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(1)

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

(2)

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.

(3)

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.

(4)

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

(5)

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;

(6)

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.

(7)

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

(8)

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

(9)

(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;

(10)

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

(11)

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.

(12)

[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.

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