Jose Bertin 1
and PolVanhaeke 2
Abstrat
In this paper we study a generalised Kummer surfae assoiated to the Jaobian of those omplex
algebraiurvesofgenus twowhihadmit anautomorphismof orderthree. Suh aurveanalwaysbeen
writtenasy 2
=x 6
+2x 3
+1and 2
6=1isthemodularparameter. Theautomorphismextendslinearlyto
anautomorphism oftheJaobianandweshowthat this extensionhasa9
4
invariantonguration,i.e., it
has9xed pointsand9invariantthetaurves,eahof theseurvesontains4xedpointsand 4invariant
urves passthrough eah xed point. The quotient of the Jaobianby this automorphism has 9singular
pointsoftypeA
2
andthe9
4
ongurationdesendstoa9
4
ongurationofpointsandlines, reminisentto
thewell-known16
6
ongurationontheKummersurfae. Our\generalisedKummersurfae"embedsinIP 4
andisaompleteintersetionofaquadriandaubihypersurfae. Equationsforthese hypersurfaesare
omputedand takeaverysymmetriform in well-hosenoordinates. This omputation isdone by using
anintegrablesystem,the\evenmastersystem".
1
UniversitedeGrenoble I,Institut Fourier,Laboratoirede Mathematiques,AssoieauCNRS(URA 188),
BP74,38402SaintMartinD'Heres,Frane
2
UniversitedesSienesetTehnologiesdeLille,U.F.R.deMathematiques,59655Villeneuved'AsqCedex,
Frane
Reently several studies were published on the geometri aspets of Hamiltonian systems whih are
algebraiallyompletelyintegrable. For ageneralintrodution,see[1℄. Fromthepointofviewofalgebrai
geometry,these integrablesystemsleadto anoriginalapproah tostudy projetiveembeddingsofAbelian
varietiesandtheirKummervarieties,expliitequationsforaÆnepartsofthesevarieties,::: Itfollowsthat
integrablesystemsmaybeusedtostudyandsolvesomequestionsin algebraigeometry,espeiallyinurve
theoryandthetheoryofAbelianvarieties;thepresentpaperisapartiularexampleofsuh aquestion.
In order to state this question, let us reall from the lassial literature some basi fats about the
Kummer surfae of the Jaobian of a urve of genus two. Suh a urve being always hyperellipti, it
arriesan involution with sixxed points (the Weierstrass points of ); this involutionextends linearly
to theJaobianof whereit hassixteenxed pointsand sixteeninvarianttheta urves(i.e.,translatesof
theRiemann thetadivisor), eah invariant urveontains sixxed points andeah xed pointbelongs to
sixinvarianturves,givingaso-alled16
6
ongurationofurvesandpoints. ThequotientoftheJaobian
bythisinvolutionisasingularsurfae,theKummersurfae,anditembedsin IP 3
asaquartisurfae. An
equation forthis surfaehas lassiallybeenobtained(in several forms) bypurely algebraimethods (see
[7℄).
Somethingveryanalogoushappenswhenthegenustwourve hasanautomorphism oforderthree,
in whihasethe urvehasanequation y 2
=x 6
+2x 3
+1(here 2
6=1is themodularparameteraswe
will show). The symmetry of order three extends to the Jaobianand leads now to a9
4
onguration as
wewillprovebothdiretlyand byusingananalogueofthetheta harateristi,whihexpressesingeneral
theobstrutionforaline bundletodesendto aquotient. Inthepresentasethis harateristiturnsout
to beaquadrati formwhih takesvaluesin IF
3
(the eld ofthreeelements). Suhaonguration,whih
hasessentially onlyoneprojetiverealisationhas beenonsidered bySegre and Castelnuovo(see [12℄ and
[5℄). The singularsurfaeobtainedasthe quotientof the Jaobianof by theorder three automorphism
will be shown to embed now in IP 4
as the intersetion of a quadri and a ubi hypersurfae. The nine
singularpointsare oftypeA
2
andare partofa9
4
ongurationof linesand points onthissurfaewhih,
afterdesingularisation,isaK-3surfae.
The question now is to ompute expliit equations for the quadri and ubi hypersurfae. To this
aim weneed tointrodue well-adapted oordinates andthis is wherethe integrablesystemomesin. The
systemishosenin suhawaythatamongitsinvariantsurfaeswendtheJaobiansorrespondingtothe
genustwourveswithanautomorphismof orderthree. Suhasystemwasrstonstrutedbytheseond
authorin[13℄in analogywithasystemintroduedbyMumford(see[10℄). Itgivesontheonehandexpliit
equationsforaÆnepartsoftheJaobianswhihonernushere,ontheotherhanditallowsustoonstrut
an expliitbase for thefuntions with apole of order three at one of the invariant thetaurves. Among
thosefuntionstheoneswhihareinvariantby areeasilydeterminedandtheimageoftheJaobianinIP 4
bythesefuntions isomputedfrom theseexpliitdata. Thenalresultisthat intermsof anappropriate
basefor IP 4
|formed bythevexedpointswhihdonotbelongto oneoftheinvariantthetaurves|
theequationforthequadrihypersurfaeisgivenby
(y
1 +y
4 )(y
2 +y
3 +y
4 y
0 )+(y
2 +y
3 )(y
1 +y
3 +y
4 y
0 )=y
2
4 +y
2
3
;
whiletheequationfortheubihypersurfaeisgivenby
2
y
1 y
4 (y
2 +y
3 y
0 )
2
y
2 y
3 (y
1 +y
4 y
0 )=0;
where=+1and=1 .
IthasbeenpointedouttousbyI.Dolgaevthatsuhequationsalreadyappearedin thebasiworkof
Enriques and Severi onhyperellipti surfaes. Theyproved that theassoiatedKummer surfaeis hyper-
elliptiin their sense withthe aidof resultsofSegre andCastelnuovo. Theindiret argumentsused there
seemnottobeomplete. Ourdiretmethodshowsalsothepreiserelationshipbetweenthetwoparameters
andanbeusedinothersimilar situations.
ThespeialurvesonsideredhereareatuallythehiralPottsN-stateurvesorrespondingtoN =3
(see[11℄). Theresultsandtehniquesin thispapergeneraliseto allhiralPottsurves. Wehopetoreturn
tothisin thefuture.
Weonsideraurve ofgenustwo,equippedwithanautomorphismoforderthree, denotedby. By
the Riemann-Hurwitz formula the quotient = has genus zeroand has four xed points. Sine has
genus twoitis alsohyperellipti;the hyperelliptiinvolutionwill be denoted by andits xedpointsare
thesixWeierstrasspointson . Wehavethefollowingdiagram
!
3:1 IP
1
?
?
y2:1
IP 1
neessarilymapsWeierstrasspointstoWeierstrasspoints,henetheommutator[;℄xesallthesepoints
andweseethat = sinetheonlyautomorphisms whih xallWeierstrass pointsare andidentity.
Itfollowsontheonehand that induesonIP 1
afrationallineartransformation~ oforderthree,andon
theother handthat thefour xedpointsof onsistof two-orbits. Wemaythereforesuppose that ~ is
given by(x)~ =x;=exp(
2i
3
), byhosing aoordinate x onIP 1
suh that these twoorbitsorrespond
to x=0andx =1. Theimages oftheWeierstrass pointsform twoorbitsofthree pointsunder ~,whih
orrespond to theroots of theequation x 3
= 3
andx 3
= 3
, possiblyafter aresalingof x. Obviously
6=0;sinebothorbitsaredierent, 3
6=
3
,i.e., 6
6=1. Thisshowsthat hasanequation
y 2
=(x 3
3
)(x 3
3
);
=x 6
+2x 3
+1;
(1)
with6=1.
Clearly, every equation of the form (1) with 6= 1, denes a smooth urve of genus two with an
automorphism (x;y)7!(x;y) oforder three; also, if in (1)is replaed by then anisomorphiurve
isobtained. Conversely,letthere begiventwoisomorphi urves and 0
withrespetiveautomorphisms
and 0
oforder three. Wemaysupposethat theisomorphism: ! 0
respetstheautomorphism, i.e.,
=
0
:Welaimthatif and 0
arewrittenasaboveas
:y 2
=x 6
+2x 3
+1;
0
:y 2
=x 6
+2 0
x 3
+1;
then 2
= 02
. Tosee this, remark that obviously ommutes with ; hene there is an indued linear
transformation
~
whihsatises
~
(x)=
~
(x),forallx2IP 1
. Thus
~
(x)=xand(x;y)=(x;y),giving
6
=1. Itfollowsthat 2
6=1an betakenasmodularparameter.
The automorphism group of ontains a subgroup whih is isomorphi to S
3
ZZ=2ZZ, as is seen
immediately from (1); it atually oinides with this group, unless = 0 (in whih ase the group of
automorphismsjumpstoD
6
ZZ=2ZZ). Namely,thereis,apartfromthehyperelliptiinvolution,anation
ofS
3
bymeansofwhih theWeierstrasspointsbelonging toone-orbitanbeat randompermuted. For
futureusewehooseanelementofordertwointhissymmetrygroupS
3
orrespondingtoatransposition
in S
3
, say(x;y)=(x 1
;yx 3
) andremark that it ommuteswith but notwith. Its xedpointsare
thetwopointsin 1
f1g,hene =isanelliptiurve.
Wewill ndit onvenientto denote thexed pointsof, whihare mappedby
to 0(resp.1)by
o
1 and o
2
(resp. 1
1
and 1
2
). Then (o
1 ) =o
2
;(1
1 ) = 1
2
and wemay suppose (o
1 ) = 1
1 giving
also (o
2 ) = 1
2
. In the same way wedenote the Weierstrass points orresponding to the x 3
= 3
-orbit
by
i
;(
i ) =
i+1
(indies are taken modulo 3) and the ones orresponding to the x 3
= 3
-orbit by
i
;(
i )=
i
. ThentheationofS
3
ZZ=2ZZonthesepointsisontainedin thefollowingtable.
order o
1 o
2 1
1 1
2
i
i
3 o
1 o
2 1
1 1
2
i+1
i 1
2 o
2 o
1 1
2 1
1
i
i
2 1
1 1
2 o
1 o
2
i
i
Table1
4
LetJ( )denotetheJaobianof andforadivisorDofdegree0,let[D℄denotetheorrespodingpoint
in J( ) (i.e.,its linearequivalene lass). A usefulfat about theJaobianof aurveof genustwois the
following: for any xed Q
1
;Q
2
2 ; everyelement ! 2J( ) anbewritten as! =[P
1 +P
2 Q
1 Q
2
℄;
moreoverthisrepresentationisuniqueiP
1 6=(P
2
),allP+(P)and Q+(Q) (P;Q2 )beinglinearly
equivalent,P+(P)
l
Q+(Q). Inthepresentaseofurves(1)whihhaveanautomorphismoforder
three,theover
assoiatedto providesin addition(usingthenotationsof theprevioussetionforthe
xedpointsof)thefollowinglinearequivalenes
3o
1
l 3o
2
l 31
1
l 31
2
: (2)
Theautomorphism extendsin anaturalwaytoanautomorphismonJ( ), alsodenoted by. Itisgiven
andwell-dened for!=[P
1 +P
2 Q
1 Q
2
℄asfollows: (!)=[(P
1 )+(P
2
) (Q
1
) (Q
2 )℄.
Proposition1. The automorphism has nine xedpointsandnine invarianttheta urveson J( ).
Proof
TheprinipalpolarisationonJ( )isinvariantunderAut ( ),henetheisomorphismJ( )!
^
J( )from
J( )toitsdual
^
J( )isAut( )-invariantandtheseond statementfollowsfrom therstone.
Weountthenumberofxedpointsofintwodierentways. AtrstweusetheholomorphiLefshetz
xedpointformula
X
p ( 1)
p
traef
j
H p;0
(M)
= X
f(p)=p 1
det(I B
)
; (3)
foraholomorphimapf:M !M,whereB
isthelinearpartoff at thexedpointp
. Weapply itfor
f = andM =J( );in thisaseH p;0
(J( ))maybeidentied withthep-thanti-symmetripowerofthe
otangentbundle at any pointof J( ). Forthe left-handside in (3), thebase of H p;0
(J( ))maythus be
takeninapoint[P
1 +P
2 Q
1 Q
2
℄asf
1
;
2 g=f!
1 (P
1 )+!
1 (P
2 );!
2 (P
1 )+!
2 (P
2
)g;where!
i
=x i 1
dx=y
and
1
^
2
isageneratorforH 2;0
(J( )). Sine
i
= i
i
;(i=1;2), thelefthand sidein(3)gives
2
X
p=0 ( 1)
p
trae
j
H p;0
(J( ))
=1 trae
0
0 2
+1=3:
Asfortherighthandside,obviouslyallB
areequal,infat
B
=
0
0 2
(4)
whenloaloordinatesdualto
1 and
2
arepikedaroundthepointP
. Therefore
det(I B
)=(1 )(1 2
)=3;
andthenumberofxedpointsof isindeed nine.
A seond wayto determine the numberof xed points of is by writingdown anexpliit list: if we
writeeverypoint!2J( )as!=[P
1 +P
2 21
1
℄then!=!i(P
1 )+(P
2 )
l P
1 +P
2 ,i.e.,P
1
=(P
2 )
orP
1 andP
2
arebothxedpointsfor. Using(2)wearriveat thefollowinglist
fO;o
1 o
2
;o
2 o
1
; 1
1 1
2
;1
2 1
1
;o
1 1
1
;o
2 1
2
;o
1 1
2
;o
2 1
1
g: (5)
Thenine invarianturvesarethengivenbythenine translatesoverthesepointsofthe imageof inJ( )
bythemap x7![x 1
1
℄:Sinethis urveobviouslyontainsexatlythefour xedpoints
fO;1
2 1
1
;o
1 1
1
;o
2 1
1 g;
fourinvarianturvessinetheoriginO belongstothefoururves
fx7![x 1
i
℄;x7![x o
i
℄;i=1;2g
NotethatthexedpointsformagroupF (isomorphitoZZ=3ZZZZ=3ZZ)whihisasubgroupofJ
3 ( ),
thethree-torsionsubgroupof J( ). OnJ
3
( )there isanon-degeneratedalternatingform(;)induedby
theRiemann form orrespondingto theprinipal polarisation. The subgroupF J
3
( ) hasthefollowing
property.
Proposition 2. The group F of xed points of on J( ) is a totally isotropi subgroup of J
3
( ) with
respet totheRiemann form(;).
Proof
isasympletiautomorphismofJ
3 ( )
=
(ZZ=3ZZ) 4
,whihsatises1++ 2
=0;alsodimker( 1)=
2. ItfollowsthatF onsistsexatlyoftheelementsoftheform(x) x wherex2J
3
( ). Finally,ify2F,
thenobviously(y;(x) x)=0.
Apartfrom theRiemann form,whihoinidesonJ
3
( )withWeil'spairinge
3
(see[8℄)afuntionan
bedenedonF withvaluesinthegroupofubirootsofunity. ItisanalogoustoMumford'squadratiform
(thetaharateristi)onthetwo-torsionsubgroup J
2
( )ofJ( )andanbedened in ompletegenerality
(see[4℄). It measuresthe obstrutionforaline bundleto desendto thequotient J( )=. Oneandene
it asfollows. Choose alinearisation of L with respet to the yli groupZZ=3ZZ generated by , i.e., an
isomorphism:L
!
(L)with(0)=Id
L(0)
. Whenxisaxedpointof, theninduesanisomorphism
ofL(x)whihismultipliationbyarootofunitye(x),ande:x7!e(x)isthedesiredfuntion. Itdependson
thehoieofLitselfandnotonlyonthepolarisation. Ifisthe(theta)divisorwhihorrespondstoL;i.e.,
L=[℄;thentheorrespondinge=e
maybeomputedasfollows. Letf =0bealoaldening funtion
forin x. Sinethedivisorisnon-singular,theleadingparthoff islinearandwehave
(h)=e(x)h.
Sine thesingularpointsareof typeA
2
, asis seenfrom (4), there existloaloordinates fu;vgin x suh
that
(u)=uand
(v)= 2
v. Thereforewehaveeitherh=uande(x)=,orh=vande(x)= 2
:Also
ifx2=thene(x)=1. Itfollowsthate
isexpliitelygivenforallx2F bye
(x)=
jTx
,orequivalently
e
(x)v=
v forallv2T
x
. (6)
Theautomorphismsand at onF aswellasonthesetofinvarianttheta urves. Itisdesirableto
havea\totallysymmetri"thetaurve,i.e.,invariantby; and. Themainobservationofthisparagraph,
fromwhihthe9
4
-ongurationis aonsequene,isthefollowing.
Proposition 3. There isa uniquetotally symmetri theta urve among the nine invariant theta urves.
Thefuntion e
assoiatedtothis urveisaquadratiformonF;itisgiven inasuitablebasefor F and
uponidentiationof the group ofubi rootsof 1withIF
3 by e
(r;s)=r 2
s 2
(mod3).
Proof
Theexisteneoftheurveislear: sinethepolarisationisinvariantbythegroupAut ( ),wemaynd
aninvariantinvertiblesheafwhihgivesthis polarisation,henealsoaninvariantdivisor. Itisuniquesine
iftherearetwoAut ( )-invarianturves,thentheir(two)intersetionpointsmustbeinvariantunderAut ( )
whihis impossiblebyTable1. Itis easytoidentify : it isgivenby theimageof P 7![P+1
1 21
2
℄.
Toseethis,remark thatthisimagean bewrittenas
P 7![P+S
1 +(S
1 ) 3S
2
℄;
independent of the hoie of S
1
;S
2 2 fo
1
;o
2
;1
1
;1
2
g. From this representation it is also lear that
ontainsthefour points[(S
1 ) S
1
℄; S
1 2fo
1
; o
2
;1
1
;1
2 g.
1 2 1 2 1 2 2 1
= and 2
=1itfollowsusingthehainrulethatif(x)=x andv2T
(x) then
e
((x))v=
v=
v=e
(x)
v=e
(x)v;
henee
((x))=e
(x). Inthesamewayitfollowsfrom = 1
that e
((x))=e
(x)
1
. Therefore,
ifweidentifythegroupofubirootsofunitywithIF
3 bye
(
1
)=1thene
isgivenby
e
(r
1 +s
2 )=r
2
s 2
(mod3)):
The9
4
ongurationisnowdesribedasfollows. if!and! 0
aretwoxedpoints,then
!2+! 0
ie
(! !
0
)6=0:
Itfollowsthateveryinvariantthetaurvepassesthroughfourxedpointsandthateveryxedpointbelongs
to fourinvariantthetaurves. Moreoverwehaveseenthat thefuntione
determinesthediretion ofthe
tangenttointhexedpointsof. Therefore,if!; ! 0
2F then+!and+!
0
aretangentinaommon
pointx2F ifandonlyif
e
+!
(x)=e
+!
0
(x):
Sinee
+!
(x)=e
(x !),this onditionisrewrittenas
e
(x !)=e
(x !
0
)
whih is satised for ! 0
= 2x ! (only). We onlude that the four invariant urves running through
onexed point ome in two pairs: sineany two theta urves always interset in twopoints (whih may
oinide), the urves of onepair are tangent in their uniqueintersetion pointand the urves ofopposite
pairsintersetintwodierentpoints(seeFigure1,whihalsoontainsthedualpiture,equallypresentin
the9
4
-onguration).
dual
!
Figure1: Theinideneofpointsandlinesonthe9
4
-onguration
4. Equations for J( )= in IP
Inthissetion wewill omputeexpliitequationsforthequotient S=J( )=asanalgebraisurfae
inIP 4
. Sine hasnine xedpoints,S hasninesingularpointsandwehaveseenthat theyareoftypeA
2 .
Theminimal resolution of these singularities ofS leadsto aK-3surfae(a generalised Kummer surfae),
whihwewilldenote byX (see[3℄). Let:J( )!S bethe quotientanddenote bytheuniquedivisor
givenbyProposition3.
Proposition4. LetM bethe divisor onS for whih
(M)=[3℄. Then M isveryample andallowsto
embedS astheompleteintersetion of aquadriandaubi threefold inIP 4
.
Proof
Sinemostoftheproofisstandard,weonlygivefewdetails. Usingthequadratiforme
wesee that
L 3
=[3℄desendstoaninvertiblesheafM onS,i.e.,
(M)=L 3
. LetusdenotebyN thelinebundle
onX whihisthepull-bakofM bytheanonialmapfromX toS. ThenusingLL=2wend
18=L 3
L 3
=(deg)MM =3MM;
sothatMM =6;whihisalsotheself-intersetionofN. Therefore,wendbytheRiemann-RohTheorem
(forK-3surfaes),
(N)=(O
X )+
NN
2
=2+3=5:
Itfollowsmoreoverfrom SerredualityandKodairavanishing(forK-3surfaes)thatdimH i
(X;O(N))=0
fori>0,sothatdimH 0
(X;O(N))=(N)=5.
Themorphism
N
orrespondingtoN fatorisesviatheblow-upp:X !S andisshowntoprovidean
injetivemorphism:S!IP 4
. Morepreisely,itanbeseenbyanalysingthetaurvesonJ that
N isone
tooneawayfromtheexeptionalurves. Ifweonsidernowthesurjetivemap
SymH 0
(X;N)!
t0 H
0
X;N t
;
whosekernelleadstothedeningequationsfortheimageofSinIP 4
,weseebyadimensionountasabove
that thekernelontainsaquadratiaswellasan(independent)ubiform. Sine thedegreeof N equals
six,weseethat theimage istheompleteintersetionofaquadriandaubihypersurfaein IP 4
.
Wewillnowusetheso-alledevenmastersystem,introduedandstudiedbytheseondauthorin[13℄.
Letus shortlyreallwhat is neededfor ourpurposes. Letus denoteby 0
oneof thefour invarianttheta
urveswhih istangent to, say 0
=+[1
2 1
1
℄. Theneverypoint! 2J( )n(+ 0
) is written
uniquelyas[P+Q 21
2
℄andP;Q2=f1
1
;1
2
g. Itfollowsthat wemayassoiateto ! threepolynomials
u(x)=x 2
+u
1 x+u
2
, v(x)=v
1 x+v
2
andw(x)=x 4
u
1 x
3
+w
0 x
2
+w
1 x+w
2 by
y
u
1
= x(P) x(Q); u
2
=x(P)x(Q);
v
1
=
y(P) y(Q)
x(P) x(Q)
; v
2
=
x(P)y(Q) x(Q)y(P)
x(P) x(Q)
andw(x)isdenedbythefundamentalrelationu(x)w(x)+v 2
(x)=f(x),wheref(x)istherighthandside
ofourequation y 2
=x 6
+2x 3
+1fortheurve . TheoeÆientsof thisfundamental equationatually
leadtoaÆneequationsfortheaÆnepartJ( )n(+ 0
)and areeasilywrittenoutas
w
0 u
2
1 +u
2
=0;
w
1 +w
0 u
1 u
1 u
2
=2;
w
2 +w
1 u
1 +u
2 w
0 +v
2
1
=0;
w
1 u
2 +w
2 u
1 +2v
1 v
2
=0;
u
2 w
2 +v
2
2
=1:
(7)
y
ifx(P)=x(Q)thenthedenitionsofv
1 andv
2
areadjusted inanappropriateway
(u
1
;u
2
;v
1
;v
2
;w
0
;w
1
;w
2
)theweights(1;2;2;3;2;3;4),thenallequationsin(7)areweighthomogenousand
eahvariable ismultipliedby as oftenasgivenbyits weight. Clearly theationleavesthe equations(7)
invariant.
Aseondingredientwhihweneedfrom[13℄ isthatwemayndexpliitely intermsofthesevariables
(a base for) the funtions whih haveapole of order three along andare holomorphi elsewhere. This
isdonebyusingavetoreldonJ( )andits Laurentsolutionswhihare writtendownthere(wereferto
[13℄Set.6.bformoredetails). Obviouslyaweighthomogeneousbaseforthesefuntionsanbehosenand
funtionswhihareinvariantby aretheoneswhoseweightisamultipleofthree. Thelististhefollowing.
z
0
=1;
z
1
=u
1 u
2 v
2
;
z
2
=2u
1 (u
2 +v
1 u
2
1 );
z
3
=2u
2 v
2
1 +2v
2
2 +2u
2 v
1 (2u
2 u
2
1 )+2u
1 v
2 (u
2
1 v
1 3u
2 )+2u
3
2
;
z
4
=2v 3
1 2(u
2
1 +4u
2 )v
2
1 +10v
2 (u
1 v
1 v
2 )+2v
1 (7u
2 u
2
1 u
4
1 11u
2
2 )
+2v
2
(2+15u
1 u
2 5u
3
1 )+2(u
2
1 u
2 )
3
10u 3
2 4u
1 u
2 :
(8)
TondtheimageofJ( )inIP 4
itsuÆestoeliminatethevariablesu
i
;v
i andw
i
from(7)and(8). Infat,
fromtherstthreeequationsof(7) thevariablesw
i
areeliminatedlinearlyandtheotherequationsredue
to
2(u
2 u
2
1 )+3u
1 u
2
2 u
1 v
2
1 4u
2 u
3
1 +2v
1 v
2 +u
5
1
=0;
2u
1 u
2 +u
2 u
4
1 u
2 v
2
1 3u
2
1 u
2
2 +u
3
2 +v
2
2
=1;
(9)
soitsuÆestoeliminateu
1
;u
2
;v
1 andv
2
from(9)and(8)(wehavealreadyeliminatedthew
i
-variablesin
(8)). Inthelatterz
1 andz
2
aresolvedlinearlyforv
1 andv
2 ,
v
1
=u 2
1 u
2 +
z
2
2u
1
;
v
2
=u
1 u
2 z
1
;
(10)
andthenewequationforz
3
,obtainedbysubstituting (10)in (8)isthensolvedlinearlyforu
2 as
u
2
= 2u
2
1
z 2
2 z
3 z
1 z
2 2z
2
1
: (11)
After substitutionof (10)and(11)in thelast equation of(8) andin the equationsof (9),weareleft with
three linearequationsin u 3
1
, whih reetsthefat that J( )willbea3:1overofits imagein IP 4
. Ifwe
eliminateu 3
1
wearriveat thefollowingtwoequations:
8z 3
1
24z 2
1
4(2z
2 +6z
3 +z
4 )z
1 +4z
3 2z
2
2 3z
2 z
3 z
2 z
4
=0;
8z 4
1
16z 3
1
4(2+2z
2 +6z
3 +z
4 )z
2
1
+ 8+4z
3 2z
2
2 4z
2 3z
2 z
3 z
2 z
4
z
1
+2z
2 z
3 +14z
3 +2z
4 2z
2
2 +z
3 z
4 +5z
2
3
=0:
Usingtherstequation,theseondequationanbereplaed by
8 1 3 2
z 2
1
+4 2+ 1 2
2
z
2 6z
3 z
4
z
1
+2 1 2
z 2
2
5z 2
3 z
2 (5z
3 +z
4 )+2z
3 2
2
7
z
3 z
4 2z
4
=0;
orequivalentlyby ZAZ=0,where
A= 0
B
B
B
B
B
B
B
0 8 0 2(2
2
7) 2
8 16(1 3
2
) 4(1 2 2
) 24 4
0 4(1 2
2
) 4(1
2
) 5
2(2 2
7) 24 5 10 1
2 4 1 0
1
C
C
C
C
C
C
C
A :
Althoughat this pointthese equationsfor thequadriand ubihypersurfaeswhihdene S asasubset
ofIP 4
(whih wewill identify in thesequelwithS) maynot seemveryattrative,wewill see that natural
oordinatesanbepikedforIP 4
inwhihtheseequationstakeaverysymmetriform. Indeed,thebasewe
used forIP 4
israther arbitrary: forexample, theoordinatesof thenine xed points for donot possess
speialoordinatesintermsofthepresentbase. Therstinterestingobservationisherethatifthevexed
points for whih donot lieon aretakenasbase points forIP 4
then thefour xed pointson takea
simpleformandare independentof . Toseethis,let=1
1 1
2
and =o
1 o
2
andremark thatthe
points
fO; ; +; ;+g
arethevepoinswhihdonotlieon. Tondtheiroordinates,usealoal parametertandtakex=t,
y= 1+t 3
+O t 6
;
pikingeither signaroundo
1 oro
2
,andin thesameway,x=t 1
and
y=
t 3
++
1 2
2
2 t
3
+O t 4
for1
1 and1
2
. Thenaarefulomputationyieldsthefollowingoordinates:
O :(0:0:0:0:1);
:(0:0:1:1:3 2);
:(1:1:2 2:2:4 2
14+4):
Wetakethepoints
fO; ; +; ;+g
as base points for IP 4
(in that order), i.e., O = (1:0:0:0:0), et., with assoiated oordinates y
0
;:::;y
4 .
Thenthefourxedpointsonhaveasoordinates
=(1:1:1:0:0); =(1:0:0:1:1);
=(1:1:0:1:0); =(1:0:1:0:1);
andweseethat theylieonthe(2-dimensional!) plane
y
0
=y
2 +y
3
=y
1 +y
4
;
and it is easy to see that in fat is ontained in this plane. The translations
and
orrespond to
projetivetransformationsofthesurfaeandtakeintermsoftheseoordinatesthesimpleform
t
= 0
B
B
B
1 1 1 0 0
1 0 1 1 0
1 1 0 0 1
0 0 1 0 0
0 1 0 0 0 1
C
C
C
A
and t
= 0
B
B
B
1 1 0 1 0
1 0 1 1 0
0 0 0 1 0
1 1 0 0 1
0 1 0 0 0 1
C
C
C
A
This onguration of nine points in IP 4
is haraterised by the fat that there exist nine planes with the
property that eah ofthese planes ontains four ofthe nine points and everypoint belongs to four of the
planes. Thus wehavereoveredin adiret wayaongurationthat hasbeenstudied in thework ofSegre
andCastelnuovoonnetsofubihypersurfaesinIP 4
(see[5℄and[12℄).
TheequationsofthequadriandubihypersurfaesQandCtakeintermsofthenewoordinatesthe
followingsymmetriform.
Q:(y
1 +y
4 )(y
2 +y
3 +y
4 y
0 )+(y
2 +y
3 )(y
1 +y
3 +y
4 y
0 )=y
2
4 +y
2
3
;
C: 3
y 2
2 (y
1 +y
3 +y
4 y
0 )+
2
y
2 ( (y
1 +y
4 )(y
2 y
0 )+y
0 y
3 +y
1 y
4 )
3
y 2
1 (y
2 +y
3 +y
4 y
0 )
2
y
1 ( (y
2 +y
3 )(y
1 y
0 )+y
0 y
4 +y
2 y
3 )=0;
where=+1and=1 . Theubiequationanbesimpliedinasigniantwaybyaddingtoitthe
equationforQmultipliedwith 2
y
1 2
y
2
. Theresultis
2
y
1 y
4 (y
2 +y
3 y
0 )
2
y
2 y
3 (y
1 +y
4 y
0 )=0:
Ifwedene
x
1
= y
1
;
x
2
= y
4
;
x
3
=y
0 y
2 y
3
;
x
4
=y
2
;
x
5
=y
1 +y
4 y
0
;
x
6
=y
3
;
thenS isgivenasanalgebraivarietyinIP 5
by
C: 2
x
1 x
2 x
3 +
2
x
4 x
5 x
6
=0;
Q:(x
1 x
2 +x
2 x
3 +x
1 x
3 )+(x
4 x
5 +x
5 x
6 +x
4 x
6 )=0;
H:x
1 +x
2 +x
3 +x
4 +x
5 +x
6
=0;
(12)
andthe9
4
ongurationispresentedintheformusedbySegreandCastelnuovo. Namely,thesingularpoints
of S are nowthepoints
ij
(i;j =1;:::;3) witha 1onthei-th plae, a 1on position 3+j andzeroes
elsewhere; the nine planes theybelong to are given by H\(x
i
= x
j+3
= 0) for i;j = 1;:::;3. Moreover
thethetaurvesaremappedto thenine onisC
ij
;(1i;j3), givenbyH
ij
=3C
ij
Forexample,C
16 is
givenas
x
2 x
3 +x
4 x
5
=0;
x
2 +x
3 +x
4 +x
5
=0:
Notethatifonehangesthesignofintheequations(12)thenanisomorphisurfaeisobtained(interhange
$andx
i
$x
i+3
fori=1;:::;3), inagreementwiththefatthat 2
isthemodularparameter.
[1℄ M.AdlerandP.vanMoerbeke.Algebraiompletelyintegrablesystems: asystematiapproah. Per-
spetivesinMathematis(Aademi Press,to appear).
[2℄ M.AdlerandP.vanMoerbeke.TheomplexgeometryoftheKowalewski-Painleveanalysis. Invent.
Math.97(1987),3{51.
[3℄ A. Beauville. Complex Algebrai Surfaes. London Math. So. Leture Note Series 68 (Cambridge
UniversityPress,1982).
[4℄ J.BertinandG.Elenwajg.Manusript.
[5℄ G.Castelnuovo.Sulleongruenzedel3 Æ
ordinedelspazioa4dimensioni. MemoriaAttiIstitutoVenoto
VI(1888).
[6℄ P. Griffithsand J.Harris.Priniples of Algebrai Geometry. Pure &Applied Mathematis (Wiley-
Intersiene,1978).
[7℄ R.W.H.Hudson.Kummer'squarti surfae. Cambridge MathematialLibrary (CambridgeUniversity
Press,1990; rstpublished in1905).
[8℄ H. Lange and C. Birkenhake. Complex Abelian Varieties. Grundlehren der mathematishen Wis-
senshaften(Springer-Verlag,1992).
[9℄ D.Mumford.OntheequationsdeningAbelianvarietiesI.Invent.Math.1(1966),287{354.
[10℄ D.Mumford.TataleturesonTheta2.Progressin Mathematis,(Birkhauser,1984).
[11℄ S. Roan. A Charaterization of\Rapidity" Curvein the ChiralPotts Model. Commun. Math. Phys.
145(1992),605{634.
[12℄ Segre.Sullevariataubihedellospazioa4dimensioni. MemorieAad.Torino II39(1888).
[13℄ P. Vanhaeke. Linearising two-dimensional integrable systems and the onstrution of ation-angle
variables. Math.Z.211(1992),265{313.