C OMPOSITIO M ATHEMATICA
A. S OUCARIS
The ampleness of the theta divisor on the compactified jacobian of a proper and integral curve
Compositio Mathematica, tome 93, n
o3 (1994), p. 231-242
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The ampleness of the theta divisor
onthe compactified jacobian of
aproper and integral
curveA. SOUCARIS
C. N. R. S., URA D0752, Mathématiques, Bât. 425, Université de Puris-Sud, F-91405 Orsay Cédex.
Received 20 January 1993; accepted in revised form 29 July 1993
0. Introduction
Let X be a proper and
integral
curve over analgebraically
closed field k.If we suppose the curve to be
smooth,
the set made upby
theisomorphism
classes of the invertible sheaves of
degree
0 over X is the set of rationalpoints
of thejacobian,
an abelianvariety. But,
when we allowsingularities
for
X,
thejacobian
is not proper anymore.Nevertheless,
we get a naturalcompactification by considering
theisomorphism
classes of rank 1 torsion free sheaves over X. The relatedcompactified
functorPic(X)
is representa- ble.Let g
be the areithmetic genus of X.Then,
we can define the theta divisor on thereduced g -
1 component, thepoints
of whichcorrespond
to the rank 1 torsion free sheaves
having
non-zeroglobal
sections. Thiswork endeavours to prove that the theta divisor on
Pic (X) 9 -1
1 isample.
In
the firstchapter,
we recall the definition and basicproperties
ofPic(X).
In thesecond,
we define the thetadivisor,
and we show that two times theta isgenerated by
its sections on the normalization ofPic(X).
Then,
in the thirdchapter,
theampleness
of theta isproved by
demonstrat-ing
that no proper curve is included in thecomplement
of its support.Finally,
wegive
someapplications.
I. The
compactified
Picard functorWe suppose that all schemes are
locally
noetherian. We recall the definition and basicproperties
of thecompactified
Picard schemefollowing [AK 1].
For this
sake,
it is natural to work in the relative situation.Let
f :
X --> S be amorphism
of finite type, flat andprojective,
whosegeometric
fibers areintegral
curves.Moreover,
let us denoteby (Qx(l)
aninvertible and
S-ample
sheaf on X. In thischapter,
T will be an S-scheme.DEFINITION 1. A
sheaf M
onX T
= X x s T is a(relative) quasi-invertible sheaf, if
it iscoherent, T flat,
andif
its restriction to every geometricfiber of f is
rank 1 torsionfree.
It is useful to have a definition for the
degree
of a sheaf on a proper andintegral
curve, even withsingularities.
This definition willyield
de facto theRiemann-Roch formula.
DEFINITION 2. Let C be an
integral
andprojective
curve on analgebrai- cally closed field k,
and M aquasi-invertible sheaf on
C. Wedefine
thedegree of M
and we writed(M),
orsimply d,
theinteger
such that:where g is the arithmetic genus
of
X, andhi’(M)
denotesdimkHi(X, M).
LEM MA 3. Let C be a proper and
integral
curve on analgebraically
closedfield k,
M aquasi-invertible sheaf,
and 2 an invertiblesheaf corresponding
to a Cartier divisor with support in the smooth locus. We then have:
Proof.
Weonly
needproving
this result for 2 =(9c(x)
where x is asmooth
point.
We have the exact sequence:This
yields
thefollowing
exact sequence:where V is a sheaf in
k(x)-vectorial
spaces of dimension1,
concentrated at thepoint
x.By
thelong
exact sequence, we getLet
P’
be the functor that takes every S-scheme T to the set of ailquasi-invertible
sheaves onX T.
Moreover, let d be aninteger.
ThenP’d
denotes the subfunctor of
P’
made upby
thequasi-invertible
sheaves ofdegree
d.DEFINITION 4. The
compactified
Picard scheme is thesheaf.f’or
the étaletopology
associated tothe functor P’.
We shall denote itby Pic(X/S).
In thesame way,
Picd(X/S)
is associated toP".
Pic d(X/S)
is an open and closed subfunctor ofPic(X/S).
The
following
theorem has beenproved by
Altman and Kleiman[AK 1 (6.6)], following
Grothendieck’s sketch[G].
THEOREM 5. Let d be an
integer. The functor Picd(X/S)
isrepresentable by
ascheme,
that isprojective, locally
on S.We shall denote this scheme
again by Picd(X/S).
Suppose
now that there is a section 8 with values in the smooth locus:03B5: S --> X.
Consider the functor
P
of thequasi-invertible rigidified
sheaves: for anyS-point T, P(T)
is the set ofisomorphism
classesa)
such that A isquasi-invertible
onX T
and a is anisomorphism
between8*(A)
and(QT-
PROPOSITION 6.
The,functor P
is asheaf for
the étaletopology.
More-over, the natural
composite
mapis an
isomorphism of
étale sheaves.Proof
First ofall,
let us show thatrigidifying
cancels the non-trivialautomorphisms.
Indeed,
allquasi-invertible
sheaves aresimple
i.e. for anyS-point
T, there is a canonicalisomorphism
where A is any
quasi-invertible
sheaf on X[AK 1 (5.2)].
Now, to show that
P
is asheaf,
let T be anS-point,
and suppose we havean exact
diagram:
where T’ covers and
We get the
following diagram:
Take a
point
inP(T’)
whoseimage by
both arrowscorresponding
to theprojections coincide,
then thecocycle
condition is fullfilled inP(T"’),
because, as we saw, a
rigidified
sheaf hasonly
trivialisomorphisms. By
étaledescent of coherent
sheaves,
we get aT point
ofP making
commutative theabove
diagram.
DII. The thêta divisor
Now,
the base S is the spectrum of analgebraically
closed field k. The choice of a smoothpoint
x in Xgives
us arigidification
e. We writesimply Pic(X)
forPic(X/k).
Let z be a rational
point
of X x kPic(X)9-1.
Itcorresponds
to aquasi-invertible
sheaf A on X ofdegree g -
1. Thus we haveh°(M) = h1(M).
We aregoing
to provethat,
if weneglect
the embedded components ofPic(X) 9-1,
the set of thepoints
z such thathO(M) = h1(M)
do not
vanish,
is the support of a Cartierdivisor, namely
the theta divisor.Let 5’ be the universal sheaf of
Pic(X)g-l.
We consider an affine open setSpec(A)
ofPic(X)9-1.
There exists acomplex (Kj)
oflocally
freeA-modules of finite type such that the
following
holds for any A-module N(see
e.g.[H] III 12.2):
In
particular
thiscomplex
computes thecohomology
of thequasi-invertible
sheaves
corresponding
to the variouspoints
ofSpec(A):
where a is a
point
ofSpec(A),
andk(«)
denotes its residual field.Universally
there is nocohomology
indegree
greater than one, thus wecan concentrate the
complex
K ’ indegree (0, 1). Moreover,
as we work onthe component
of degree g - 1,
K° and K’ have the same rank.Working
for the momentlocally,
we introduce the determinant -9 of the mapK° ~ K1
1 definedby det(K 1)
Qdet(K°)-’.
It is an invertible sheaf onSpec(A). Tensorizing by det(K°)-’,
themap K° - K1 gives
rise to amap (OA -D.
This last map defines in turn a section ô of -9 whose set ofzeros we denote
by
0. 0 is a closed subschemelocally
definedby
oneequation,
but we do not know whether 0 is a Cartier divisor or not.We do not
modify
0 if wereplace
K ’by
aquasi-isomorphic complex
ofthe same type.
Glueing
up, we get 0globally
onPic(X)9-’.
REMARK.
Making
use of the(quasi)-projectivity
ofPic(X)9-1,
we couldhave constructed
globally
onPic(X)9-1
acomplex
KO ~ K1 1 which com-putes
universally
thecohomology
of f . This is an other way ofproving
the
global
existence of 0.We denote
by
U thecomplement
of the support of 0. It is an open subscheme ofPic(XY- 1.
We first notice that there is a natural action of the Picard group scheme
Pic(X)
onPic(X) given by
thefollowing
formula:In
particular, Pic°(X)
acts onPic(X)9 - 1 .
PROPOSITION 7. The saturation
PicO(X)
+ Uof the
open set U under the actionof Pico(X)
is the whole schemePic(X)9-1.
Proof.
Let A be aquasi-invertible
sheaf on X ofdegree g -
1. We aregoing
to prove that there exists an invertible sheaf 2 on X ofdegree
0 suchthat
To see
this,
let JV be an invertible sheaf ofdegree n
on Xsufficiently ample
such that:
Lemma 3
yields: d[M
QN]
=d(M)
+ n.Let us denote now
by Mo
the sheaf JI Q ae. IfAo
has noglobal section,
the
problem
is solved with 2 = ae. Otherwise we consider a non-zeroglobal
section s ofMo
and apoint
x, of the smooth locus where s does notvanish
(i.e.
where s generatesMo).
ConsiderMo
QOx ( - x 1 ); s
is not anymore a
global
section ofM1
Q(9x( -X 1).
Thus we have:On the other
hand,
we have:Hence the two
equalities
below hold:We set
M1
=Mo
O(OX ( - x 1 ). It
is a sheaf ofdegree (g - 1)
+(n - 1).
We now go on
by descending
induction. We denoteby
xi the variouspoints
appearing
in the process, andby Ai
the various sheaves obtained. We findthat
An belongs
to U and arises from Atensorizing by
the sheafN Q9
O(- Il, i,,, Xi)
we shall denoteby 2
which is ofdegree
0. DLet us notice that we do not know in
general
ifPic(X)
is reduced orwhether it has embedded components. Let Y be the sheaf of those
nilpotent
ideals of the structural sheaf of
Pic(X)9 -1
1generated by
the sections whose support are closed setswith empty
interior. From now on pg-1 will denote the closed subscheme ofPic(X)g-
1 definedby
the sheaf J.THEOREM AND DEFINITION 8. The closed subscheme 0 induces on
pg- 1 a
Cartierdivisor, namely
the theta divisor that we denoteagain by
0.Moreover we have
D/pg-1
=(opg-1( 0).
Proof. Pic°(X) being connected,
wefind,
as acorollary
ofProposition
7, that U does not contain anygeneric point
ofPic(X)g-1.
Thus b induces anon-zero divisor on
D/pg-1
DWe endeavour to show that 0 is an
ample
divisor. Let P,g- 1 be thenormalization of
(Pg - 1 )red.
Noticethat,
asPicO(X)
issmooth,
the action ofPic°(X)
on Pg-1 1 lifts to an action ofPic°(X)
on P’9-’. We shall denotePic°(X) by
J.The
analog
ofProposition
7 holds on P,g- 1. Infact,
if x’ is apoint
of P,g- over apoint
x ofpg- 1,
J + x meets U, hence J + x’ meets thepullback
U’ of U in P’9- l. Set also 0’ thepullback
of O in P,g- 1.PROPOSITION 9. The invertible
sheaf (9(20’)
on the scheme P’9-1 isgenerated by
itsglobal
sections.Proof.
Let éP be the Picard scheme of P’9- 1. For anypoint
a of J, letTa
be the translation
by a operating
on P,g - 1 .The map: a -
Ta(0’) -
0’,taking
divisorclasses,
defines amorphism
ofschemes h: J --+
Po,
where f!JJ° denotes the neutral component of P.The next result is a variation on the theorem of the square.
LEMMA 10. h is a
morphism of
group schemes.Let us prove this lemma.
Studying
therepresentability
of J we findby ([BLR] 9.2) that,
if we start withthe jacobian B
of the normalization of X(an
abelianvariety),
J arises as an extension of Bby
a linear group,namely
a successive extension of additive groups
Ga
andmultiplicative
groupsGm.
Thus J is an extension of B
by
a smooth and connected group H which isa rational
variety.
On the other handYroed
is an abelianvariety
as P’9-’ 1 isnormal
([G]
Th.2.1). Now,
every mapgoing
from a rationalvariety
to anabelian
variety
is constant. Hence themap h
from J to’pr’ed
factorizesthrough
B. We get amorphism
of schemes from B toYr’ed
which sends theorigin
to theorigin. By
therigidity
lemma(see [M]
Ch. 6 Cor.6.4),
thismorphism
is amorphism
of group schemes. Thus h has the same property.COROLLARY 11. For any
point
a inJ, Ta(O’ )
+T-a(0’)
islinearly equivalent
to 20’.End
of
theProof of Proposition
9. To show that 20’ isgenerated by
itssections,
we reduce toproving
thefollowing
fact: for anypoint
x inP,g- 1,
there is a
point a
in J such that x does not lie inTa (O’ )
+T-a(0’).
Let us then fix x in J and let us consider the subset
V+ (resp. V_ )
of theelements a of J for which x + a
(resp.
x -a) belongs
to U’.By Proposition 7, V+
and V are non-empty open sets of J.But,
Jbeing irreducible, V+
meets V_. We take a to be any element in
V+
n V_. DIII. The
ampleness
of the thêta divisorIn this
chapter,
we work over analgebraically
closed field k. We fix asection c with values in the smooth locus of X and we call x its
image.
Firstof
all,
notice that 0 isample
on pg-1 if andonly
if 0’ isample
on P’g-1.PROPOSITION 12.
The followiny properties
areequivalent:
(i)
0’ isample (ii)
U’ isaffine
(iii)
No irreduciblecomplete
curve lies in U’.Proof (i)
=>(ii)
=>(iii):
well known.(iii)
=>(ii):
20’ isgenerated by
itssections,
so there exists amorphism
9 from P’9-’ to aprojective
space pn definedby
the linear system20’.
20’appears as the
pullback by
ç of ahyperplane
section H of pn. Denoteby
Q its
complement.
ThenU’ = qJ - 1 (Q),
and themorphism ~’
from U’to
Q,
restriction of 9 to U, is proper.ç’
is alsoquasi-finite,
otherwise atleast one of its fibers would contain an irreducible curve C. C would be closed in
p,g-l,
and that would contradict(iii).
Themorphism ~’
is nowproper and
quasi-finite,
hence it is alsofinite,
and inparticular
affine. Weconclude that U’ is
affine, being
thepullback
of an affine open setby
anaffine
morphism.
(ii) => (i):
As U’ isaffine,
we have the same property for7§(U’),
wherea is any
point
in J. P’9-’being separated, Ta(U’)
nT-a(U’)
isagain
affine.In other
words,
this holds also for thecomplement
of the divisorT (0’)
+T-a(0’). Moreover,
we saw before that these open sets cover P,g - 1 .Thus any fiber of 9 is contained in an affine open set. From
this,
we getthat any fiber of 9 is finite because it is proper. Hence 9 is finite and 0’ is
ample.
QWe are
going
to prove theamplitude
of 0’using
condition(iii)
ofProposition
12.So,
let us assume that U’ contains a proper andintegral
curve, the normalization of which we denote
by
C.Then,
thecomposite morphism
C --- >U’ ---> U --> Pg -
1corresponds
to aquasi-invertible
C-sheaf-9Y on X x
C, rigidified along
the section c. Thisrigidification corresponds
to an
isomorphism
a from(9c
to8*(A).
In the
sequel,
S will be the surface X xC,
andp: S --+ X,
q: S - C will denote the twoprojections. By hypothesis,
C is aboveU,
soq*(M)
andR 1 q* (M) vanish;
this holds also afterbase-change,
and inparticular by
restriction to the fibers of q. We are
going
to show that under thesehypotheses
the sheaf -eV is p-constant i.e. is in the formp*(M 0),
where-97o
is a
quasi-invertible
sheaf on X. The map from C to pg-l 1being
non-constant, we will get a contradiction.
The
proof
runsthrough
six steps.Step
1. Recall that x denotes theimage
of the section e. Let(Os(x)
be thepullback
of(9x(x) by p
i.e. the invertible sheaf ofdegree
1 on Xconsisting
of rational functions
having
at most apole
of order 1along
x.Let us write the exact sequence:
Taking
tensorproducts by (Ox(x),
we get:where V is a vector space over k of dimension 1 concentrated at x.
Taking
thepullback by
p of the above exact sequence, we now get anexact sequence on S:
Next,
we take the tensorproduct by
M of this sequence. As M is invertible in theneighbourhood
of E, the sequence we obtain on S remains exact:As A is
rigidified along
the last term can be identified with VQk (9c,
making
use of theisomorphism
a.Finally
we get the exact sequence:Step
2. Asq* (M) = R1q*(M) = 0,
we getby
thecohomological
exactsequence
for q
anisomorphism
between q*[M(x)]
and VQk (9c.
Choose abase e of V. It then
corresponds
to aglobal
section ofM(x)
we shall alsodenote
by e,
which in turn induces a non-zero section on the fibers ofM(x)
over
C,
and which generatesaV(x)
in theneighbourhood
of c. From nowon we set a7’ =
A(x).
Thus the section e defines aninjective morphism from
into M that remainsinjective
after anybase-change.
Let fi’ bethe
quotient
sheafM/os.
Its support is closed inS,
hence is proper. As it does not meet[x]
xC,
it is finite over C.Step
3. Here we want to show that the support of 1’ is horizontal i.e. it is thepullback by
p offinitely
manypoints
of X.Let
Ci
be an irreducible component of the support of 1’.Projecting Ci by
p onX,
we get asingle
closedpoint
of X.Otherwise,
the restriction of p toCi
would be asurjective
map fromCi
ontoX,
becauseCi
is proper. Inparticular, Ci
would intersect the curve[x]
x C at onepoint (x, c)
at least.A contradiction.
Step
4. We are nowgoing
to show thatq*(N’)
is constant.The section e
gives
rise to the exact sequence:The
long
exact sequence thenyields:
Now
q.«(Os) - q.(M’) is
anisomorphism by
construction. On the otherhand, R1q*(M’)
vanishes. Hence the map fromq* (N’ ) to R1q*(Os) is
anisomorphism. By
flatbase-change,
we get theequality
We set E =
H1(X, (Ox).
Thus we find thatq*(N’)
is the sheaf E0, (9c.
Step
5. We shall now embed JI’ into an invertible and p-constant sheaf on S, that is to say a sheaf in the formp*(2),
where 2 is invertible on X. Let{X1...,Xn}
be theprojection by p
of the support of 1’. We can find anaffine open set W of X
containing
xl, ... , Xn, and a functionf
on W whosezero-locus,
looked upon from the set-theoreticalviewpoint,
is(xi, ... , ,Xn}.
We consider the open set
p-1(W)
and the functionf ’
=f Qk 1
=p -1 ( f ).
As the support of fi’ is contained in the zero-locus of
f’,
1’ is annihilatedby
a power off’,
sayf’m.
Let
Ox(D)
be the invertible sheafon X, generated
on Wby f - m
andoutside W
by
1. Wejust
saw that -e7’ is contained inp* [(9x(D)] (that
weshall
simply
denoteby OS(D)).
Hence we get the inclusions:Step
6. We now prove that N’ and A’ are constant sheaves.Taking quotients by Os, the
inclusions(Os
4 -l7’ q(OS(D) correspond
to an inclu-sion from 1’ =
-97’/Ws
intoO9s(D)/(9s.
Directimage by q yields:
By
flatbase-change,
the sheafq*[(Os(D)/(Os] is isomorphic
to the sheafHO[X, (9x(D)/(9 x ] Q9k (OC.
Set F =
HO[X,(Ox(D)/O9x],
so thatq*[(Os(D)/(Os]
=F ©k Oc.
Thus wehave got two constant sheaves over
(9c namely q* (N )
andq*[(Qs(D)/Os].
The curve C
being
proper, the canonicalembedding
fromq* (N’ )
toq*[(Os(D)/Os]
is forced constant and arises from a k-linearinjective
map from E into F. In otherwords,
theglobal
sections of 1’ are constantsections of
Os(D)/Os.
Fiber tofiber, they
generateN’,
hence N’ is a p-constant subsheaf of(9s(D)/(9s,
that is to say N’ is thepullback by
p ofa subsheaf
%0
of(9x(D)/(9x.
We now define..,/(0
to be the subsheaf(9x(D) making
thediagram
below commutative with exact rows:Pullback
by p
over Syields
thefollowing diagram:
We now see that A’ =
p*(M 0).
Hence aV’ is constant, and that ends theproof
of theamplitude
of the theta divisor. DIV.
Applications
and additional remarks 1. The relative caseSuppose
we work in the relative situation wheref:
X -S is amorphism
offinite type, flat and
projective,
whosegeometric
fibers areintegral
curves ofgenus g.
Suppose
thatf
has a section e: S --> X with values in the smooth locus.In a similar way to that
of part II,
we define the sheaf -9 oPic9 -1 (X/S)
to be the determinant of the relative
cohomology
of the universal sheaf onTHEOREM 13.
Thesheaf -9
onPicg-1(X/S)
isS-ample.
Proof
AsPicg-1(X/S)
is proper, to prove theamplitude
of -9 onPicg - l(X /S),
it suffices to show it for the restriction of D toPic9 -1 (X Os k(s)/k(s))
where s is anygeometric point
of S.So,
we reduceto the case worked out in
chapter
III. D 2. The caseof’ locally planar
curvesRecall that an
integral
curve X on a field k is said to belocally planar (for
the étale
topology),
if the Zariski tangent space at anypoint
x of X is ofdimension not
exceeding
2. This condition isequivalent
tosaying
that thecompleted
localring
at x is thequotient
of thering k[[u, v]] by
a reducednon-zero
equation.
In viewof ([AIK],
Theorem9) (see
also[R]), Pic9 -1 (X)
is then the schematic closure of
Picg-1(X)
and is a localcomplete
intersection. Thus we get a theta divisor on
Pic(X}q - 1 .
COROLLARY 14. Let X be a
planar
curve.Then,
the theta divisor is anample positive
Cartier divisor onPicg-1(X),
the schematic closureof
When one accepts curves that are not
locally planar,
recall(see [R])
thatthe
compactified jacobian
is not anymore the closure of the usualjacobian
and may have in
particular
components of dimensionsexceeding
g.Acknowledgement
I wish to thank M.
Raynaud
for valuable advice andstimulating
dis-cussions.
References
[AIK] A. Altman, A. Iarrobino, S. Kleiman, Nordic Summer School/NAVF, Symposium in mathematics, Oslo 1976.
[AK1] A. Altman, S. Kleiman, Compactifying the Picard Scheme, Advances in Mutheniatics, Vol.
35, p. 50-112.
[BLR] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, Springer 1990.
[G] A. Grothendieck, Fondements de la Géométrie Algébrique, second exposé sur le foncteur de Picard.
[H] R. Hartshorne, Introduction to Algebraic Geometry, Springer 1978.
[M] D. Mumford, Geometric Invariant Theory, Springer 1965.
[R] C. J. Rego, The compactified Jacobian, Annales scientifiques E.N.S., t. 13 1980, p. 211-223.