CHRISTINABIRKENHAKEANDPOLVANHAECKE
Abstrat. Wegiveageometriproofofaformula,duetoSegalandWilson,
whihdesribestheorderof vanishingof theRiemannthetafuntioninthe
diretionwhihorrespondstothediretionofthetangentspaeofaRiemann
surfaeat amarkedpoint. Whilethisformulaappearsinthe workofSegal
andWilsonasaby-produtofsomenon-trivialonstrutionsfromthetheory
of integrable systems(loop groups, innite-dimensionalGrassmannians, tau
funtions,Shurpolynomials,:::)ourproofonlyusesthelassialtheoryof
linearsystemsonRiemannsurfaes.
Contents
1. Introdution 1
2. Preliminaries 2
3. Geometridesriptionoftheorderofvanishing 4
4. Thedivisor
L;p; n
5
5. Formula(s)fortheorder ofvanishing 6
Referenes 7
1. Introdution
Thefundamental paper[SW ℄bySegalandWilsononsoliton equationsleadsto
anexpliitformulafor omputingthevanishing oftheRiemann theta funtionin
adiretionwhihis naturalfrom thegeometripointofview. Inorderto present
this formula, let C be a ompat Riemann surfae (of genus g > 1), let p 2 C
and let denote thetheta divisor Pi g 1
(C). Alsolet # denote Riemann's
theta funtion, for whih (#) =. For apoint L 2 , onsider the embedding
C ! Pi g 1
(C) : q 7! L(q p). The natural diretion alluded to above is the
tangentspae X
p
to this embedded urveat L. Following[SW ℄ the vanishing of
#atL in thediretionofX
p
,denotedord
L (#;X
p
)isobtainedbyonsideringthe
innitesubsetofZ,denedby
S
L
=fs2Zjh 0
(L((s+1)p))=h 0
(L(sp))+1g:
Infat,thesetionsofLoverCnfpgdeneaninnite-dimensionalplaneW,whih
is anelementof the SatoGrassmannian andthe vanishing ofthe taufuntion in
theKP-diretionis,aordingto [SW,Prop.8.6℄,givenbytheodimensionofW,
1991MathematisSubjetClassiation. 14K25,14H40,14H55.
Keywordsandphrases. Thetafuntions,Jaobians,Gapsequenes.
SupportedbyDFG-ontratsHu337/5-1.
whihis expliitly givenby thenite sum
i0 i s
i
, upon writingS
L
=fs
0
<
s
1
< s
2
< g. The tau funtion oinides, up to an exponential fator, with
the Riemann theta funtion of C ([SW, Th. 9.11℄)and thetangentdiretion X
p
oinides with the KP-diretion (see [S , Lemma 5and Appendix 0℄). Therefore,
theorderofvanishingisgivenby
ord
L (#;X
p )=
X
i0 i s
i :
Thepurposeofthis paperisto giveanalgebrai-geometriproof ofthisresult.
Our proof uses (only) the lassialtheory of linear systems onRiemann surfaes
and it highlights thegeometri meaning of the order of vanishing. As is pointed
outin [SW, footnotep. 51℄ anindependent (analytial)proofof this formulahas
alsobeengivenbyJohnFay(see[F℄),byusingthetheoryofthetafuntions.
Therststepofourapproahonsistsofaninterpretationoftheorderofvanish-
ingas theintersetionmultipliityofthethetadivisorwith aopyofC,properly
embedded (at leastaroundL) in Pi g 1
(C). If wepull bakthetheta divisorus-
ingthis embedding wendadivisorR onC whih isthe sumof theramiation
divisorsofthemaps
' k
L
:C !Grass
k +1 (H
0
(L)
);
whih are the natural generalizations of the morphisms '
L
: C ! P(H 0
(L)
)
denedbythelinearsystemjLj(assumedherebasepointfree). Itfollowsthatthe
orderofvanishingisgivenbythemultipliityofpin R ,leadingto
ord
L (#;X
p )=
n
X
i=1 m
i i:
where fm
1
< < m
n
g is the gap sequene G
p
(L(np)) of L(np) at p. This
formula is independent of n, whih is assumed suÆientlylarge (e.g. n = g will
do). Notiingthatforn=gonehass
i
=g m
g i
(fori=0;:::;g 1)fromwhih
Formula(1)followsatone.
NotiethattheSegal-Wilsonformulaforthevanishingof thetaufuntion may
also be applied in the ase of tau funtions that ome from singular urves. It
would be interestingto adapt ourgeometri argumentsto this ase, leadingto a
formulaforthevanishingofthethetafuntionsfor singularurves,asproposedin
[SW,Remark6.13℄.
Thestruture of this paper isasfollows. InSetion 2wex thenotation and
wereall the notionsof gap numbers for arbitraryline bundles. In Setion 3we
translatetheorderofvanishingofthethetafuntionintermsofintersetiontheory
and we show that this order is given asan inetionary weight. This is used in
Setion 5 to obtain an expliit formula, whih we show to be equivalent to the
formulabySegalandWilson.
2. Preliminaries
Inthissetionweintroduethenotationandolletsomeresultsonurvetheory.
ThroughoutthewholepaperCdenotesaompatRiemannsurfaeofgenusgand
p2C amarkedpoint.
For adivisorD on C we denote byO
C
(D) theorresponding line bundle and
foralinebundleL onC itslinearsystemisdenotedby jLj. Weuse thestandard
abbreviationsh 0
(L)fordimH 0
(C ;L)andL(D)forLO (D),whereLisanyline
bundleandD is anydivisoronC. WewillusetheRiemann-Rohtheorem inthe
form
h 0
(L)=h 0
(!
C L
1
) g+deg (L)+1;
whereL isanylinebundleonCand !
C
istheanonialbundleofC.
Wenowreallthenotionsofgapnumbersandinetionaryweights. Forproofs
anddetailswereferto[Mi,Set.VII.4℄andto [ACGH,Ch. 1Ex.C℄.
LetLbealine bundleonC ofpositivedegreeandletq2C. Anintegerm1
isalledagapnumber forL atqif
h 0
(L( mq))=h 0
(L( (m 1)q)) 1;
andthe setG
q
(L) ofgap numbersfor Lat qis alled thegapsequene ofL at q;
itsardinalityisr=h 0
(L)andnogapnumberislargerthandegL+1. Writing
G
q
(L)=f1m
1
<m
2
<<m
r
degL+1g;
wehavethatm
1
>1ifandonlyifqisabasepointofLand thatm
r
=degL+1
ifandonlyifL=O
C
(degLq). Forageneralpointq2C thegapsequeneof L
at q is f1;2;:::;h 0
(L)g; apointq for whih thegap sequene of L at q isnot of
this formis alledan inetion pointfor L. Notie that qisan inetionpointif
andonlyifh 0
(L( rq))6=0,wherer=h 0
(L).
IfthelinearsystemjLjisbasepointfreetheinetionpointshavethefollowing
geometri interpretation. Considerthe morphism '
L
: C ! P(H 0
(L)
)dened
by the linear system jLj. For a generi q 2 C there is a unique k-dimensional
osulatingplaneto'
L
(C)at'
L
(q),yieldingawell-denedmorphism
(1) '
k
L
:C !Grass
k +1 (H
0
(L)
);
alledthek-thassoiatedmap. Thiswayonearrivesat h 0
(L) 1assoiatedmaps
' i 1
L
; i =1;:::;h 0
(L) 1;(' 0
L
= '
L
). In these terms apointq is aninetion
pointifandonlyifqisaramiationpointofoneofthemaps' k
L
. Wedenotethe
ramiationdivisorof' k
L byR
k
(L)andwedene
R (L)= h
0
(L) 1
X
k =1 R
k 1 (L):
The multipliity w
q
(L) of q in R (L) is alled the inetionary weight of q with
respettoL andisgivenby
(2) w
q (L)=
h 0
(L)
X
i=1 (m
i i):
When L is not base point free we dene the inetionary weights w
q
(L) by (2)
and theramiation divisorR (L) byR (L) = P
q w
q
(L)q. Thisdivisor admitsan
alternativedesriptionasthezerodivisorofW =W(z)(dz) n(n 1)
2
wherezisaloal
oordinate,n=h 0
(L)andW(z)=W(f
1
;:::;f
n
)istheWronskianwithrespetto
anybasisf
1
;:::;f
n ofH
0
(L). InpartiularW isaholomorphisetionoftheline
bundleL n
! n(n 1)
2
C .
TakingL=!
C
onereoversthewell-knownnotionofthegapsequeneofq2C,
denoted by G
q
, and the above denition of the inetionary points and weights
redues, by a simple appliation of Riemann-Roh, to the standard denition of
Finally let us x our onventions about the Jaobian J(C) of C. By deni-
ton J(C) =H 0
(!
C )
=H
1
(C ;Z), sothat the vetorspae H 0
(!
C )
is anonially
identied with the tangent spae of J(C) at every point. By the Abel-Jaobi
theorem there is a anonial isomorphismJ(C) ' Pi 0
(C). Moreoverevery line
bundle L on C of degree g 1 indues an isomorphism J(C) ' Pi 0
(C) !
Pi g 1
(C);P 7!LP. Forourpurposesitisonvenientto workwithPi g 1
(C)
rather than Pi 0
(C). So in the sequel we identify J(C) with Pi g 1
(C) with-
out further notie; theunderlying isomorphism(respetivelyline bundledening
the isomorphism) will always be evident from the ontext. The main advan-
tage working with J(C) = Pi g 1
(C) is, that we have a anonial theta divisor
=fL 2J(C) j h 0
(L)> 0g. ByRiemann-Roh, is invariant with respet to
thenaturalinvolution
(3) :J(C) !J(C);(L)=!
C L
1
:
Morepreiselywehaveh 0
(L)=h 0
((L))foranyL2.
Wedenote by#the Riemanntheta funtion onH 0
(!
C )
forwhih
isthe
zerodivisorof#,where isthenaturalprojetionH 0
(!
C )
!J(C).
ForanyL2J(C)wehaveanembedding
L;p
ofCintoJ(C),givenby
L;p (q)=
L(q p). Clearlyfor dierentL andpthemaps
L;p
onlydierbyatranslation
onJ(C).
3. Geometri desription of the order of vanishing
Let L 2 J(C) and let X be aone-dimensional subvetor spae of the tangent
spaeH 0
(!
C )
at L. Chooseanypointlintheberof overL andonsiderthe
aÆneline l+X whih passesthroughland whih hasdiretion X. Theorder of
vanishing of #j
l+X
at the pointl is independent of the hoie of l 2 1
(L). So
denetheorderofvanishing of #at Linthediretion of X,denotedord
L (#;X),
asord
l
#j
l+X
. Ifdoesnotontainthestraightline
X =(l+X)thenthereexists
asmallneighborhoodU oflinl+X suhthat(U)\=fLgandord
L
(#;X)=
((U))
L
,theintersetion multipliityofwith(U)atL.
LetX
p
denote thetangentspae to
L;p
(C)at L. Notie that,asasubvetor
spae of H 0
(!
C )
, X
p
does not depend on L but only on the point p 2 C. We
wish to omputeord
L (#;X
p
)foran arbitraryL 2; ifL2= thenthis order is
triviallyzero. Itanbeshown 1
that forall C, L andpthat
X
p
isnotontained
in ,whih islearlytrueassoonasC, L orpis generi. Forthis theideais to
replae(U)byaaompleteurvewhih,aroundL,looks like(U). Notiethat
ifL2weannotuse
L;p
(C)beausethelatterdoesnotneessarilyinterset
properly. Considerforanyintegern6=0themorphism
(4)
L;p;n
:C !J(C);
L;p;n
(q)=L(nq np):
Notie that
L;p;n
(p) = L and that for a small neighborhood V of p in C the
tangentspaeto
L;p;n
(V)at LispreiselyX
p .
Lemma3.1. ForallL2J(C)and n>0wehave
(1)
L;p;n
(C)andinterset properly ifandonly ifh 0
(L( np))=0;
(2)
L;p; n
(C)andintersetproperly if andonlyif h 0
((L)( np))=0.
1
Thisfollowsfrom[SW , Prop. 8.6 andTh.9.11℄ butageometriproofofthis(geometri!)
Proof. We prove(1), the proof of (2) is similar. Reall that anirreduible urve
intersetsadivisorproperlypreiselywhentheurveisnotontainedinthesupport
ofthedivisor. So
L;p;n
(C)anddonotintersetproperlyifandonlyifh 0
(L(nq
np))>0forallq2C. Welaimthatthisisequivalenttoh 0
(L( np))>0. Indeed,
byRiemann-Roh
h 0
(L(nq np))=h 0
((L)(np nq)); and
h 0
((L)(np))=h 0
(L( np))+n:
Soh 0
(L(np nq))>0forallq2C ifandonlyifh 0
((L)(np))>n,leadingtoour
laim.
Inpartiular,whenjnjg then
ord
L (#;X
p
)=(
L;p;n (V))
L
;
whereV isasmallneighborhoodofpin C. PullingthisintersetionbaktoC we
getthatforanyjnjg
ord
L (#;X
p
)=mult
p (
L;p;n ):
Thismultipliitywillbeomputedinthenextsetion.
4. The divisor
L;p; n
Theaimofthissetionistoprovethefollowing
Theorem4.1. ForallL2J(C), p2C,and n>0with h 0
((L)( np))=0
L;p; n
=R (L(np)):
For theproofweneedthefollowing
Proposition4.2. ForallL2J(C);p2C andn>0
L;p; n O
J(C)
()=(L(np)) n
! n(n 1)
2
C :
Proof.
StepI:Theasen=1followsexatlyfrom[LB℄Lemma11.3.4withx=0;=
L, and=p.
StepII:Forn1onsiderthedierenemap
Æ n
L :C
2n
!J(C); Æ n
L (p
1
;q
1
;:::;p
n
;q
n )=L(
P
i p
i q
i )
anddenoteby n
i :C
2n
!Cthei-thprojetion. Weshowbyindutiononnthat
foralln1andallL2J(C)
Æ n
L
O
J(C) ()=
n
O
i=1
n
2i 1
(!
C L
1
) n
2i
L
O
C 2n
X
1i<j2n ( 1)
i+j+1
( n
i
; n
j )
; (5)
where denotesthediagonalin C 2
.
Forn=1wehavetoshowthat
Æ 1
L
O
J(C)
()= 1
1
(!
C L
1
) 1
2
LO
C 2
()
forallL2Pi g 1
(C). AordingtotheSeesawPriniple(see[LB℄A.9)itsuÆes
oinideforallq2C. ButsinetheompositionofÆ 1
L
withthenaturalembedding
C'Cfqg !CC isthemap Æ
!CL 1
;q; 1 and
=wehave,using
StepI,
Æ 1
L
O
J(C)
()jCfqg=
!
C L
1
;q; 1 O
J(C)
()=!
C L
1
(q)
= 1
1
(!
C L
1
) 1
2
LO
C
2()jCfqg;
andsimilarlyfortherestritionto fqgC.
Now suppose n > 1 and equation (5) holds for all n 0
< n. Restriting both
sides of equation (5) to C 2n 2
fp;qg and fp
1
;q
1
;:::;p
n 1
;q
n 1 gC
2
for all
p;q;p
1
;q
1
;:::;p
n 1
;q
n 1
2C, andusing the indution hypothesis for n 0
=n 1
andn 0
=1respetively,theSeesawPrinipleimpliesthatalsoequation (5)holds.
StepIII:Considertheembedding |
p
:C !C 2n
; |
p
(q)=(p;q;:::;p;q)and
notiethatÆ n
L Æ|
p
=
L;p; n
,sothat
L;p; n O
J(C) ()=|
p Æ
n
L
O
J(C) ():
Itfollowsthat
L;p; n O
J(C)
()anbeomputedfrom(5). Sine
( n
i
; n
j )Æ|
p (q)=
8
<
:
(p;p) i;j odd
(q;q) i;j even
(p;q)or(q;p) otherwise,
wehavethat
|
p (
n
i
; n
j )
O
C 2()=
8
<
: O
C
i;j odd
! 1
C
i;j even
O
C
(p) otherwise.
It followsthat the pullbak by|
p
of theright hand sideof (5)equalsL n
(n 2
p)
! n(n 1)
2
C
. This ompletestheproof.
Proofof Theorem 4.1. Bythe hoie of n we haveh 0
(L(np)) = n and the urve
L;p; n
(C)intersetsthedivisor properly. Theline bundle(L(np)) n
! n(n 1)
2
C
hastwodistinguisheddivisors,namely
L;p; n
(aordingtoProposition4.2)and
R (L(np))(aordingtoSetion2). Moreoverthesedivisorshavethesamesupport,
sinebydenitionq2
L;p; n
ifandonlyif
L;p; n
(q)2,i.e.,h 0
(L(np nq))>
0,andthisistheaseifandonlyifqisaninetionpointforthelinebundleL(np).
Forgeneri L and pthe line bundle L(np) admits only normal inetion points,
so R (L(np)) = P
n 2
g
i=1 q
i
with pairwise dierent points q
i
, and hene R (L(np)) =
L;p; n
. Thisequality extendsto allL andpfor whih
L;p; n
exists andby
Lemma3.1this isexatlythesetf(L;p)2J(C)Cjh 0
((L)( np))=0g.
Remark 1. Similarly one an show that if
L;p;n
(C) intersets properly, then
L;p;n
=R ((L)(np)).
5. Formula(s)forthe order of vanishing
Inthissetionweprovethefollowingtheorem:
Theorem5.1. Forevery L2
(6) ord
L (#;X
p )=
X
m g+h 0
(L((g m)p));
wherethe sum runsoverthe g integers msatisfying h 0
(L((g m)p))=h 0
(L((g
m+1)p)) 1.
Wewillobtainitas adiret onsequeneofthefollowingproposition.
Proposition5.2. Withnhosensuhthat
L;p; n
(C)intersetsproperly(e.g.,
n = g), the order of vanishing of # at L in the diretion X
p
is the inetionary
weightof pwithrespettoL(np). Therefore,
(7) ord
L (#;X
p )=
n
X
i=1 m
i i:
wherefm
1
<<m
n
gisthe gapsequeneG
p
(L(np))of L(np)atp.
Proof. AordingtoSetion3andTheorem4.1wehavethat
ord
L (#;X
p
)=mult
p (
L;p; n )=w
p
(L(np))= n
X
i=1 m
i i:
Proofof Theorem 5.1. By denition the sum in equation (6) runs over the set
G
p
(L(gp)) = fm
1
< < m
g
g of gap numbers of L(gp) at p. An immediate
omputationshowsthath 0
(L((g m
i
)p))=g ifori=1;:::;g. Sotheassertion
followsfromProposition5.2withn=g.
We now relate Theorem 5.1 to the Formula (1), given by Segal and Wilson.
Reallfromtheintrodutiontheinniteset
S
L
=fs2Zjh 0
(L((s+1)p))=h 0
(L(sp))+1g:
Proposition5.3. DenoteS
L
=fs
0
<s
1
<s
2
<g. Then
ord
L (#;X
p )=
X
i0 i s
i :
Proof. Note rstthat s
0
degL 1= g. For g sg 1wehavethat
s2S
L
ifandonlyifg s2G
p
(L(gp))=fm
1
<<m
g
g,sothats
i
=g m
g i
fori =0;:::;g 1. Onthe otherhand n2 S
L
for any ng so that s
n
=nfor
anyng. Summingupwend
X
i0 i s
i
= g 1
X
i=0
i g+m
g i
= g
X
i=1 m
i i:
HeneFormula(1)followsfrom Proposition5.2.
Referenes
[ACGH℄ Arbarello,E.,Cornalba, M.,GriÆths,P.A.,Harris,J.,GeometryofAlgebraiCurves,
VolumeI,SpringerGrundlehren,Bd.267(1984)
[F℄ Fay,J.,On theeven-ordervanishing of Jaobian thetafuntions,DukeMath. J.,51,
109{132(1984)
[LB℄ Lange,H.,Birkenhake,Ch.,ComplexAbelianVarieties,SpringerGrundlehren,Bd.302
(1992)
[Mi℄ Miranda,R.,AlgebraiCurvesandRiemannSurfaes,AMS,GraduateStudiesinMath-
ematis,Vol5(1995)
[S℄ Shiota,T.,CharaterizationofJaobianvarietiesintermsofsolitonequations,Invent.
[SW℄ Segal,G.,WilsonG.,Loopgroupsandequationsof KdVtype,Publ.I.H.E.S.,61,5{65
(1985)
Christina Birkenhake, Mathematishes Institut, Bismarkstrae11/2,D-91054Er-
langen
E-mailaddress: birkenmi.uni-erlangen .de
URL:http://www.mi.uni-erlang en. de/~ bir ken/
PolVanhaeke,
Universit
edePoitiers, D
epartementdePoitiers, Tel
eport2,Boule-
vardMarieetPierreCurie,BP30179,F-86962FuturosopeChasseneuilCedex
E-mailaddress: Pol.Vanhaekemathlabo .un iv-p oit iers .fr
URL:http://wwwmathlabo.sp2mi .un iv-p oit iers .fr /~va nha ek/
