx
3 +Æ
2 dx
2
^dx
4 :
1
(L)isalledthepolarization determined byLanddependsonlyonLuptoalgebraiequivalene;
Æ
1 and Æ
2
are invariants of
1
(L): The pair (Æ
1
;Æ
2
) is alled the type of L; (or the type of the
polarization
1
(L)). Loosely speaking we often say that the Abelian surfaeT 2
has type (Æ
1
;Æ
2 ).
T 2
is said to be prinipal polarized if it has type (1;1). A prinipal polarizedAbelian surfae is
either isomorphito a produtof elliptiurves(eah taken with its prinipal polarization), orto
theJaobian ofa smoothurveof genustwo, polarizedbyits thetadivisor .
ForageneriAbeliansurfaethelinebundleL=[D℄orrespondingto anyeetive divisorD
isample andone hasthefollowingusefulstringof identities:
g(D) 1=dimH 0
T 2
;O(L)
=Æ
1 Æ
2
; (4)
where g(D) is the virtual genus of D; whih an (for Abelian surfaes) be dened in terms of
intersetion of divisorsby
g(D)= DD
2
+1; (5)
ifDisnon-singular,g(D)isjustthetopologialgenusofD:ToLthereisassoiatedarationalmap
L :T
2
!IP Æ
1 Æ
2 1
whihis denedbymeansof the setionsof thesheaf O(L), or equivalentlyby
meansof theelements of L(D),where
L(D)=ff jf meromorphion T 2
and(f)+D0g:
In this paperwe onentrate on Abelian surfaes of type (1;4): These Abeliansurfaes have
a very rih geometry, whih we desribe now (see [BLS℄). As in [BLS℄ we will without further
mentionalwaysrestrit ourselvestothose Abeliansurfaesof type(1;4) whih arenotisomorphi
onan AbeliansurfaeT 2
:Itfollows from(4)that dimH 0
(T 2
;O(L))=4and Linduesa rational
map
L :T
2
!IP 3
:
In thegeneri ase, the image of this map O =
L (T
2
) IP 3
is an oti and
L
is birational
on its image. Let K(L)bethe kernelofthe isogeny
I
L :T
2
!
^
T 2
a7!t
a LL
1
betweenT 2
anditsdual
^
T 2
(denedasthesetofalllinebundlesonT 2
ofdegree0;t
a
istranslation
bya2T 2
),then K(L)is a groupof translations,isomorphito ZZ=4ZZZZ=4ZZ. Pikinganysuh
isomorphism,letand begeneratorsofthesubgroupsorrespondingtothisdeomposition. Then
homogeneousoordinates(y
0 :y
1 :y
2 :y
3 )forIP
3
anbepiked,suhthat; andthe( 1)-involution
{ onT 2
(dened as{(z
1
;z
2
)=( z
1
; z
2
) for(z
1
;z
2 )2C
2
=)atasfollows (see [M1℄):
(y
0 :y
1 :y
2 :y
3 )=(y
2 :y
3 :y
0 : y
1 );
(y
0 :y
1 :y
2 :y
3 )=(y
1 :y
0 :iy
3 :iy
2 );
{(y
0 :y
1 :y
2 :y
3 )=(y
0 :y
1 :y
2 : y
3 );
(6)
(stritlyspeakingitmaybeneessaryto replae by3;itiseasilyheked thattheseoordinates
existonlyfor(;) and(3;3) orfor(;3)and (3;)). [BLS℄showthattheotiO isgivenin
these oordinatesby
2
0 y
2
0 y
2
1 y
2
2 y
2
3 +
2
1 (y
4
0 y
4
1 +y
4
2 y
4
3 )+
2
2 (y
4
0 y
4
2 +y
4
1 y
4
3 )+
2
3 (y
4
0 y
4
3 +y
4
1 y
4
2 )+
2
1
2 (y
2
0 y
2
1 +y
2
2 y
2
3 )(y
2
1 y
2
3 y
2
0 y
2
2 )+2
1
3 (y
2
0 y
2
3 y
2
1 y
2
2 )(y
2
0 y
2
1 y
2
2 y
2
3 )+
2
2
3 (y
2
1 y
2
2 +y
2
0 y
2
3 )(y
2
1 y
2
3 +y
2
0 y
2
2 )=0;
(7)
for some (
0 :
1 :
2 :
3 ) 2 IP
3
nS where S is some divisor of IP 3
whih we will determine later
(Setion 6.4). Remark that for any
i
= 1; the oordinates (
0 y
0 :
1 y
1 :
2 y
2 :
0
1
2 y
3
) will also
satisfy (6) and these are the only oordinates with this property. It is also seen that, if (;) is
replaedby(3;3),thentheoordinates(y
0 :y
1 :y
2 :y
3
)arereplaed by(y
0 :y
1 :y
2 : y
3
):Sinethe
equationofOdependsonlyony 2
i
thesehoiesdonotaettheequation(7), sothereisassoiated
to a deomposition K(L) = K
1 K
2
(where K
1
and K
2
are yli of order 4) an equation for
O: [BLS℄ also show that the polarized Abelian surfae as well as the deomposition of K (L) an
be reovered from (7) and that every oti of thetype (7) (with (
0 :
1 :
2 :
3
)2= S) is theimage
L (T
2
) of some (1;4)-polarizedAbeliansurfae(T 2
;L).
If we denote by
~
A 0
(1;4)
the modulispae of (isomorphism lasses of) (1;4)-polarized Abelian
surfaesforwhih
L
isbirational,equippedwithadeompositionofK(L)asabove,thenitfollows
that
~
A 0
(1;4)
= IP
3
nS
0
0
: (8)
Moreover, ifwe denotebyK thesubgroupofK(L)of two-torsionelements,
K=f0;2;2;2 +2g;
thenT =K isa prinipalpolarizedAbeliansurfae,whihistheJaobianof aurve ofgenustwo;
we allT 2
=K theanonial Jaobian assoiatedto T 2
:Reallthatforatwo-dimensionalJaobian
J its Kummer surfae is the image of
[2℄
IP 3
, where is the theta divisor of J. Then it
is seen from (6) that an equation forthe Kummer surfae of T 2
=K is given by the quarti Q in
IP 3
, obtained by replaing y 2
i by z
i
in the equation (7) for O and there is an obvious projetion
p:O!Q. Infat, hoosingtheoriginof T 2
suhthatL beomessymmetri,L isthepull-bakof
a linebundleN onT 2
=K of type (1,1) via theanonial projetion
p:T 2
!T 2
=K ;
and
N
2 induestheKummermapping;[BLS℄ prove thatthefollowing diagramommutes
T 2
L
! O
?
?
y p
?
?
y p
T 2
=K
N 2
! Q
(9)
If
L
isnotbirational,thenitis2:1and
L (T
2
)isaquartiinIP 3
;givenbyoneoftheequations
1 (y
2
0 y
2
1 +y
2
2 y
2
3 )+
2 (y
2
1 y
2
3 y
2
0 y
2
2 )=0;
1 (y
2
2 y
2
3 y
2
0 y
2
1 )+
3 (y
2
1 y
2
2 y
2
0 y
2
3 )=0;
2 (y
2
1 y
2
3 +y
2
0 y
2
2 )+
3 (y
2
1 y
2
2 +y
2
0 y
2
3 )=0;
depending on the hoie of the deomposition; in this ase the Abelian surfae as well as the
deompositionof K(L)an onlypartly bereovered fromthese equationsand T 2
=K isa produt
of elliptiurves (inpartiular T 2
is isogeneousto a produtof elliptiurves). Squaring eah of
these equationswendequation (7)respetivelywith
(
2
0
=2(
2
2 +
2
3 )
1
=0
2
3
6=0;
2
2
2
3 6=0;
(
2
0
= 2(
2
1 +
2
3 )
2
=0
1
3
6=0;
2
1
2
3 6=0;
(
2
0
=2(
2
1
2
2 )
3
=0
1
2
6=0;
2
1 +
2
2 6=0;
(10)
Summarizing,intherst ase (thegeneri ase),
L (T
2
) is anoti,T 2
=K is aJaobian and
T 2
as well as the deomposition of K(L) an be reonstruted from the oti; in the other ase
L (T
2
) isaquarti,T 2
=K isaprodutof elliptiurvesandT 2
annotbereonstruted fromthe
quarti. The rationalmap
L
providesuswitha naturalsurjetive map
0
:A 0
(1;4)
!
IP 3
nS
[
(threerationalurvesinS,eah missingeight points)
Æ
(
0
0 );
whereA 0
(1;4)
denotesthemodulispae of(isomorphismlassesof) (1;4)-polarizedAbeliansurfaes
together witha deompositionof K(L) (as above). The map 0
extends thebijetion(8) dened
on the dense subset
~
A
(1;4) of A
(1;4)
and maps the (two-dimensional)omplement of
~
A
(1;4) to the
three rational urves, whih are thought of as lying inside the boundary of 0
(
~
A 0
(1;4)
), i.e., in S;
thegeneri pointof S however doesnotorrespond to Abeliansurfaes,butto surfaes whih an
beinterpretedas degenerationsof Abeliansurfaes (see[BLS℄).
2.2. Two-dimensional a..i. systems
We now reall thebasi tools to study two-dimensionala..i. systems (see [AvM1℄, [V1℄). At
rst, an integrable system on (IR 2n
;!) (! may be any sympleti struture on IR 2n
butthe ase
that ! is the standard sympleti struture will suÆe for this paper) onsists of a Hamiltonian
vetor eld X
H
,denedas
!(X
H
;)=dH();
forwhih thereexist n 1 additionalinvariants,i.e., there arenindependent,Poisson-ommuting
funtions H
1
;:::;H
n on IR
2n
; Poisson-ommuting funtions F;G 2 C 1
(IR 2n
), are by denition
funtions forwhihtheir Poisson braket fF;Gg
!
=!(X
F
;X
G
) vanishes. Theintersetion
n
\
i=1
x2IR 2n
jH
i
(x)=
i
is by Poisson-ommutativityinvariant for theows of all X
Hi
and is smooth forgeneri values of
=(
1
;:::;
n
). Bythewell-knownArnold-LiouvilleTheorem,theompatonnetedomponents
of these invariant manifolds are dieomorphi to real tori (the non-ompat omponents being
dieomorphi to ilindres, assuming that the ow of the vetor eldsX
Hi
is omplete on them);
moreover the ows of the vetor elds X
H
i
are linear, when seen as ows on the tori (ilindres)
usingthedieomorphism. n isalled thedimensionof thesystem.
A notable ase | whih appears most often in both the lassial and reent, mathematial
and physis literature | is the ase that there exist oordinates q
1
;:::;q
2n for IR
2n
, in whih
all H
i
;(i = 1;:::;n) as well as all brakets fq
i
;q
j g
!
;(i;j = 1:::;2n) are polynomials (stritly
speaking,forthelargerlassoftheseexamples(IR 2n
;f;g
!
)isreplaedbythemoregeneralPoisson
manifold (IR m
;f;g), where f;g does not neessarily ome from a sympleti struture). Then
the sympleti strutureand thevetor eld are easilyomplexied, giving a Poisson ommuting
familyof funtions onC 2n
and for generi=(
1
;:::;
n
)(where the
i
maynowalso take values
inC) theinvariant manifolds
A
= n
\
i=1
x2C 2n
jH
i
(x)=
i
are aÆne (algebrai) varieties. In suh a situation, the integrable system will be alled algebrai
ompletely integrableifthese generi invariant manifoldsA
areaÆne partsof an Abelianvariety
T n
,A
=T n
nD
, where D
is the minimaldivisor where theoordinate funtions (restritedto
theinvariant manifolds)blowup,and ifthe(omplex) owof thevetor eldson T
islinear (see
[AvM3℄).
Inthetwo-dimensionalase(n=2)theinvariantmanifoldsompleteintoAbeliansurfaesby
addingoneorseveral(possiblysingular)urvestotheaÆnesurfaesA
. Inthisase, thefollowing
algorithm, proposed in[V1℄ leads to an expliit linearization(i.e., integration) of the vetor eld
X
H
(steps (1)and (2)are dueto Adlerand van Moerbeke, see[AvM1℄).
these to onstrut an embedding of the generi invariant manifolds in projetive spae (see
[AvM3℄, [V1℄ and [V2℄).
(2) Dedue from the embedding the struture of the divisors D
to be adjoinedto the (generi)
aÆne invariant manifolds A
in order to ompletethem into Abeliansurfaes. At this point
thetypeof polarizationinduedbyeah irreduibleomponent ofD
an alsobedetermined.
(3) a) If one of the omponentsof D
is a smooth urve
of genus two, ompute the image of
therationalmap
[2
℄ :T
2
!IP 3
whih isa singularsurfaeinIP 3
,theKummer surfaeK
of Ja (
).
b) Otherwise, ifone of theomponents of D
is a d:1 unramiedover C
of a smooth urve
of genus two, p:C
!
, the map p extends to a map p: T 2
! Ja (
). In thisase, let
E
denotethe(non-omplete) linearsystemp
j2
jj2C
jwhihorrespondsto theomplete
linearsystemj2
jand omputenowthe KummersurfaeK
ofJa (
) astheimage of
E
:T
2
!IP 3
:
)Otherwise,hangethedivisorat innitysoasto arrive inasea) orb). Thisanalwaysbe
done fora generi Abelian surfae (i.e., for an Abelian surfae whih has no automorphisms
exeptidentity andtheobvious ( 1)-involution).
(4) Choosea WeierstrapointW on theurve
and oordinates(z
0 :z
1 :z
2 :z
3
) forIP 3
suhthat
[2
℄
(W)=(0:0:0:1) inase (3)a) and
E
(W)=(0:0:0:1) inase (3) b). Then thispoint
willbe a singularpoint(node)for K
and K
hasan equation
p
2 (z
0
;z
1
;z
2 )z
2
3 +p
3 (z
0
;z
1
;z
2 )z
3 +p
4 (z
0
;z
1
;z
2 )=0;
where the p
i
are polynomials of degree i. After a projetive transformation whih xes
(0:0:0:1) we mayassume that
p
2 (z
0
;z
1
;z
2 )=z
2
1 4z
0 z
2 :
(5) Finally,letx
1 andx
2
betherootsofthequadratiequationz
0 x
2
+z
1 x+z
2
=0,whosedisrim-
inant is p
2 (z
0
;z
1
;z
2
), withthez
i
expressed interms ofthe originalvariables q
1
;:::;q
4
. Then
the dierential equations desribing the vetor eld X
H
are rewritten by diret omputation
inthelassialWeierstra form
dx
1
p
f(x
1 )
+ dx
2
p
f(x
2 )
=
1 dt;
x
1 dx
1
p
f(x
1 )
+ x
2 dx
2
p
f(x
2 )
=
2 dt;
where
1
and
2
depend on (i.e., on the torus) only. From it, the symmetri funtions
x
1 +x
2 (= z
1
=z
0
) and x
1 x
2 (=z
2
=z
0
) and henealso theoriginal variables q
1
;:::;q
4
an be
writteninterms ofthe Riemanntheta funtionassoiated to theurvey 2
=f(x).
The best way to see that this algorithm is very eetive and easy to apply is to look at one or
severalof theworked-out examplesin[V1℄. Inthepresentpaperthisalgorithmwillnotbeusedas
itstands, sinewe donotknowinadvanethat oursystemisa..i.; insteadwe willseehowit an
be helpfulwhen provingalgebraiomplete integrability. We remarkthat itis shown in[V1℄ how
a Laxpair forthesystemderivesfrom theabove linearization.
Itis shownin [CC℄thatforany=(
1
;:::;
n
);thepotential
V
= n
X
i=1 q
2
i
!
2
+ n
X
i=1
i q
2
i
; (11)
denes an integrable system on IR 2n
= f(q
1
;:::;q
n
;p
1
;:::;p
n ) j q
i
;p
i
2 IRg; equipped with the
standardsympleti struture!= P
dq
i
^dp
i
;whentheHamiltonianis taken asthetotal energy
H
=T+V
;T = 1
2 n
X
i=1 p
2
i
;
(T isthekineti energy). Thisresultalso followsimmediatelyfromtheintegrabilityof theGarnier
system, whih will be realled in the Appendix. We study here the ase n = 2 (two degrees of
freedom) writing
V
=(q 2
1 +q
2
2 )
2
+q 2
1 +q
2
2 :
Itwouldbeinterestingtostudyalsothehigher-dimensionalpotentialsaswellasotherases ofthe
Garniersystemfrom thepoint of viewof algebraigeometry.
Fixingarbitrary parameters 6=; let H =T +V
: Then theequations forthe vetor eld
X
H
;denedby!(X
H
;)=dH()are givenby
_ q
1
=p
1
;
_ q
2
=p
2
;
_ p
1
= 2q
1 (2q
2
1 +2q
2
2 +);
_ p
2
= 2q
2 (2q
2
1 +2q
2
2 +):
(12)
Foranyf;gonsidertheaÆne surfaeA
fg
denedby
F (q
1 p
2 q
2 p
1 )
2
+( )(p 2
1 +2q
4
1 +2q
2
1 q
2
2 +2q
2
1 )=f;
G(q
1 p
2 q
2 p
1 )
2
+( )(p 2
2 +2q
4
2 +2q
2
1 q
2
2 +2q
2
2 )=g;
(when the dependene on and is important we will denote this surfae by A
where =
(;;f;g)). Then A
fg
is invariant under the ow of X
H
sine both F and G Poisson ommute
withH. Sine
F G=2( )H
and 6=,anypairoffuntionstakenformfF;G;Hganbetakenasamaximalsetofindependent
Poisson ommuting funtions; in order t o simplify some of the formulas inthe sequel we let, for
given f and g;theonstant h be determinedbyf g=2( )h:
Thesurfae A
fg
has thefollowingindependentinvolutions:
{
1 (q
1
;q
2
;p
1
;p
2
)=( q
1
;q
2
; p
1
;p
2 );
{
2 (q
1
;q
2
;p
1
;p
2 )=(q
1
; q
2
;p
1
; p
2 );
whih bothpreservethevetor eld,andone other (independent)involution
|(q
1
;q
2
;p
1
;p
2 )=(q
1
;q
2
; p
1
; p
2 );
phi to (ZZ=2ZZ) 3
. Moreover one sees that for xed ;;f and g all A
(;;
3
f;
3
g)
; 2C
are
isomorphi. It is therefore natural to onsider(;;f;g) asbelonging to the weightedprojetive
spae 1
IP (1;1;3;3)
. A trivialobservation whih will turn outto be important is that also A
(;;f;g)
and A
(;;g;f)
areisomorphi.
Remarkthat if= thenF(=G) is justthesquare ofthemomentum
q=q
1 p
2 q
2 p
1
; (13)
whih obviously Poisson-ommutes withthe energy orresponding to a entral potential. What is
remarkablehoweveristhatif6=thentheequationsdeningA
fg
an berewritten(birationally)
interms ofq
1
;q
2
and themomentumq;giving preiselytheequations(7) oftheoti O with
2
0
=4( ) 2
(+) 2(f+g);
2
1
=g;
2
2
=2( ) 3
;
2
3
=f;
y
0
=1;
y
1
=q
1 4 p
2( )=f;
y
2
=q=
4 p
fg;
y
3
=q
2 4 p
2( )=g:
(14)
It follows that for generi f;g the surfae A
fg
is birationally equivalent to the aÆne part O
0
=
O\fy
0
6= 0g of the oti O whih is itself birationally equivalent to an Abelian surfae of type
(1;4). We show inthe followingtheorem thatA
fg
atually is(isomorphi to)an aÆne part of an
Abeliansurfaeof type (1;4).
Theorem 1 Fixing any 6= 2C,the aÆne surfae A
fg C
4
dened by
(q
1 p
2 q
2 p
1 )
2
+( )(p 2
1 +2q
4
1 +2q
2
1 q
2
2 +2q
2
1 )=f;
(q
1 p
2 q
2 p
1 )
2
+( )(p 2
2 +2q
4
2 +2q
2
1 q
2
2 +2q
2
2 )=g;
is for generi 2
f;g 2 C isomorphi to an aÆne part of an Abelian surfae T 2
fg
; of type (1;4),
obtained by removing a smooth urve D
fg
of genus5,
A
fg
=T 2
fg nD
fg
;
and the vetor eld X
H
extends to a linear vetor eld on T 2
fg :
Proof
(i) Let G be the group generated by the involutions {
1
;{
2
, and |. Our rst aim is to show that
A
fg
=G is (isomorphito) an aÆne part of a Kummersurfae. Sine f and g aregeneri, we may
supposethat (
0 :
1 :
2 :
3
) given by(14) do not belongto S. For these
i
, let Qbe the quadri
(Kummer surfae)
2
0 z
0 z
1 z
2 z
3 +
2
1 (z
2
0 z
2
1 +z
2
2 z
2
3 )+
2
2 (z
2
0 z
2
2 +z
2
1 z
2
3 )+
2
3 (z
2
0 z
2
3 +z
2
1 z
2
2 )+
2
1
2 (z
0 z
1 +z
2 z
3 )(z
1 z
3 z
0 z
2 )+2
1
3 (z
0 z
3 z
1 z
2 )(z
0 z
1 z
2 z
3 )+
2
2
3 (z
1 z
2 +z
0 z
3 )(z
1 z
3 +z
0 z
2 )=0;
(15)
1
aquikintrodutionto weightedprojetivespaes isgiven inan appendixto[AvM3℄
2
preise onditionswillbe given later(Theorem6)
whih is obtainedfrom (7) bysetting z
i
= y
i
;i.e., there is an unramied 8:1 over O ! Q; this
maprestritstoamapp
0 :O
0
!Q
0
;whereQ
0
=Q\fz
0
6=0g:Alsotherationalmap:A
fg
!O
0
given by (13) and (14) indues a birational map
~
:A
fg
=G ! Q
0
; giving rise to a ommutative
diagram
A
fg
! O
0
?
?
y
?
?
y p
0
A
fg
=G
~
! Q
0
(16)
SineQ
0
isnormal,itsuÆesto showthat
~
isbijetive. Obviously
~
issurjetive: if(x
1
;x
2
;x
3 )2
Q
0
;let(y
1
;y
2
;y
3
)besuhthaty 2
i
=x
i
andletq
1
;q
2
;qbedeterminedfrom(14). Thenthesesatisfy
the ondition under whih p
1
;p
2
exist suh that (q
1
;q
2
;p
1
;p
2 ) 2 A
fg
and q =q
1 p
2 q
2 p
1 : Then
~
(q
1
;q
2
;p
1
;p
2 )=(x
1
;x
2
;x
3
). Attheotherhand,if(
~
Æ)(q
1
;q
2
;p
1
;p
2 )=(
~
Æ)(q 0
1
;q 0
2
;p 0
1
;p 0
2 )then
q
1
=
1 q
0
1
;q
2
=
2 q
0
2
;q=q 0
; (whereq 0
=q 0
1 p
0
2 q
0
2 p
0
1 ) for
1
;
2
;2f 1;1g: Thenone sees that
(q
1
;q
2
;p
1
;p
2 )={
1
1 {
2
2 {
(q 0
1
;q 0
2
;p 0
1
;p 0
2 );
where i k
k
means {
k
in ase
k
= 1 and identity for
k
= 1: It follows that (q
1
;q
2
;p
1
;p
2 ) =
(q 0
1
;q 0
2
;p 0
1
;p 0
2 );and
~
isinjetive. Thisshowsthat
~
isanisomorphism,heneA
fg
=Gisisomorphi
to the(aÆne) KummersurfaedenedbyQ
0 :
(ii) We proeed to show that A
fg
is isomorphi to an aÆne part of an Abelian surfae, more
preisely to the normalization A of O
0
(the oti is singular along the oordinate planes). This
normalization an beobtained via thebirational map
L :T
2
!O:Inpartiular, byrestrition of
(9)to an aÆne piee we geta ommutative diagram
A L
! O
0
?
?
y p0
?
?
y p
0
K
0
N 2
! Q
0
(17)
where
N 2
is anisomorphism. If we ombine bothdiagrams(16)and (17) we get
A
fg '
! A
?
?
y8:1
?
?
y8:1
A
fg
=G
~ '
! K
0
with'thebirationalmap 1
L
and '~ theisomorphism 1
N 2
~
. Nowthetwoovers A
fg
!A
fg
=G
and A ! K
0
are only ramiedin disrete points; the same holdstrue if A and A
fg
are replaed
bytheir losures: thelosure of A isjust T 2
and thelosure of A
fg
isobtained from the expliit
embeddingwhihwillbegivenin6.1. ByZariski'sMainTheoremthenormalityofT 2
impliesthat
thelifting'of '~ mustalso bean isomorphismand we get
A
fg
=T 2
fg nD
fg
forsomedivisorD
fg
ona(1;4)-polarizedAbeliansurfaeT 2
fg
:ItisseenthatD
fg
isa4:1unramied
over of a translate of the Riemanntheta divisorof theanonial Jaobian, hene D
fg
is smooth
and hasgenus5;an equation forD
fg
willbegiven inSetion6.
(iii) Finally we show that X
H
extends to a linear vetor eld on T
fg
: Letting
0
= 1;
1
= q
1
and
3
= q 2
; we have shown that an equation for theKummer surfae of theanonial Jaobian
assoiatedtoA
fg
isa quartiinthese variables. From (14)and (7)theleadingtermin 2
3
isgiven
by((+)
0 +
1 +
2 )
2
4(
0 +
1
+
2
); or,interms ofthe originalvariables,
(q 2
1 +q
2
2
++) 2
4(+q 2
2 +q
2
1
): (18)
We let x
1 and x
2
betheroots ofthe polynomial
x 2
+ q 2
1 +q
2
2
++
x++q 2
2 +q
2
1
;
as suggested by the algorithm realled in Setion 2.2 (\suggested" beause we did not prove yet
thatthe systemisa..i.). Expliitely,let
x
1 +x
2
= (q 2
1 +q
2
2
++);
x
1 x
2
=+q 2
2 +q
2
1
;
_ x
1 +x_
2
= 2(q
1 p
1 +q
2 p
2 );
x
1 _ x
2 +x_
1 x
2
=2(q
1 p
1 +q
2 p
2 );
(19)
thenit is nothardto rewritethe equationsF =f;G=g;deningA
fg
;interms of x
1
;x
2
;x_
1
;x_
2 :
Thisgives
_ x 2
i
= 8(x
i
+)(x
i
+) x 3
i
+(+)x 2
i
+( h)x
i
+(f g)=2( )
(x
1 x
2 )
2
sothat
dx
1
p
f(x
1 )
+ dx
2
p
f(x
2 )
=0;
x
1 dx
1
p
f(x
1 )
+ x
2 dx
2
p
f(x
2 )
=2 p
2dt;
(20)
where
f(x)=(x+)(x+)
x 3
+(+)x 2
+( h)x+
f g
2( )
:
Integrating(20)weseethatX
H
isalinearvetor eldonA
fg
;whihobviouslyextendsto alinear
vetor eld on T 2
fg
: From this expression the symmetri funtions x
1 +x
2
and x
1 x
2
; hene the
variablesq
1
;q
2
;p
1
;p
2
an be writtenat oneinterms of theta funtions(see [M2℄).
Remarkthat asa by-produtwe ndanequation
y 2
=(x+)(x+)
x 3
+(+)x 2
+( h)x+
f g
2( )
: (21)
fortheurve whoseJaobian istheanonial Jaobian assoiatedto T 2
fg :
Thetheorem leadsto thefollowingimportant orollary:
Corollary 2 If 6= then the potential
V
= q 2
1 +q
2
2
2
+q 2
1 +q
2
2
denes an a..i.system (inthe senseof [AvM1℄)on IR withthe anonial sympletistruture. A
Lax representation of the vetor eld X
H
, where H= 1
2 (p
2
1 +p
2
2 )+V
, isgiven by
d
dt
v(x) u(x)
w(x) v(x)
= p
2
v(x) u(x)
w(x) v(x)
;
0 1
x 2(q 2
1 +q
2
2 ) 0
;
where
u(x)=x 2
+(q 2
1 +q
2
2
++)x++q 2
2 +q
2
1
;
v(x)= 1
p
2 [ (q
1 p
1 +q
2 p
2
)x+(q
1 p
1 +q
2 p
2 )℄;
w(x)=x 3
+(+ q 2
1 q
2
2 )x
2
p 2
1 +p
2
2
2
+(+) q 2
1 +q
2
2
x
p 2
1
2 +
p 2
2
2 +q
2
1 +q
2
2
:
Proof
TheLiouvilleintegrabilityisprovenin[G℄and[CC℄;itisinouraseproveneasilybyshowing
that fF;Gg=0 (F;G Poissonommute) and that F and Gare independent on adense subset of
IR 4
: To show that for 6= the system is a..i. we need to prove in addition the following three
laims:
(i) thegeneri (omplex) aÆne invariant surfaeA
fg
is an aÆne part of an Abeliansurfae
T 2
fg
; A
fg
=T 2
fg nD
fg
;whereD
fg
issome divisor onT 2
fg ,
(ii) D
fg
istheminimaldivisorwhere thevariablesq
1
;q
2
;p
1 andp
2
blowup,
(iii) thevetor eldsX
F
andX
H
extendto holomorphi(=linear)vetor eldson T 2
fg :
(i) and half of (iii) are shown in Theorem 1. To show the other half of (iii), whih onerns the
extensionofX
F
,thelinearizingvariablesaredenedinthesameway,buttheirderivativesarenow
alulatedusingX
F
instead ofX
H
:Finally,sinethevariablesq
1
;q
2
;p
1 and p
2
do notblowup on
A
fg
;and sineD
fg
isirreduible,theyall blowupalong D
fg
,showing(ii).
To onstruta Laxpair,notethatifu(x) isdenedasu(x)=(x x
1
)(x x
2
) and v(x)is its
derivative (suitablenormalised),then
f(x) v 2
(x) isdivisiblebyu(x);
wheref(x) isthepolynomialintroduedintheproof ofTheorem1. Thequotient
w(x)=
f(x) v 2
(x)
u(x)
iseasily alulated. The formof theLax pairthen follows from[V1℄.
Inthissetionwedesribeamap fromthemodulispaeA
(1;4)
ofpolarizedAbeliansurfaes
of type (1;4) into an algebrai one M 3
in some weighted projetive spae. To be preise we
reallthat(1;4)-polarizedAbeliansurfaeswhihareprodutsof elliptiurves(withtheprodut
polarization)areexludedfrom A
(1;4)
. The mapwillbe bijetive onthedensesubset
~
A
(1;4) whih
isthemodulispaeofpolarizedAbeliansurfaes(T 2
;L)forwhihtherationalmap
L :T
2
!IP 3
isbirational. An alternative wayto onstrut themap andtheone M 3
willome up later.
Reallfrom Setion2 thatA 0
(1;4)
mapsonto
P = IP
3
nS
0
0 [
(threerationalurvesinS,eah missingeight points),
bijetivelyontherstomponent(whihisdense);thethreerationalurvesarethoughtofaslying
inIP 3
=(
0
0
)attheboundaryofthisomponent. A 0
(1;4)
isa24:1(ramied)overingofA
(1;4) :
let and be elementsof order 4 suh thatK(L)=hihi,and dene
K
1
=f0;;2;3g;
K
2
=f0;;2;3g;
K
3
=f0;+;2+2;3+3g;
K
4
=f0;+2;2;3 +2g;
K
5
=f0;2+;2;2+3g;
K
6
=f0;+3;2+2;3+g:
These are the only ylisubgroups of order 4 of K(L). It is easy to see that taking all possible
isomorphismsK(L)
=
ZZ=4ZZZZ=4ZZwe ndexatlythe24 deompositions
K(L)=K
i K
j
; (1i;j 6;ji jj6=0;3):
We desribethe over
A 0
(1;4) 24:1
! A
(1;4)
andonstrut a24:1 overP !M 3
anda map :A
(1;4)
!M
3
,whereM 3
isan algebraivariety
(lyinginweightedprojetive spae IP
(1;2;2;3;4)
),suhthat there resultsaommutative diagram
A 0
(1;4) 24:1
! A
(1;4)
~
A
(1;4)
?
?
y 0
?
?
y
?
?
y
~
P
24:1
! M
3
M
3
nD
(22)
inwhihtherestrition
~
of to
~
A
(1;4)
isabijetion(DisadivisoronM 3
whihwillbedetermined
expliitely).
Themainideainthisonstrutionis tosee how theGaloisgroupoftheover A 0
(1;4)
!A
(1;4)
ats on P and dene M 3
to be thequotient. This quotient will be easy to alulate sine it is a
quotientof(aZariskiopensubsetof)IP 3
byagroupwhihatslinearly. Thefatthatthisationis
sosimpleissurprizingandwassuggested tousbytheobviousobservationthattheaÆneinvariant
surfaesA
and A
0,
with=(;;f;g) and 0
=(;;g;f) areisomorphi,showingby(14)that
1 and
2
an (insome way) be interhanged.
thatK(L)=hihi and
a b
d
2Gthen
a b
d
(;)=(a+b;+d);
giving anew deompositionK(L)=ha+bih+di: We denote byH thenormalsubgroup
of G whih onsists of those elements of G whih are ongruent to the identity matrix, modulo
2. Then H ats on the set of deompositions of K(L), thus H ats on A 0
(1;4)
; to determine the
orresponding ation on the isomorphi spae P, it is suÆient to take any element of H, at to
obtaina newbaseanddeterminethenewoordinates(y
0 :y
1 :y
2 :y
3
)aording to(6). Substituting
these in (7) the new parameters (
0 :
1 :
2 :
3
) are found immediately. The result is ontained
in the following table (sine diagonal matries at trivially only one representative of eah oset
modulo diagonalmatries isshown):
H base K(L) oo. forIP
3
moduliin P
1 0
0 1
(;) K
1 K
2
(y
0 :y
1 :y
2 :y
3
) (
0 :
1 :
2 :
3 )
1 2
0 1
(+2;) K
4 K
2
(y
0 :y
1 :iy
2 :iy
3
) (
0 :
1 :
2 :
3 )
1 0
2 1
(;2+) K
1 K
5
(y
0 :iy
1 :y
2 :iy
3
) (
0 :
1 :
2 :
3 )
1 2
2 1
(+2;2+) K
4 K
5 (y
0 :iy
1 :iy
2 : y
3
) (
0 :
1 :
2 :
3 )
Table1
The upshot of the table is that all (
0 :
1 :
2 :
3
) orrespond to the same Abelian
surfae. Thequotient spaeis given by
P 0
=
P
(
0 :
1 :
2 :
3
)(
0 :
1 :
2 :
3 )
= IP
3
nS 0
[
(threerationalurvesinS 0
,eah missingthree points),
(23)
upon dening
i
= 2
i
as oordinates forthe quotient IP 3
, from whih inpartiular equationsfor
the three rational urves, as wellas for the three points are immediately obtained (the fat that
there are three missing points instead of two is due to ramiation of the quotient map at two
of the three points). The divisors S and S 0
will be alulated later. We will also interpret this
\intermediate" modulispae P 0
.
Remark thatG=H isisomorphi to thepermutationgroup S
3
;so we have an ationof S
3 on
P 0
(whihextends to all ofIP 3
sineit islinear). ChoosingsixrepresentativesforG=H we ndas
above the followingtable:
S
3
G=H base K(L) oo.forIP moduliinP
0
()
1 0
0 1
(;) K
1 K
2
(y
0 :y
1 :y
2 :y
3
) (
0 :
1 :
2 :
3 )
(12)
0 1
3 0
(;3) K
2 K
1
(y
0 :y
2 :y
1 :iy
3
) (
0 :
2 :
1 :
3 )
(13)
1 0
1 1
(;+) K
1 K
3
( p
i y
2 :y
1 :
p
iy
0 :y
3
) (
0 :
3 :
2 :
1 )
(23)
1 1
0 1
(+;) K
3 K
2
(y
1 :y
0 :
p
iy
2 :
p
iy
3
) (
0 :
1 :
3 :
2 )
(123)
0 3
1 1
(3;+) K
2 K
3 (
p
iy
1 :iy
2 :
p
iy
0 :y
3
) (
0 :
3 :
1 :
2 )
(321)
1 1
3 0
(+;3) K
3 K
1 (
p
iy
2 :
p
iy
0 : y
1 : iy
3
) (
0 :
2 :
3 :
1 )
Table2
The tables 1 and 2 together show how to reonstrut expliitly the deomposition of K(L)
from the equation of the oti. More important, it allows us to onstrut the quotient spae M 3
asis showninthefollowingtheorem.
Theorem 3 There isa bijetive map
~
:
~
A
(1;4)
!M
3
nD, where M 3
is the one dened by
f 2
4
=f
1 (4f
3
2 27f
2
3 )
in weighted projetive spae IP
(1;2;2;3;4)
(with oordinates (f
0 ::f
4
)) and D = D
1 +D
2 is the
divisor whosetwo irreduible omponents areut o from M 3
by the hypersurfaes
D
1 :f
4
=f
1 (f
1 3f
2 );
D
2 :512f
4
= 16 16f 2
2 +72f
1 f
2 27f
2
1 48f
0 f
3
+3f 2
0 f
2
0 +24f
1 32f
2
:
(24)
In partiular the moduli spae
~
A
(1;4)
has the struture of an aÆne variety. Themap
~
extends in
a natural way toa map
:A
(1;4)
!M
3
;
theimageofthe(two-dimensional)boundaryA
(1;4) n
~
A
(1;4)
beingCnfP;Qg;whereC istherational
urve (inside D) given by
C:3f 2
0
=4(4f
2 f
1 );
and P;Q2C are given by P =(4:0:3:2:0); and Q=(2:1:1:0: 2): Moreover, apart from its top
(1:0:0:0:0), all points in the one M 3
orrespond to some invariant surfae A
(;;f;g)
for some
;;f and g, with6=:
Proof
Firstwe desribe the quotient of IP 3
bythe ationof S
3
, and show that it is(isomorphi to)
the algebrai varietyM 3
given by an equation f 2
4
= f
1 (4f
3
2 27f
2
3
) inweighted projetive spae
IP . To do this we use the (indued) ation of S
3
onC whih is given in terms of aÆne
oordinatesx
i
=
i
=
0 forC
3
by
(1;2)(x
1
;x
2
;x
3
)=( x
2
; x
1
; x
3 );
(1;2;3)(x
1
;x
2
;x
3
)=( x
3
;x
1
; x
2 ):
Sinetheationisorthogonal,itmustbereduible,havingan invariantlineandaninvariantplane
orthogonal to it. Indeedlet
u
1
=x
1 +x
2 x
3
;
u
2
=x
1 x
2
;
u
3
=x
1 +x
3
;
(25)
thenu
1
isanti-invariant for(1;2) and is invariantfor(1;2;3); u
2 andu
3
arehosen orthogonalto
u
1
. Then invariants
f
2
=u 2
2 u
2 u
3 +u
2
3
;
f
3
=u
2 u
3 (u
2 u
3 );
fortheationof S
3
arefound. Alsothere is
=u 2
2 (2u
2 3u
3 )+u
2
3 (2u
3 3u
2 )
whih is(1;2)-anti-invariant and(1;2;3)-invariant, givinga newinvariant f
4
=u
1
. Sine f
2 and
f
3
generate the invariants dependingon u
2
;u
3
the invariant 2
is expressible in terms of f
2 and
f
3
;
2
=4f 3
2 27f
2
3
;
i.e., 2
isnothingelsethanthedisriminantof theubipolynomialx 3
f
2 x+f
3
:Itfollowsthat
f 2
4
=f
1 (4f
3
2 27f
2
3
); (26)
wheref
1
=u 2
1
. Remarkthat(f
1
;f
2
;f
3
;f
4
) havedegree (2;2;3;4) sothatthequotientofIP 3
bythe
ation of S
3
is given by (26) viewed as an equation in weighted projetive spae IP
(1;2;2;3;4)
with
respetto oordinates(f
0 :f
1 :f
2 :f
3 :f
4
):Inonlusionwe haveestablishedtheoverP !M 3
and
there isan induedmap :A
(1;4)
!M
3
whih makes
A 0
(1;4) 24:1
! A
(1;4)
?
?
y 0
?
?
y
P
24:1
! M
3
(27)
into a ommutative diagram(sine theationson A 0
(1;4)
arethesame byonstrution).
The reduible divisor D is easily omputed one expliit equtions for S (or S 0
) are known.
Sineweknowof noeasy diretwaytodetermineS,we postponetheomputationof S toSetion
6.4, where the potentials willbe used to ompute S ina straightforward way; we will show there
that S 0
breaks upin fourirreduiblepiees
1
=0;
2
=0;
3
=0and dis(P
3
(x))=0 whereP
3
isthe polynomial
P
3
=4
2 x
3
(
0 +2
1 +6
2 +2
3 )x
2
+(
0 2
1 +2
2 6
3
)x 4
3
;
and dis(P
3
(x))=0denotes its disriminant(in x). Grantedthis, wetake
1
=0;let x
1
=0 and
eliminatex
2 and x
3
from f
1
;f
2 and f
4
. Then therelation
f
4
=f
1 (f
1 3f
2 );
is found at one; obviously the same equation is found for
2
= 0;
3
=0: The omputation for
dis(P
3
(x))=0 islongerbutalso straightforward. Namely,byasimpletranslationinxthemoni
polynomialP
3
(x)=(4
2
) an be written asx 3
ax+b, with disriminant 4a 3
27b 2
. When this
disriminant(dependingon
i
)iswrittenintermsofu
i
usingtheinverseof(25),theequation(24)
forD
2
is readoimmediately.
As for the urve to be added to
~
(
~
A
(1;4)
) to obtain (A
(1;4)
) remark that the ation of S
3
identiesthethree rationalurvesin(23),leadingto asingleurve. Toomputeitsequation (asa
subvarietyofD
1
)intermsoftheoordinatesf
i
,letaording to(10),
1
=0and
0
=2(
2 +
3 ).
Then intermsof
0 and
2
weget
f
0
=
0
;
f
1
=(2
2
0
=2) 2
;
f
2
= 2
2
0
2
2 +
2
0
4
;
leadingto
3f 2
0
=4(4f
2 f
1 );
byeliminationof
0 and
2
. AsforthetwospeialpointsP andQonthisurve,itiseasytohek
thatpiking
1
=0;
2
=
3 and
0
=2(
2 +
3
)leadsto thepoint (4:0:3:2:0) andalternatively
taking
1
=
2
=0;
0
=2
3
leadsto thepoint(2:1:1:0: 2). Thisgivesexpliitequationsforall
these spaesand provesthe announedresultin(22).
Finally, let (f
0 ::f
4
) 2 M 3
be any point dierent from the top (1:0:0:0:0) of this one.
Then
2
6=0foratleastone of thesixpoints(
0 :
1 :
2 :
3
) lyingoverthispoint. Dene;;f;g
by
=
0 +2
1 +2
2 +2
3
;
=
0 +2
1 2
2 +2
3
;
f =128 2
2
3
;
g=128 2
2
1
;
(28)
then 6= and ;;f and g satisfy (14). This shows that, apart from the top, all points inthe
one M 3
orrespond to some invariant surfae A
(;;f;g)
for some 6= ;f and g. This nishes
theproofof thetheorem.
Inthissetionwewantto show thata (1;4)-polarizedAbeliansurfaeT 2
2
~
A
(1;4)
is intimely
related to its anonial Jaobian,denoted byJ(T 2
) (introduedinSetion 2), henealso to some
urveofgenustwo,denoted (T 2
). Infatthereismore: attheleveloftheJaobian, letJ(T 2
)be
representedasC 2
=, thenT 2
induesa non-degenerate deompositionof thelattie and at the
leveloftheurve,T 2
induesadeompositionofthesetofWeierstrapointsof (T 2
)whihinturn
orresponds to an inidenediagram for the 16
6
onguration on its Kummer surfae; moreover,
theAbeliansurfaean be reonstrutedfrom either ofthese data(Theorem 4).
Reallthat theanonial Jaobianof a(1;4)-polarizedAbeliansurfaeT 2
=(T 2
;L)2
~
A
(1;4)
isdenedasthe(irreduibleprinipallypolarized)AbeliansurfaeJ(T 2
)=T 2
=K,whereK isthe
(unique) subgroupof two-torsion elements of K(L). As is well-known suh an Abelian surfae is
theJaobianofasmoothurve ofgenustwo,i.e.,itisgivenasC 2
=,whereistheperiodlattie
= I
~
! j 2H
1 ( ;ZZ)
onsisting of all periodsof ~! = t
(!
1
;!
2
), the !
i
being (independent) holomorphidierentials on
. The Abelian group H
1
( ;ZZ) has an (alternating) intersetion form
℄
() and H
1
( ;ZZ) an be
deomposedinto non-degenerateplanes(in manydierent ways),
H
1
( ;ZZ)=H
1 H
2
;
℄
()
H1 and
℄
()
H2
non-degenerate:
Suh adeompositionleadsto a deomposition=
1
2
upon dening
i
= I
~
! j 2H
i
; (29)
bothH
1
( ;ZZ)=H
1 H
2
and=
1
2
willbeallednon-degenerate deompositions. Theyare
alled in addition simple if eah H
i
is generated by yles whih ome from simple losed urves
(Jordanurves) inIP 1
undersome (hene any)doubleover: !IP 1
.
We also reall from the lassial literature the 16
6
onguration on the Kummer surfae of
Ja ( ),where is aurve of genus two. Let W
1
;:::;W
6
be theWeierstra pointson , thenthe
points
W
ij
= Z
Wj
Wi
~
! (mod )
are half-periodsof Ja( ), sixteen in total sine W
ij
=W
ji
and W
ii
= W
jj
for all i;j = 1;:::;6.
Thereare alsosixteengenustwo urves
ij
inJa( ),thetranslates W
ij +
kk
ofthesingle urve
11
==
66
,whihhavethepropertythat
11
,heneall
ij
passthroughsixpointsW
kl . Then
alsoeahpointbelongstosixlines
ij
. ThiswholeongurationgoesdowntotheKummersurfae
inIP 3
and gives there a 16
6
onguration, lassially alled Kummer's onguration. The sixteen
points are nodes (singular points) and the sixteen planes the lines belong to are tropes (singular
planes)of theKummersurfae. The16
6
ongurationisbestvisualizedbytheinidenediagram,
whih onsistsof apair of squarediagrams, suh as
W
11 W
12 W
23 W
13
W
45 W
36 W
16 W
26
W
46 W
35 W
15 W
25
W
56 W
34 W
14 W
24
11 12 23 13
45 36 16 26
46 35 15 25
56 34 14 24
pointsinthe m-throw and n-tholumn,butnotinboth,of therst square diagram. Dually,the
sameappliesforthelinesinidentwithapoint. The24 2
inidenediagramsobtainedbypermuting
therowsorolumnsofbothsquarediagramsinaninidenediagram(inthesameway)aredened
to bethesame astheoriginalinidenediagram (we willseethat thereare 20inidenediagrams
whih aredierent inthissense).
The relevane of simple, non-degenerate deompositions and inidene diagrams for (1;4)-
polarizedAbeliansurfaes is seenfrom thefollowingtheorem.
Theorem 4 There is a natural orrespondene between the following (isomorphism lasses) of
data:
(1) a (1;4)-polarized Abelian surfaeT 2
2
~
A
(1;4) ,
(2) a Jaobi surfae J =C 2
= + a simple, non-degenerate deomposition =
1
2 of ,
(3) a smooth genus two urve + a deomposition W =W
1 [W
2
;#W
1
=#W
2
=3, of its
Weierstra points.
(4) a smooth genus two urve + an inidene diagram for the 16
6
onguration on its
orresponding Kummer surfae.
Theorrespondene (1)$(2) isestablished intwo ways, namely J maybetaken asthe quotient of
T 2
using
2
or asa overof T 2
using
1 (orW
1
). Moreover,interhangingtheomponents of the
deomposition in (2) amounts to taking the dual
^
T 2
of T 2
in (1). J is the Jaobian of the urve
whih appears in (3) and (4) and interhanging
1 and
2
in (2) amounts to interhangingW
1
and W
2
in (3) and taking the transpose of bothsquarediagrams in the inidenediagram in (4) .
Summarizing we have the following ommutative diagram, determined by T 2
(only),
J
2
!
^
T 2
?
?
y
1 2
J
&
?
?
y
1
T 2
2
! J
(30)
where 2
J
denotes multipliation by 2 in J and a
i
labeling an arrow means that a projetion is
onsidered on the quotient torus that is obtained by doublingthe sublattie
i .
Proof
(3) ! (2) Given a genus two urve and a deomposition W = W
1 [W
2
of its Weierstra
points, with#W
i
=3, let: !IP 1
be anytwo-sheeted over ofIP 1
. It iswellknownthat has
branhpointsexatlyat W;thepointsinW aswellastheirprojetionsunder willbedenotedby
W
1
;:::;W
6
, also (W
i
) willjust be written asW
i . If IP
1
is overed with onneted open subsets
U
1 and U
2
forwhihW
i U
i and U
1
\U
2
\W =; thenH
1
( ;ZZ)deomposesas H
1 H
2 where
H
1
and H
2
aredenedas
H
i
=f2H
1
( ;ZZ)j
2H
1 (U
i nW
i
;ZZ)g:
AmongtheylesinH
i
therearethosewhihomefrom simplelosedurvesinU
i nW
i
enerling
twopointsinW
i
andthesegenerateH
i
. Sineany(dierent)oftheseinterset(one)therestrition
℄
()
H
i
isnon-degenerate,heneleads(upon using(29)) to anon-degenerate simpledeomposition
=
1
2
fortheperiodlattie. ThusC 2
= and=
1
2
providetheorrespondingdata.
Wenowshowthattheonstruteddataonlydepend(upto isomorphism)ontheisomorphism
lass of the data ;W = W
1 [ W
2
. Let : ! be an automorphism whih permutes the
Weierstrapoints(suhanautomorphismonlyexistsforspeialurves ). Thenextendslinearly
