Jacobian Varieties
J.S. Milne
June 12, 2021

Abstract
This is the original TEX file for my article Jacobian Varieties,
published as Chapter VII of Arithmetic geometry (Storrs, Conn.,
1984), 167–212, Springer, New York, 1986. The table of contents
has been restored, some corrections and minor improvements to the
exposition have been made, and an index and a some asides added.
The numbering is unchanged.

Contents
1
2
3
4
5

Definitions . . . . . . . . . . . . . . . . . . .
The canonical maps from C to its Jacobian variety . . . . .
The symmetric powers of a curve . . . . . . . . . . .
The construction of the Jacobian variety. . . . . . . . .
The canonical maps from the symmetric powers of C to its
Jacobian variety . . . . . . . . . . . . . . . . .
6 The Jacobian variety as Albanese variety; autoduality . . . .
7 Weil’s construction of the Jacobian variety . . . . . . . .
8 Generalizations . . . . . . . . . . . . . . . . .
9 Obtaining coverings of a curve from its Jacobian . . . . .
10 Abelian varieties are quotients of Jacobian varieties . . . .
11 The zeta function of a curve . . . . . . . . . . . . .
12 Torelli’s theorem: statement and applications . . . . . . .
13 Torelli’s theorem: the proof . . . . . . . . . . . . .
14 Bibliographic notes . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . .
Index . . . . . . . . . . . . . . . . . . . . . . .

2
5
8
14
16
19
24
27
30
33
35
37
40
45
49
50

This article contains a detailed treatment of Jacobian varieties. Sections
2, 5, and 6 prove the basic properties of Jacobian varieties starting from the
definition in Section 1, while the construction of the Jacobian is carried out
in Sections 3 and 4. The remaining sections are largely independent of one
another.
It is a companion to my article “Abelian Varieties” (Milne 1986), which
is cited as “AVs”. The conventions are the same as in those listed at the start
of AVs (see also the start of Section 5 of AVs). In particular, k is a field.

1

1

1

DEFINITIONS

2

Definitions

Let C be a complete nonsingular curve over a field k. We would like
to define a variety J , called the Jacobian variety of C , such that J.k/ D
Pic0 .C / functorially. Unfortunately, this is not always possible: clearly, we
would want that J.k sep / D Pic0 .Ck sep /; but then
J.k sep / D J.k/ D Pic0 .Ck sep / ;

D Gal.k sep =k/;

and it is not always true that Pic0 .Ck sep / D Pic0 .C /. However, this is true
if C.k/ is nonempty.
Recall that for a scheme S, Pic.S / denotes the group H 1 .S; OS / of
isomorphism classes of invertible sheaves on S, and that S 7! Pic.S / is a
functor from the category of schemes over k to that of abelian groups.
Let
PC be a completePnonsingular curve over k. The degree of a divisor
D D i ni Pi on C is i ni Œk.Pi /W k. Since every invertible sheaf L on
C is of the form L.D/ for some divisor D, and D is uniquely determined
up to linear equivalence, we can define deg.L/ D deg.D/. Then deg.Ln / D
deg.nD/ D n  deg.D/, and the Riemann-Roch theorem says that
.C; Ln / D n  deg.L/ C 1

g:

This gives a more canonical description of deg.L/W when .C; Ln / is written
as a polynomial in n, deg.L/ is the leading coefficient. We write Pic0 .C /
for the group of isomorphism classes of invertible sheaves of degree zero
on C .
Let T be a connected scheme over k, and let L be an invertible sheaf
on C  T (by which we mean C Spec.k/ T /. Then (AVs, 4.2(b)) shows
that .Ct ; Lnt /, and therefore deg.Lt /, is independent of t ; moreover, the
constant degree of Lt is invariant under base change relative to maps T 0 !
T . Note that for a sheaf M on T , .q  M/t is isomorphic to OCt and, in
particular, has degree 0. Let
PC0 .T / D fL 2 Pic.C  T / j deg.Lt / D 0 all t g=q  Pic.T /:
We may think of PC0 .T / as being the group of families of invertible sheaves
on C of degree 0 parametrized by T , modulo the trivial families. Note that
PC0 is a functor from schemes over k to abelian groups. It is this functor
that the Jacobian attempts to represent.
T HEOREM 1.1. There exists an abelian variety J over k and a morphism
of functors W PC0 ! J such that W PC0 .T / ! J.T / is an isomorphism whenever C.T / is nonempty.
Let k 0 be a finite Galois extension of k such that C.k 0 / is nonempty, and
let G be the Galois group of k 0 over k. Then for every scheme T over k,
C.Tk 0 / is nonempty, and so .Tk 0 /W PC0 .Tk 0 / ! J.Tk 0 / is an isomorphism.
As
def
J.T / D Mork .T; J / D Mork 0 .Tk 0 ; Jk 0 /G D J.Tk 0 /G ,

1

DEFINITIONS

3

we see that J represents the functor T 7! PC0 .Tk 0 /G , and this implies that
the pair .J; / is uniquely determined up to a unique isomorphism by the
condition in the theorem. The variety J is called the Jacobian variety of C.
Note that for any field k 0  k in which C has a rational point,  defines an
isomorphism Pic0 .C / ! J.k 0 /.
When C has a k-rational point, the definition takes on a more attractive
form. A pointed k-scheme is a connected k-scheme together with an element
s 2 S.k/. Abelian varieties will always be regarded as being pointed by the
zero element. A divisorial correspondence between two pointed schemes
.S; s/ and .T; t/ over k is an invertible sheaf L on S  T such that LjS  ft g
and Ljfsg  T are both trivial.
T HEOREM 1.2. Let P be a k-rational point on C . Then there is a divisorial
correspondence MP between .C; P / and J such that, for every divisorial
correspondence L between .C; P / and a pointed k-scheme .T; t /, there
exists a unique morphism 'W T ! J such that '.t / D 0 and .1  '/ MP 
L.
Clearly the pair .J; MP / is uniquely determined up to a unique isomorphism by the condition in (1.2). Note that each element of Pic0 .C / is
represented by exactly one sheaf Ma , a 2 J.k/, and the map 'W T ! J
sends t 2 T .k/ to the unique a such that Ma  Lt .
Theorem 1.1 will be proved in 4. Here we merely show that it implies
(1.2).
L EMMA 1.3. Theorem 1.1 implies Theorem 1.2.
P ROOF. Assume that there is a k-rational point P on C . Then for any
k-scheme T , the projection qW C  T ! T has a section s D .t 7! .P; t //,
which induces a map
s  D .L 7! LjfP g  T /W Pic.C  T / ! Pic.T /
such that s  ı q  D id . Consequently, Pic.C  T / D Im.q  / ˚ Ker.s  /,
and so PC0 .T / can be identified with
P 0 .T / D fL 2 Pic.C  T / j deg.Lt / D 0 all t, LjfP g  T is trivialg.
Now assume (1.1). As C.k/ is nonempty, C.T / is never empty, and
J represents the functor PC0 D P 0 . This means that there is an element
M of P 0 .J / (corresponding to idW J ! J under / such that, for every
k-scheme T and L 2 P 0 .T /, there is a unique morphism 'W T ! J such
that .1  '/ M  L. In particular, for each invertible sheaf L on C of
degree 0, there is a unique a 2 J.k/ such that Ma  L. After replacing M
with .1  ta / M for a suitable a 2 J.k/, we can assume that M0 is trivial,
and therefore that M is a divisorial correspondence between .C; P / and J .
It is clear that M has the universal property required by (1.2).
2
E XERCISE 1.4. Let .J; MP / be a pair having the universal property in
(1.2) relative to some point P on C. Show that J is the Jacobian of C .

1

DEFINITIONS

4

We next make some remarks concerning the relation between PC0 and
J in the case that C does not have a k-rational point.
R EMARK 1.5. For all k-schemes T , .T /W PC0 .T / ! J.T / is injective.
The proof of this is based on two observations. Firstly, because C is a
complete variety H 0 .C; OC / D k, and this holds universally: for any kscheme T , the canonical map OT ! q OC T is an isomorphism. Secondly,

for any morphism qW X ! T of schemes such that OT ! q OX , the
functor M 7! q  M from the category of locally free OT -modules of
finite-type to the category of locally free OX -modules of finite-type is
fully faithful, and the essential image is formed of those modules F on X
such that q F is locally free and the canonical map q  .q F/ ! F is an
isomorphism. (The proof is similar to that of AVs, 5.1.)
Now let L be an invertible sheaf on C  T that has degree 0 on the
fibres and which maps to zero in J.T /; we have to show that L  q  M
for some invertible sheaf M on T . Let k 0 be a finite extension of k such
that C has a k 0 -rational point, and let L0 be the inverse image of L on
.C  T /k 0 . Then L0 maps to zero in J.Tk 0 /, and so (by definition of J / we
must have L0  q  M0 for some invertible sheaf M0 on Tk 0 . Therefore q L0
is locally free of rank one on Tk 0 , and the canonical map q  .q L0 / ! L0 is
an isomorphism. But q L0 is the inverse image of q L under T 0 ! T (see
AVs, 4.2a), and elementary descent theory (cf. 1.8 below) shows that the
properties of L0 in the last sentence descend to L; therefore L  q  M with
M D q L.
R EMARK 1.6. It is sometimes possible to compute the cokernel to W PC0 .k/ !
J.k/. There is always an exact sequence
0 ! PC0 .k/ ! J.k/ ! Br.k/
where Br.k/ is the Brauer group of k. When k is a finite extension of
Qp , Br.k/ D Q=Z, and it is known (see Lichtenbaum 1969, p. 130) that
the image of J.k/ in Br.k/ is P 1 Z=Z, where P (the period of C / is the
greatest common divisor of the degrees of the k-rational divisors classes on
C.
R EMARK 1.7. Regard PC0 as a presheaf on the large étale site over C ;
then the precise relation between J and PC0 is that J represents the sheaf
associated with PC0 (see Grothendieck 1968, 5).
Finally we show that it suffices to prove (1.1) after an extension of
the base field. For reference, we first state a result from descent theory.
Let k 0 be a finite Galois extension of a field k with Galois group G, and
let V be a variety over k 0 . A descent datum for V relative to k 0 =k is a
collection of isomorphisms ' W V ! V , one for each  2 G, such that
'  D ' ı ' for all  and . There is an obvious notion of a morphism
of varieties preserving the descent data. Note that for a variety V over k ,
Vk 0 has a canonical descent datum. If V is a variety over k and V 0 D Vk 0 ,
then a descent datum on an OV 0 -module M is a family of isomorphisms
' W  M ! M such that '  D ' ı  ' for all  and  .

2

THE CANONICAL MAPS FROM C TO ITS JACOBIAN VARIETY5

P ROPOSITION 1.8. Let k 0 =k be a finite Galois extension with Galois group
G.
(a) The map sending a variety V over k to Vk 0 endowed with its canonical descent datum defines an equivalence between the category of
quasi-projective varieties over k and the category of quasi-projective
varieties over k 0 endowed with a descent datum.
(b) Let V be a variety over k, and let V 0 D Vk 0 . The map sending an OV module M to M0 D OV 0 ˝ M endowed with its canonical descent
datum defines an equivalence between the category of coherent OV modules and that of coherent OV 0 -modules endowed with a descent
datum. Moreover, if M0 is locally free, then so also is M.
P ROOF. See Serre 1959, V, 20, or Waterhouse 1979, 17. (For the final
statement, note that being locally free is equivalent to being flat for a
coherent module, and that V 0 is faithfully flat over V .)
2
P ROPOSITION 1.9. Let k 0 be a finite separable extension of k; if (1.1) is
true for Ck 0 , then it is true for C .
P ROOF. After possibly enlarging k 0 , we may assume that it is Galois over
k (with Galois group G, say) and that C.k 0 / is nonempty. Let J 0 be the
Jacobian of Ck 0 . Then J 0 represents PC0 0 , and so there is a universal M
k
in PC0 .J 0 /. For any  2 G, M 2 PC0 .J 0 /, and so there is a unique map
' W J 0 ! J 0 such that .1  ' / M D M (in PC0 .J 0 //. One checks
directly that '  D ' ı  ' ; in particular, ' ı ' 1 D 'id , and so the '
are isomorphisms and define a descent datum on J 0 . We conclude from
(1.8) that J 0 has a model J over k such that the map PC0 .Tk 0 / ! J.Tk 0 /
is G-equivariant for all k-schemes T . In particular, for all T , there is a

map PC0 .T / ! PC0 .Tk 0 /G ! J.k 0 /G D J.k/. To see that the map is an
isomorphism when C.T / is nonempty, we have to show that in this case
PC0 .T / ! PC0 .Tk 0 /G is an isomorphism. Let s 2 C.T /; then (cf. the proof
of (1.3)), we can identify PC0 .Tk 0 / with the set of isomorphism classes of
pairs .L; ˛/ where L is an invertible sheaf on C  Tk 0 whose fibres are

of degree 0 and ˛ is an isomorphism OTk0 ! .s; 1/ L. Such pairs are
rigid—they have no automorphisms—and so each such pair fixed under G
has a canonical descent datum, and therefore arises from an invertible sheaf
on C  T .
2

2 The canonical maps from C to its Jacobian variety
Throughout this section, C will be a complete nonsingular curve, and J
will be its Jacobian variety (assumed to exist).
P ROPOSITION 2.1. The tangent space to J at 0 is canonically isomorphic
to H 1 .C; OC /; consequently, the dimension of J is equal to the genus of
C.

2

THE CANONICAL MAPS FROM C TO ITS JACOBIAN VARIETY6

P ROOF. The tangent space T0 .J / is equal to the kernel of J.kŒ/ ! J.k/,
where kŒ is ring in which  2 D 0 (see Hartshorne 1977, II, Ex.2.8). Analogously, we define the tangent space T0 .PC0 / to PC0 at 0 to be the kernel of
PC0 .kŒ/ ! PC0 .k/. From the definition of J , we obtain a map of k-linear
vector spaces T0 .PC0 / ! T0 .J / which is an isomorphism if C.k/ ¤ ˚.
Since the vector spaces and the map commute with base change, it follows
that the map is always an isomorphism.
Let C D CkŒ ; then, by definition, PC0 .kŒ/ is equal to the group of
invertible sheaves on C whose restrictions to the closed subscheme C
of C have degree zero. It follows that T0 .PC0 / is equal to the kernel of


H 1 .C ; OC
/ ! H 1 .C; OC
/. The scheme C has the same underlying

topological space as C , but OC D OC ˝k kŒ D OC ˚ OC . Therefore


we can identify the sheaf OC
on C with the sheaf OC
˚ OC  on C , and



1
1
1
so H .C ; OC / D H .C; OC / ˚ H .C; OC / . It follows that the map
a 7! exp.a/ D 1 C a;


OC ! OC
;


induces an isomorphism H 1 .C; OC / ! T0 .PC0 /.

2

Let P 2 C.k/, and let LP be the invertible sheaf L. C  fP g
fP g  C / on C  C , where  denotes the diagonal. Note that LP is
symmetric and that LP jC  fQg  L.Q P /. In particular, LP jfP g  C
and LP jC  fP g are both trivial, and so LP is a divisorial correspondence
between .C; P / and itself. Therefore, according to (1.2) there is a unique
map f P W C ! J such that f P .P / D 0 and .1  f P / MP  LP . When
J.k/ is identified with Pic0 .C /, f P W C.k/ ! J.k/ becomes identified with
the map Q 7! L.Q/ ˝ L.P / 1 (or, in terms of divisors, the map sending
Q to the linear equivalence class ŒQ P  of Q P /. Note that the map
hX
i
X
X
nQ 7!
nQ f P .Q/ D
nQ Q
Q

Q

Q

from the group of divisors of degree zero on C to J.k/ induced by f P
is simply the map defined by . In particular, it is independent of P , is
surjective, and its kernel consists of the principal divisors.
From its definition (or from the above descriptions of its action on the
0
points) it is clear that if P 0 is a second point on C , then f P is the composite
of f P with the translation map tŒP P 0  , and that if P is defined over a
Galois extension k 0 of k, then f P D f P for all  2 Gal.k 0 =k/.
If C has genus zero, then (2.1) shows that J D 0. From now on we
assume that C has genus g > 0.
P ROPOSITION 2.2. The map .f P / W .J; ˝J1 / ! .C; ˝C1 / is an isomorphism.
P ROOF. As for any group variety, the canonical map hJ W .J; ˝J1 / !
T0 .J /_ is an isomorphism Shafarevich 1994, III, 5.2. Also there is a well
known duality between .C; ˝C1 / and H 1 .C; OC /. We leave it as an

2

THE CANONICAL MAPS FROM C TO ITS JACOBIAN VARIETY7

exercise to the reader (unfortunately rather complicated) to show that the
following diagram commutes:
.J; ˝J1 /

f

.C; ˝C1 /


hJ 

T0 .J /_



H 1 .C; OC /_ :

The bottom isomorphism is the dual of the isomorphism in 2.1.

2

P ROPOSITION 2.3. The map f P is a closed immersion (that is, its image
f P .C / is closed and f P is an isomorphism from C onto f P .C //; in
particular, f P .C / is nonsingular.
We first need a lemma.
L EMMA 2.4. Let f W V ! W be a map of varieties over an algebraically
closed field k, and assume that V is complete. If the map V .k/ ! W .k/ defined by f is injective and, for all closed points Q of V , the map on tangent
spaces TQ .V / ! Tf Q .W / is injective, then f is a closed immersion.
P ROOF. The proof is the same as that of the “if” part of Hartshorne 1977,
II, 7.3. (Briefly, the image of f is closed because V is complete, and the
condition on the tangent spaces together with Nakayama’s lemma shows
that the maps Of Q ! OQ on the local rings are surjective.)
2
It suffices to prove (2.3) in the case that k is algebraically closed. We
apply the lemma to f D f P . If f .Q/ D f .Q0 / for some Q and Q0 in C.k/,
then the divisors Q P and Q0 P are linearly equivalent. This implies
that Q Q0 is linearly equivalent to zero, which is impossible if Q ¤ Q0
because C has genus > 0. Consequently, f is injective, and it remains
to show that the maps on tangent spaces .df P /Q W TQ .C / ! Tf Q .J / are
injective. Because f Q differs from f P by a translation, it suffices to do this
in the case that Q D P . The dual of .df P /P W TP .C / ! T0 .J / is clearly
f

.J; ˝ 1 / !

hC

.C; ˝ 1 / ! TP .C /_ ;

where hC is the canonical map, and it remains to show that hC is surjective.
The kernel of hC is f! 2 .C; ˝ 1 / j !.P / D 0g D .C; ˝ 1 . P //, which
is dual to H 1 .C; L.P // . The Riemann-Roch theorem shows that this last
group has dimension g 1, and so Ker.hC / ¤ .C; ˝ 1 /: h is surjective,
and the proof is complete.

We now assume that k D C and sketch the relation between the abstract and classical definitions of the Jacobian. In this case, .C.C/; ˝ 1 /
(where ˝ 1 denotes the sheaf of holomorphic differentials in the sense of
complex analysis) is a complex vector space of dimension g, and
R one shows
in the theory of abelian integrals that the map  7! .! 7!  !/ embeds

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THE SYMMETRIC POWERS OF A CURVE

8

H1 .C.C/; Z/ as a lattice into the dual space .C.C/; ˝ 1 /_ . Therefore
def
J an D .C.C/; ˝ 1 /_ =H1 .C.C /; Z/ is a complex torus, and the pairing
H1 .C.C/; Z/  H1 .C.Z/; Z/ ! Z
defined by Poincaré duality gives a nondegenerate Riemann form on J an .
Therefore J an is an abelian variety over C. For each P there is a canonical
P
an
by .! 7!
Rmap g W C ! J sending a point Q to the element represented
1
_
!/, where is any path from P to Q. Define eW .C.C/; ˝ / ! J .C/
to be the surjection in the diagram:
.C.C/; ˝ 1 /_

J.C/

f _ 

exp


.J; ˝ 1 /_

T0 .J /:

Note that if .C.C/; ˝ 1 /_ is identified with TP .C /, then .de/0RD .df P /P .
It follows that if is a path from P to Q and ` D .! 7!
!/, then
P
e.`/ D f .Q/.
T HEOREM 2.5. The canonical surjection eW .C.C/; ˝ 1 /_  J.C / induces an isomorphism J an ! J carrying g P into f P .
P ROOF. We have to show that the kernel of e is H1 .C.C/;Z/, but this
follows from Abel’s theorem and the Jacobi inversion theorem.
(Abel): Let P1 ; :::; Pr and Q1 ; :::; Qr be elements of C.C/; then there
is a meromorphic function on C.C/ with its poles at the Pi and its zeros at
the Qi if and only if for any paths i from P to Pi and i0 from P to Qi
there exists a in H1 .C.C/; Z/ such that
Z
XZ
XZ
!
! D ! all !.
i

i

i

0
i

(Jacobi) Let ` be a linear mapping .C.C/; ˝ 1 / ! C. Then there exist
g pointsP
P1R; :::; Pg on C.C/ and paths 1 ; :::; g from P to Pi such that
`.!/ D i i ! for all ! 2 .C.C/; ˝ 1 /.
Let ` 2 .C.C/; ˝ 1 /_ ; we may assume that it is defined by gPpoints
P1 ; :::; Pg . Then ` maps to zero in J.C/ if and only if the divisor Pi
gP is linearly equivalent to zero, and Abel’s theorem shows that this is
equivalent to ` lying in H1 .C.C/; Z/.
2

3

The symmetric powers of a curve

Both in order to understand the structure of the Jacobian, and as an aid in
its construction, we shall need to study the symmetric powers of C .
For any variety V , the symmetric group Sr on r letters acts on the
product of r copies V r of V by permuting the factors, and we want to define
the rth symmetric power V .r/ of V to be the quotient Sr nV r . The next
proposition demonstrates the existence of V .r/ and lists its main properties.

3

THE SYMMETRIC POWERS OF A CURVE

9

A morphism 'W V r ! T is said to be symmetric if ' ı  D ' for all 
in Sr .
P ROPOSITION 3.1. Let V be a variety over k. Then there is a variety V .r/
and a symmetric morphism W V r ! V .r/ having the following properties:
(a) as a topological space, .V .r/ ; / is the quotient of V r by Sr I
(b) for any open affine subset U of V , U .r/ is an open affine subset of
V .r/ and .U .r/ ; OV .r/ / D .U r ; OV r /Sr (set of elements fixed by
the action of Sr /.
The pair .V .r/ ; / has the following universal property: every symmetric
k-morphism 'W V r ! T factors uniquely through .
The map  is finite, surjective, and separable.
P ROOF. If V is affine, say V D Spec A, define V .r/
S to be Spec..A ˝k
::: ˝k A/Sr /. In the general case, write V as a union i Ui of open affines,
.r/
and construct V by patching together the Ui . See Mumford 1970, II, 7,
p, 66, and III, 11, p. 112, for the details.
2
The pair .V .r/ ; / is uniquely determined up to a unique isomorphism
by the conditions of by the proposition. It is called the rth symmetric power
of V .
P ROPOSITION 3.2. The rth symmetric power C .r/ of a nonsingular curve
is nonsingular for all r.
P ROOF. We may assume that k is algebraically closed. The most likely
candidate for a singular point on C .r/ is the image Q of a fixed point
.P; :::; P / of Sr on C r , where P is a closed point of C. The completion
yP of the local ring at P is isomorphic to kŒŒX , and so
O
y.P;:::;P /  kŒŒX  y̋ ::: y̋ kŒŒX   kŒŒX1 ; :::; Xr .
O
yQ  kŒŒX1 ; :::; Xr Sr where Sr acts by permuting the
It follows that O
variables. The fundamental theorem on symmetric functions says that, over
any ring, a symmetric polynomial can be expressed as a polynomial in the
elementary symmetric functions 1 ; :::; r . This implies that
kŒŒX1 ; :::; Xr Sr D kŒŒ1 ; :::; r ,
which is regular, and so Q is nonsingular.
For a general point Q D .P; P; :::; P 0 ; :::/ with P occurring r 0 times,
0
P occurring r 00 times, and so on. Therefore,
yQ  kŒŒX1 ; :::; Xr 0 Sr 0 y̋ kŒŒX1 ; :::; Xr 00 Sr 00 y̋    ,
O
which the same argument shows to be regular.

2

R EMARK 3.3. The reader may find it surprising that the fixed points of
the action of Sr on C r do not force singularities on C .r/ . The following
remarks may help clarify the situation. Let G be a finite group acting

3

THE SYMMETRIC POWERS OF A CURVE

10

effectively on a nonsingular variety V , and supppose that the quotient
variety W D GnV exists. Then V ! W is ramified exactly at the fixed
points of the action. A purity theorem (SGA 1, X, 3.1)1 says that W can be
nonsingular only if the ramification locus is empty or has pure codimension
1 in V . As the ramification locus of V r over V .r/ has pure codimension
dim.V /, this implies that V .r/ can be nonsingular only if V is a curve.
Let K be field containing k. If K is algebraically closed, then (3.1a)
shows that C .r/ .K/ D Sr nC.K/r , and so a point of C .r/ with coordinates
in K is an unordered r-tuple of K-rational points. This is the same thing as
an effective divisor of degree r on CK . When K is perfect, the divisors on
x
CK can be identified with those on CKx fixed under the action of Gal.K=K/.
Since the same is true of the points on C .r/ , we see again that C .r/ .K/
can be identified with the set of effective divisors of degree r on C. In the
remainder of this section we shall show that C .r/ .T / has a similar interpretation for any k-scheme. (Since this is mainly needed for the construction of
J , the reader more interested in the properties of J can pass to the Section
5.)
Let X be a scheme over k. Recall Hartshorne 1977, II, 6, p. 145, that a
Cartier divisor D is effective if it can be represented by a family .Ui ; gi /i
with the gi in .Ui ; OX /. Let I.D/ be the subsheaf of OX such that
I.D/jUi is generated by gi . Then I.D/ D L. D/, and there is an exact
sequence
0 ! I.D/ ! OX ! OD ! 0
where OD is the structure sheaf of the closed subscheme of T associated
with D. The closed subschemes arising from effective Cartier divisors
are precisely those whose sheaf of ideals can be locally generated by a
single element that is not a zero-divisor. We shall often identify D with its
associated closed subscheme.
For example, let T D A1 D Spec kŒY , and let D be the Cartier divisor
associated with the Weil divisor nP , where P is the origin. Then D is represented by .Y n ; A1 /, and the associated subscheme is Spec.kŒY =.Y n //.
D EFINITION 3.4. Let W X ! T be a morphism of k-schemes. A relative
effective Cartier divisor on X=T is a Cartier divisor on X that is flat over
T when regarded as a subscheme of X.
Loosely speaking, the flatness condition means that the divisor has no
vertical components, that is, no components contained in a fibre. When T
is affine, say T D Spec.R/, then a subscheme D of X is a relative effective
S
Cartier divisor if and only if there exists an open affine covering X D i Ui
and gi 2 .Ui ; OX / D Ri such that
(a) D \ Ui D Spec.Ri =gi Ri /,
(b) gi is not a zero-divisor, and
(c) Ri =gi Ri is flat over R, for all i .
Henceforth all divisors will be Cartier divisors.
1 Grothendieck, A, Revetements étale et groupe fondamental (SGA 1, 1960-61), Lecture

Notes in Mathematics 224, Springer, Heidelberg (1971).

3

THE SYMMETRIC POWERS OF A CURVE

11

L EMMA 3.5. If D1 and D2 are relative effective divisors on X=T , then so
also is their sum D1 C D2 .
P ROOF. It suffices to prove this in the case that T is affine, say T D
Spec.R/. We have to check that if conditions (b) and (c) above hold for
gi and gi0 , then they also hold for gi gi0 . Condition (b) is obvious, and the
flatness of Ri =gi gi0 Ri over R follows from the exact sequence
gi0

0 ! Ri =gi Ri ! Ri =gi gi0 Ri ! Ri =gi0 Ri ! 0;
which exhibits it as an extension of flat modules.

2

R EMARK 3.6. Let D be a relative effective divisor on X=T . On tensoring
the inclusion I.D/ ,! OX with L.D/, we obtain an inclusion OX ,! L.D/
and hence a canonical global section sD of L.D/. For example, in the case
that T is affine and D is represented as in the above example, L.D/jUi is
gi 1 Ri and sD jUi is the identity element in Ri .
The map D 7! .L.D/; sD / defines a one-to-one correspondence between relative effective divisors on X=T and isomorphism classes of pairs
.L; s/ where L is an invertible sheaf on X and s 2 .X; L/ is such that
s

0 ! OX ! L ! L=sOX ! 0
is exact and L=sOX is flat over T .
Observe that, in the case that X is flat over T , L=sOX is flat over T if
and only if, for all t in T , s does not become a zero divisor in L ˝ OXt . (Use
that an R-module M is flat if TorR
1 .M; N / D 0 for all finitely generated
modules N , and that any such module N has a composition series whose
quotients are the quotient of R by a prime ideal; therefore the criterion has
only to be checked with N equal to such a module.)
P ROPOSITION 3.7. Consider the Cartesian square
X

X0

T

T 0:

If D is a relative effective divisor on X=T , then its pull-back to a closed
subscheme D 0 of X 0 is a relative effective divisor on X 0 =T 0 .
P ROOF. We may assume both T and T 0 are affine, say T D Spec R and
T 0 D Spec R0 , and then have to check that the conditions (a), (b), and (c)
above are stable under the base change R ! R0 . Write Ui0 D Ui T T 0 ;
clearly D 0 \ Ui0 D Spec.Ri0 =gi Ri0 /. The conditions (b) and (c) state that
gi

0 ! Ri ! Ri ! Ri =gi Ri ! 0
is exact and that Ri =gi Ri is flat over R. Both assertions continue to hold
after the sequence has been tensored with R0 .
2

3

THE SYMMETRIC POWERS OF A CURVE

12

P ROPOSITION 3.8. Let D be a closed subscheme of X, and assume that
def
D and X are both flat over T . If Dt D D T ftg is an effective divisor on
Xt =t for all points t of T , then D is a relative effective divisor on X .
P ROOF. From the exact sequence
0 ! I.D/ ! OX ! OD ! 0
and the flatness of X and D over T , we see that I.D/ is flat over T . The
flatness of OD implies that, for all t 2 T , the sequence
0 ! I.D/ ˝OT k.t / ! OXt ! ODt ! 0
is exact. In particular, I.D/ ˝ k.t / ' I.Dt /. As Dt is a Cartier divisor,
I.Dt / (and therefore also I.D/ ˝ k.t // is an invertible OXt -module. We
now apply the fibre-by-fibre criterion of flatness: if X is flat over T and
F is a coherent OX -module that is flat over T and such that Ft is a flat
OXt -module for all t in T , then F is flat over X (Bourbaki AC, III, 5.4).
This implies that I.D/ is a flat OX -module, and since it is also coherent,
it is locally free over OX . Now the isomorphism I.D/ ˝ k.t / ! I.Dt /
shows that it is of rank one. It is therefore locally generated by a single
element, and the element is not a zero-divisor; this shows that D is a relative
effective divisor.
2
Let W C ! T be a proper smooth morphism with fibres of dimension
one. If D is a relative effective divisor on C=T , then Dt is an effective
divisor on Ct , and if T is connected, then the degree of Dt is constant; it is
called the degree of D. Note that deg.D/ D r if and only if OD is a locally
free OT -module of degree r.
C OROLLARY 3.9. A closed subscheme D of C is a relative effective divisor
on C=T if and only if it is finite and flat over T ; in particular, if sW T ! C is
a section to , then s.T / is a relative effective divisor of degree 1 on C=T .
P ROOF. A closed subscheme of a curve over a field is an effective divisor
if and only if it is finite. Therefore (3.8) shows that a closed subscheme D
of C is a relative effective divisor on C=T if and only if it is flat over T and
has finite fibres, but such a subscheme D is proper over T and therefore
has finite fibres if and only if it is finite over T (see Milne 1980, I, 1.10, or
Hartshorne 1977, III, Ex.11.3).
2
When D and D 0 are relative effective divisors on C=T , we write D  D 0
if D  D 0 as subschemes of C (i.e., if I.D/  I.D 0 //.
P ROPOSITION 3.10. If Dt  Dt0 (as divisors on Ct / for all t in T , then
D  D0.
P ROOF. Represent D as a pair .s; L/ (see 3.6). Then D  D 0 if and only
if s becomes zero in L ˝ OD 0 D LjD 0 . But L ˝ OD 0 is a locally free OT module of finite rank, and so the support of s is closed subscheme of T .
The hypothesis implies that this subscheme is the whole of T .
2

3

THE SYMMETRIC POWERS OF A CURVE

13

Let D be a relative effective S
divisor of degree r on C=T . We shall
say that D is split if Supp.D/
D si .T / for some sections si to . For
P
example, a divisor D D i ni Pi on a curve over a field is split if and only
if k.Pi / D k for all i .
P ROPOSITION 3.11. Every split relative
P effective divisor D on C=T can
be written uniquely in the form D D i ni si .T / for some sections si .
S
P ROOF. Let Supp.D/ D i si .T /, and suppose that
Pthe component of D
with support on si .T / has degree
P ni . Then Dt D . i ni si .T //t for all t ,
and so (3.10) shows that D D i ni si .T /.
2
E XAMPLE 3.12. Consider a complete nonsingular curve C over a field
k. For each i there is a canonical section si to qW C  C r ! C r , namely,
.P1 ; :::; Pr / 7! .Pi ; P1 ; :::; Pr /. Let Di be si .C rP
/ regarded as a relative
effective divisor on C  C r =C r , and let D D i Di . Then D is the
unique
relative effective divisor C  C r =T whose fibre over .P1 ; :::; Pr / is
P
i Pi . Clearly D is stable under the action of the symmetric group Sr , and
Dcan D Sr nD (quotient as a subscheme of C  C r / is a relative effective
divisor on C  C .r/ =C .r/ whose fibre over D 2 C .r/ .k/ is D.
For C a complete smooth curve over k and T a k-scheme, define
DivrC .T / to be the set of relative effective Cartier divisors on C  T =T of
degree r. Proposition 3.7 shows that DivrC is a functor on the category of
k-schemes.
T HEOREM 3.13. For any relative effective divisor D on C  T =T of
degree r, there is a unique morphism 'W T ! C .r/ such that D D .1 
'/ 1 .Dcan /. Therefore C .r/ represents the functor DivrC .
P
P ROOF. Assume first that D is split, so that D D i ni si .T / for some
sections si W T ! C  T . In this case, we define T ! C r to be the map
.p ı s1 ; :::; p ı s1 ; p ı s2 ; :::/, where each si occurs ni times, and we take '
to be the composite T ! C r ! C .r/ . In general, we can choose a finite flat
covering W T 0 ! T such that the inverse image D 0 of D on C  T 0 is split,
and let ' 0 W T 0 ! C .r/ be the map defined by D 0 . Then the two maps ' 0 ı p
and ' 0 ı q from T 0 T T 0 to T 0 are equal because they both correspond to
the same relative effective divisor
p 1 .D 0 / D . ı p/ 1 .D/ D . ı q/ 1 .D/ D q 1 .D/
on T 0 T T 0 . Now descent theory (e.g, Milne 1980, I, 2.17) shows that ' 0
factors through T .
2
E XERCISE 3.14. Let E be an effective Cartier divisor of degree r on C ,
r
and define a subfunctor DivE
C of DivC by
r
DivE
C .T / D fD 2 DivC .T / j Dt  E all t 2 T g:

Show that DivE
C is representable by P.V / where V is the vector space
.C; L.E// (use Hartshorne 1977, II, 7.12]) and that the inclusion DivE
C ,!
DivrC defines a closed immersion P.V / ,! C .r/ .
R EMARK 3.15. Theorem 3.13 says that C .r/ is the Hilbert scheme HilbP
C =k
where P is the constant polynomial r.

4

THE CONSTRUCTION OF THE JACOBIAN VARIETY

4

14

The construction of the Jacobian variety

2 In this section, C will be a complete nonsingular curve of genus g > 0,

and P will be a k-rational point on C. Recall (1.9), that in constructing J ,
we are allowed to make a finite separable extension of k.
For a k-scheme T , let
PCr .T / D fL 2 Pic.C  T / j deg.Lt / D r all tg= ;
where L  L0 means that L  L0 ˝ q  M for some invertible sheaf M on T.
Let Lr D L.rP /; then L 7! L˝p  Lr is an isomorphism PC0 .T / ! PCr .T /,
and so, to prove (1.1), it suffices to show that PCr is representable for some
r. We shall do this for a fixed r > 2g.
Note that there is a natural transformation of functors f W DivrC ! PCr
sending a relative effective divisor D on C  T =T to the class of L.D/ (or,
in other terms, .s; L/ to the class of L/.
L EMMA 4.1. Suppose there exists a section s to f W DivrC ! PCr . Then
PCr is representable by a closed subscheme of C .r/ .
P ROOF. The composite ' D s ı f is a natural transformation of functors
DivrC ! DivrC and DivrC is representable by C .r/ , and so ' is represented
by a morphism of varieties. Define J 0 to be the fibre product,
C .r/

J0

.1;'/

C .r/  C .r/



C .r/ :

Then
J 0 .T / D f.a; b/ 2 C .r/ .T /  C .r/ .T / j a D b, a D 'bg
D fa 2 C .r/ .T / j a D '.a/g
D fa 2 C .r/ .T / j a D sc, some c 2 PCr .T /g
' PCr .T /,
because s is injective. This shows that PCr is represented by J 0 , which is a
closed subscheme of C .r/ because  is a closed immersion.
2
The problem is therefore to define a section s or, in other words, to find
a natural way of associating with a family of invertible sheaves L of degree
r a relative effective divisor. For L an invertible sheaf of degree r on C ,
the dimension h0 .L/ of H 0 .C; L/ is r C 1 g, and so there is an .r g/dimensional system of effective divisors D such that L.D/  L. One way
to cut down the size of this system is to fix a family D .P1 ; :::; Pr g / of
k-rational points on C andP
consider only divisors D in the system such
that D  D , where D D i Pi . As we shall see, this provides a partial
solution to the problem.
2 The method of construction of the Jacobian variety in this section was suggested to me

at the conference by János Kollár.

4

THE CONSTRUCTION OF THE JACOBIAN VARIETY

15

P ROPOSITION 4.2.
P Let be an .r g/-tuple of k-rational points on C ,
and let L D L. P 2 P /.
(a) There is an open subvariety C of C .r/ such that, for all k-schemes
T;
C .T / D fD 2 DivrC .T / j h0 .Dt

D / D 1, all t 2 T g:

If k is separably closed, then C .r/ is the union of the subvarieties
C .
(b) For all k-schemes T , define
P .T / D fL 2 PCr .T / j h0 .Lt ˝ L 1 / D 1, all t 2 T g:
Then P is a subfunctor of PCr and the obvious natural transformation
f W C ! P has a section.
P ROOF. (a) Note that for any effective divisor D of degree r on C , h0 .D
D /  1, and that equality holds for at least one D (for example, D D
D C Q1 C    C Qg for a suitable choice of points Q1 ; :::; Qg ; see the
elementary result (5.2b) below). Let Dcan be the canonical relative effective
divisor of degree r on C C .r/ =C .r/ . Then (AVs, 4.2c) applied to L.Dcan
p 1 D / shows that there is an open subscheme C of C .r/ such that
h0 ..Dcan /t D / D 1 for t in C and h0 ..Dcan /t D / > 1 otherwise.
Let T be a k-scheme, and let D be a relative effective divisor of degree r
on C  T =T such that h0 .Dt D / D 1. Then (3.13) shows that there is
a unique morphism 'W T ! C .r/ such that .1  '/ 1 .Dcan / D D, and it is
clear that ' maps T into C . This proves the first assertion.
S
Assume that k is S
separably closed. To show that C D C , it suffices
to show that C.k/ D C .k/, or that for every divisor D of degree r on C ,
there exists a such that h0 .D D / D 1. Choose a basis e0 ; :::; er g for
H 0 .C; L.D//, and consider the corresponding embedding W C ,! Pr g .
Then
in any hyperplane (if it were contained in
P .C / is not contained
P
a
X
D
0,
then
a
e
would
be zero on C /, and so there exist r g
i i i
i i i
points P1 ; :::; Pr g on C disjoint from D whose images are not contained
in linear subspace of codimension 2 (choose P1 ; P2 ; :.. inductively so that
P1 ; :::; Pi is not contained in a linear subspace of dimension i 2/. The
.r g/-tuple D .P1 ; :::; Pr g / satisfies the condition because
P
P
P
H 0 .C; L.D
j Pj // D f i ai ei j
i ai ei .Pj / D 0, j D 1; :::; r gg;
which has dimension < 2.
(b) Let L be an invertible sheaf on C  T representing an element of
P .T /. Then h0 .Dt D / D 1 for all t, and the Reimann-Roch theorem shows that h1 .Dt D / D 0 for all t. Now (AVs, 4.2e) shows that
def
M D q .L ˝ p  L 1 / is an invertible sheaf on T and that its formation
commutes with base change. This proves that PC is a subfunctor of PCr .
On tensoring the canonical map q  M ! L ˝ p  L 1 with q  M 1 , we
obtain a canonical map OC T ! L ˝ p  L 1 ˝ q  M 1 . The natural map
L ! OC induces a map p  L 1 ! OC T , and on combining this with

5

THE CANONICAL MAPS FROM THE SYMMETRIC POWERS OF C TO IT

the preceding map, we obtain a canonical map s W OC T ! L ˝ q  M 1 .
The pair .s ; L ˝ q  M 1 / is a relative effective divisor on C  T =T whose
image under f in P .T / is represented by L ˝ q  M 1  L (see 3.6).
We have defined a section to C .T / ! P .T /, and our construction is
obviously functorial.
2
C OROLLARY 4.3. The functor P is representable by a closed subvariety
J of C .
P ROOF. The proof is the same as that of (4.1).

2
0

Now consider two .g r/-tuples and 0 , and define P ; to be the
0
0
functor such that P ; .T / D P .T / \ P .T / for all k-schemes T. It easy
0
0
to see that P ; is representable by a variety J ; such that the maps
0
0
0
0
J ; ,! J and J ; ,! J defined by the inclusions P ; ,! P and
0
0
P ; ,! P are open immersions.
We are now ready to construct the Jacobian
of C . Choose tuples
S
.r/
i
D C . After extending k, we
1 ; :::; m of points in C.ks / such that C
can assume that the i are tuples of k-rational points. Define J by patching
together the varieties J i using the open immersions J i ; j ,! J i ; J j .
It is easy to see that J represents the functor PCr , and therefore also the
functor PC0 . Since the latter is a group functor, J is a group variety. The
natural transformations DivrC ! PCr ! PC0 induce a morphism C .r/ ! J ,
which shows that J is complete and is therefore an abelian variety. The
proof of (1.1) is complete.
2

5 The canonical maps from the symmetric
powers of C to its Jacobian variety
Throughout this section C will be a complete nonsingular curve of genus
g > 0. Assume there is a k-rational point P on C , and write f for the map
f P defined in 2.
Let f r be the map C r ! J sending .P1 ; :::; Pr / to f .P1 /C  Cf .Pr /.
On points, f r is the map .P1 ; :::; Pr / 7! ŒP1 C    C Pr rP . Clearly it
is symmetric, and so induces a map f .r/ W C .r/ ! J . We can regard f .r/
as being the map sending an effective divisor D of degree r on C to the
linear equivalence class of D rP . The fibre of the map f .r/ W C .r/ .k/ !
J.k/ containing D can be identified with the space of effective divisors
linearly equivalent to D, that is, with the linear system jDj. The image
of C .r/ in J is a closed subvariety W r of J , which can also be written
W r D f .C / C    C f .C / (r summands).
T HEOREM 5.1. (a) For all r  g, the morphism f .r/ W C .r/ ! W r is birational; in particular, f .g/ is a birational map from C .g/ onto J .
(b) Let D be an effective divisor of degree r on C , and let F be the
fibre of f .r/ containing D. Then no tangent vector to C .r/ at D maps to

5

THE CANONICAL MAPS FROM THE SYMMETRIC POWERS OF C TO IT

zero under .df .r/ /D unless it lies in the direction of F ; in other words, the
sequence
0 ! TD .F / ! TD .C .r/ / ! Ta .J /;

a D f .r/ .D/;

is exact. In particular, .df .r/ /D W TD .C .r/ / ! Ta .J / is injective if jDj has
dimension zero.
For D a divisor on C , we write h0 .D/ for the dimension of
H 0 .C; L.D// D ff 2 k.C / j .f / C D  0g
and h1 .D/ for the dimension of H 1 .C; L.D//. Recall that
h0 .D/

h1 .D/ D deg.D/ C 1

g;

and that H 1 .C; L.D//_ D H 0 .C; ˝ 1 . D//, which can be identified with
1
the set of ! 2 ˝k.C
whose divisor .!/  D.
/=k
L EMMA 5.2. (a) Let D be a divisor on C such that h1 .D/ > 0; then there
is a nonempty open subset U of C such that h1 .D C Q/ D h1 .D/ 1 for
all closed points Q in U , and h1 .D C Q/ D h1 .D/ for Q … U . P
(b) For every r  g, there is an open subset U of C r such that h0 . Pi / D
1 for all .P1 ; :::; Pr / in U .
P ROOF. (a) If Q is not in the support of D, then H 1 .C; L.D C Q//_ '
.C; ˝ 1 . D Q// can be identified with the subspace of .C; ˝ 1 . D//
of differentials with a zero at Q. Clearly therefore we can take U to be
the complement of the zero set of a basis of H 1 .C; L.D// together with a
subset of the support of D.
(b) Let D0 be the divisor zero on C . Then h1 .D0 / D g, and on applying
(a) P
repeatedly, we find that there is an open subset U of C r such that
1
h . Pi / D g r for
P all .P1 ; :::; Pr / in U . The Riemann-Roch theorem
now shows that h0 . Pi / D r C .1 g/ C .g r/ D 1 for all .P1 ; :::; Pr /
in U .
2
In proving (5.1), we can assume that k is algebraically closed. If U 0
is the image in C .r/ of the set U in (5.2b), then f .r/ W C .r/ .k/ ! J.k/ is
injective on U 0 .k/, and so f .r/ W C .r/ ! W r must either be birational or
else purely inseparable of degree > 1. The second possibility is excluded
by part (b) of the theorem, but before we can prove that we need another
proposition.
P ROPOSITION 5.3. (a) For all r  1, there are canonical isomorphisms
.C; ˝ 1 / !

.C r ; ˝ 1 /Sr !

.C .r/ ; ˝ 1 /:

Let ! 2 .C; ˝ 1 / correspond to ! 0 2 .C .r/ ; ˝ 1 /; then for any effective
divisor D of degree r on C , .!/  D if and only if ! 0 has a zero at D.
(b) For all r  1, the map f .r/ W .J; ˝ 1 / ! .C .r/ ; ˝ 1 / is an isomorphism.

5

THE CANONICAL MAPS FROM THE SYMMETRIC POWERS OF C TO IT

A global 1-form on a product of projective varieties
L is a sum1of global
r
1
1-forms on the factors. Therefore .C ; ˝ / D i pi .C; ˝ /, where
the pi are theP
projection maps onto the factors, and so it is clear that
the map ! 7! i pi ! identifies .C; ˝ 1 / with .C r ; ˝ 1 /Sr . Because
W C r ! C .r/ is separable,   W .C .r/ ; ˝ 1 / ! .C r ; ˝ 1 / is injective,
and its image is obviously fixed by the action of Sr . The composite map
.J; ˝ 1 / !

.C .r/ ; ˝ 1 / ,!

.C r ; ˝ 1 /Sr '

.C; ˝ 1 /
P
sends !
to the element ! 0 of .C; ˝ 1 / such that f r ! D i pi ! 0 . As
P
f r D i f ı pi , clearly ! 0 D f  !, and so the composite map is f  ,
which we know to be an isomorphism (2.2). This proves that both maps
in the above sequence are isomorphisms. It also completes the proof of
the proposition except for the second part of (a), and for this we need a
combinatorial lemma.
L EMMA 5.4. Let 1 ; :::; r be the elementary symmetric polynomials in
j
X1 ; :::; Xr , and let j D Xi dXi . Then
m 0

m 1 1 C    C . 1/m m D dmC1 ;

all m  r

1:

P ROOF. Let m .i/ be the mth elementary symmetric polynomial in the
variables
X1 ; :::; Xi 1 ; Xi C1 ; :::; Xr :
Then
m n D m n .i / C Xi m n 1 .i /;
and on multiplying this by . 1/n Xin and summing over n (so that the
successive terms cancel out) we obtain the identity
m

m 1 Xi C ::. C . 1/m Xim D m .i /:

On multiplying this with dXi and summing, we get the required identity.2
yQ D
We now complete the proof of (5.3). First let D D rQ. Then O
y
kŒŒX  and OD D kŒŒ1 ; :::; r  (see the proof of (3.2); by OD we mean the
local ring at the point D on C .r/ /. If ! D .a0 Ca1 X Ca2 X 2 C   /dX , ai 2
k, when regarded as an element of ˝ 1y
, then ! 0 D a0 0 Ca1 1 C   . We
OQ =k

know that fd1 ; :::; dr g is a basis for ˝ 1y

OQ =k

yD -module, but the
as an O

lemma shows that 0 ; :::; r 1 is also a basis. Now .!/  D and ! 0 .D/ D 0
are both obviously equivalent to a0 D a1 D :.. D ar 1 D 0. The proof for
other divisors is similar.
2
We finally prove the exactness of the sequence in (5.1). The injectivity
of .d i/D follows from the fact that i W F ,! C .r/ is a closed immersion.
Moreover the sequence is a complex because f ı i is the constant map
x 7! a. It remains to show that
dim Im.d i /D D dim Ker.df .r/ /D :

6

THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY19

Identify Ta .J /_ with .C; ˝ 1 / using the isomorphisms arising from (2.1).
Then (5.3) shows that ! is zero on the image of TD .C .r/ / if and only if
.!/  D, that is, ! 2 .C; ˝ 1 . D//. Therefore the image of .df .r/ /D has
dimension g h0 .˝ 1 . D// D g h1 .D/, and so its kernel has dimension
r g C h1 .D/. On the other hand, the image of .d i /D has dimension
jDj. The Riemann-Roch theorem says precisely that these two numbers are
equal, and so completes the proof.
2
C OROLLARY 5.5. For all r  g, f r W C r ! W r is of degree rŠ.
P ROOF. It is the composite of W C r ! C .r/ and f .r/ .

2

R EMARK 5.6. (a) The theorem shows that J is the unique abelian variety
birationally equivalent to C .g/ . This observation is the basis of Weil’s
construction of the Jacobian (see Section 7).
(b) The exact sequence in (5.1b) can be regarded as a geometric statement of the Riemann-Roch theorem (see especially the end of the proof). In
fact it is possible to prove the Riemann-Roch theorem this way (see Mattuck
and Mayer 1963).
(c) As we observed above, the fibre of f .r/ W C .r/ .k/ ! J.k/ containing
D can be identified with the linear system jDj. More precisely, the fibre
of the map of functors C .r/ ! J is the functor DivD
C of (3.14); therefore
the scheme-theoretic fibre of f .r/ containing D is a copy of projective
space of dimension h0 .D/ 1. Corollary 3.9 of AVs shows that conversely
every copy of projective space in C .r/ is contained in some fibre of f .r/ .
Consequently, the closed points of the Jacobian can be identified with the
set of maximal subvarieties of C .r/ isomorphic to projective space.
Note that for r > 2g 2, jDj has dimension r g, and so .df .r/ /D is
surjective, for all D. Therefore f .r/ is smooth (see Hartshorne 1977, III,
10.4), and the fibres of f .r/ are precisely the copies of Pr g contained in
C .r/ . This last observation is the starting point of Chow’s construction of
the Jacobian Chow 1954.

6 The Jacobian variety as Albanese variety;
autoduality
Throughout this section C will again be a complete nonsingular curve of
genus g > 0 over a field k, and J will be its Jacobian variety.
P ROPOSITION 6.1. Let P be a k-rational point on C . The map f P W C !
J has the following universal property: for any map 'W C ! A from C
into an abelian variety sending P to 0 , there is a unique homomorphism
W J ! A such that ' D ı f P .
P
P ROOF. Consider the map C g ! A, .P1 ; :::; Pg / 7! i .Pi /. Clearly
this is symmetric, and so it factors through C .g/ . It therefore defines a
rational map W J ! A, which (AVs, 3.1) shows to be a morphism. It is
clear from the construction that ı f P D ' (note that f P is the composite

6

THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY20

of Q 7! Q C .g 1/P W C ! C .g/ with f .g/ W C .g/ ! J /. In particular,
maps 0 to 0, and (AVs, 2.2) shows that it is therefore a homomorphism. If
0 is a second homomorphism such that 0 ı f P D ', then
and 0 agree
P
P
on f .C / C    C f .C / (g copies), which is the whole of J .
2
C OROLLARY 6.2. Let N be a divisorial correspondence between .C; P /
and J such that .1  f P / N  LP ; then N  MP (notations as in 2 and
(1.2)).
P ROOF. Because of (AVs, 6.2) we can assume k to be algebraically closed.
According to (1.2) there is a unique map 'W J ! J such that N  .1 
'/ MP . On points ' is the map sending a 2 J.k/ to the unique b such
that
MP jC  fbg  N jC  fag:
By assumption,
N jC  ff P Qg  LP jC  fQg  MP jC  ff P Qg,
and so .' ı f P /.Q/ D f P .Q/ for all Q. Now (6.1) shows that f is the
identity map.
2
C OROLLARY 6.3. Let C1 and C2 be curves over k with k-rational points
P1 and P2 , and let J1 and J2 be their Jacobians. There is a one-to-one
correspondence between Homk .J1 ; J2 / and the set of isomorphism classes
of divisorial correspondences between .C1 ; P1 / and .C2 ; P2 /.
P ROOF. A divisorial correspondence between .C2 ; P2 / and .C1 ; P1 / gives
rise to a morphism .C1 ; P1 / ! J2 (by 1.2), and this morphism gives
rise to homomorphism J1 ! J2 (by 6.1). Conversely, a homomorphism
W J1 ! J2 defines a divisorial correspondence .1  .f P1 ı // MP2 between .C2 ; P2 / and .C1 ; P1 /.
2
In the case that C has a point P rational over k, define F W C  C ! J
to be the map .P1 ; P2 / 7! f P .P1 / f P .P2 /. One checks immediately
that this is independent of the choice of P . Thus, if P 2 C.k 0 / for some
Galois extension k 0 of k, and F W Ck 0  Ck 0 ! Jk 0 is the corresponding map,
then F D F for all  2 Gal.k 0 =k/ ; therefore F is defined over k whether
or not C has a k-rational point. Note that it is zero on the diagonal  of
C C.
P ROPOSITION 6.4. Let A be an abelian variety over k. For any map 'W C 
C ! A such that './ D 0, there is a unique homomorphism W J ! A
such that ı F D '.
P ROOF. Let k 0 be a finite Galois extension of k, and suppose that there
exists a unique homomorphism W Ck 0 ! Jk 0 such that ı Fk 0 D 'k 0 . Then
the uniqueness implies that  D for all  in Gal.k 0 =k/, and so is
defined over k. It suffices therefore to prove the proposition after extending
k, and so we can assume that C has a k-rational point P . Now (AVs,

6

THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY21

2.5) shows that there exist unique maps '1 and '2 from C to A such that
'1 .P / D 0 D '2 .P / and '.a; b/ D '1 .a/ C '2 .b/ for all .a; b/ 2 C  C .
Because ' is zero on the diagonal, '1 D '2 . From (6.1) we know that
there exists a unique homomorphism from J to A such that '1 D ı f ,
and clearly is also the unique homomorphism such that ' D ı F . 2
R EMARK 6.5. The proposition says that .A; F / is the Albanese variety of
C in the sense of Lang 1959, II, 3, p. 45. Clearly the pairs .J; f P / and
.J; F / are characterized by the universal properties in (6.1) and (6.4).
Assume again that C has a k-rational point P , and let  D W g 1 . It
is a divisor on J , and if P is replaced by a second k-rational point,  is
replaced by a translate. For any effective divisor D on J , write
L0 .D/ D m L.D/ ˝ p  L.D/ 1 ˝ q  L.D/ 1
D L.m 1 .D/

D J

J  D/.

Recall (AVs, 9.1 and 10) that D is ample if and only if 'L.D/ W J ! J _ is an
isogeny, and then .1  'L.D/ / .P/ D L0 .D/, where P is the Poincaré sheaf
on J  J _ . Write  for the image of  under the map . 1/J W J ! J ,
and a for ta  D  C a, a 2 J.k/. Abbreviate . /a by a .
T HEOREM 6.6. The map 'L./ W J ! J _ is an isomorphism; therefore,
1  'L./ is an isomorphism
.J  J; L0 .// ! .J  J _ ; P/:
As usual, we can assume k to be algebraically closed. Recall (AVs,
12.13) that 'L. / D . 1/2 'L./ D 'L./ , and that 'L.a / D 'L./ for
all a 2 J.k/.
L EMMA 6.7. Let U be the largest open subset of J such that
(i) the fibre of f .g/ W C .g/ ! J at any point of U has dimension zero,
and
(ii) if a 2 U.k/ and D.a/ is the unique element of C .r/ .k/ mapping to a,
then D.a/ is a sum of g distinct points of C.k/.
Then f 1 .a / D D.a/ (as a Cartier divisor) for all a 2 U.k/, where
f D f P WC ! J .
P ROOF. Note first that U can be obtained by removing the subset over
which the fibres have dimension > 0, which is closed (see Shafarevich 1994,
I, 6, Theorem 7), together with the images of certain closed subsets of the
form   C g 2 . These last sets are also closed because C g ! J is proper
(Shafarevich 1994, II, 4.8), and it follows that U is a dense open subset of
J.
P
Let a 2 U.k/, and let D.a/ D i Pi , Pi ¤ Pj for i ¤ j . A P
point Q1
of C maps to a point of a ifP
and only if there exists a divisor giD2 Qi
P
P
on
i f .Qi / C a. The equality implies
PgC such that f .Q1 / D
P that
Q

D,
and
the
fact
that
jDj
has
dimension
0
implies
that
Qi D
i D1 i

6

THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY22

D. It follows that the support of f 1 .a / is fP1 ; :::; Pg g, and it remains
to show that f 1 .a / has degree  g for all a.
Consider the map W C   ! J sending .Q; b/ to f .Q/ C b. As the
composite of with 1  f g 1 W C  C g 1 ! C   is f g W C g ! J , and
these maps have degrees .g 1/Š and gŠ respectively (5.5), has degree g.
Also is projective because C   is a projective variety (see Hartshorne
1977, II, Ex.4.9). Consider a 2 U ; the fibre of over a is f 1 .a / (more
accurately, it is the subscheme of C associated with the Cartier divisor
1 .U / is quasi-finite and
f 1 .a /). Therefore the restriction of to
projective, and so it is finite (see Hartshorne 1977, III, Ex.11.2). As U is
normal, this means that all the fibres of over points of U are finite schemes
of rank  g (cf. Shafarevich 1994, II.5, Theorem 6). This completes the
proof of the lemma.
2
L EMMA 6.8. (a) Let a 2 J.k/, and let f .g/ .D/ D a; then f  L.a / 
L.D/.
(b) The sheaves .f  . 1/J / L0 . / and MP on C  J are isomorphic.
P ROOF. Note that (6.7) shows that the isomorphism in (a) holds for all a in
a dense open subset of J . Note also that composite of the the maps
C

Q7!.Q;a/

! C  fag

f . 1/

m

! J J ! J

is t a ı f , and so
.f  . 1// m L. /jC  fag ' L.t a1  /jf .C /
' L.a /jf .C /
' f  L.a /:
Similarly
.f  . 1// p  L. /jC  fag  f  L. /
and
.f  . 1// q  L. /jC  fag
is trivial.
On the other hand, MP is an invertible sheaf on C  J such that
(i) MP jC  fag  L.D gP / if D is an effective divisor of degree g
on C such that f .g/ .D/ D a;
(ii) MP jfP g  J is trivial.
Therefore (a) is equivalent to .f  . 1// m L. /jC  fag being
isomorphic to MP ˝ p  L.gP /jC  fag for all a. As we know this is true
for all a in a dense subset of J , (AVs, 5.3) applied to
MP ˝ p  L.gP / ˝ .f  . 1// m L. / 1
proves (a). In particular, on taking a D 0, we find that f  L. /  L.gP /,
and so .f  . 1// p  L. /  p  L.gP /. Now (AVs, 5.1) shows that
.f  . 1// .m L. / ˝ p  L. / 1 /  MP ˝ q  N

6

THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY23

for some invertible sheaf N of J . On computing the restrictions of the
sheaves to fP g  J , we find that N  . 1/ L. /, which completes the
proof.
2
Consider the invertible sheaf .f  1/ P on C  J _ . Clearly it is a divisorial correspondence, and so there is a unique homomorphism f _ W J _ ! J
such that .1f _ / MP  .f 1/ P. The next lemma completes the proof
of the Theorem 6.6.
L EMMA 6.9. The maps f _ W J _ ! J and 'L./ W J ! J _ are inverse.
P ROOF. Write

D 'L./ D 'L.

/ . We have

.1  / .1  f _ / MP  .1  / .f  1/ P
 .f  / P
 .f  . 1// .1  'L./ / P
 .f  . 1// L0 . /
 MP :
Therefore, f _ ı is a map ˛W J ! J such that .1  ˛/ MP  MP ; but
the only map with this property is the identity.
2
R EMARK 6.10. (a) Lemma 6.7 shows that f .C / and  cross transversely
at any point of U . This can be proved more directly by using the descriptions
of the tangent spaces implicitly given near the end of the proof of (5.1).
(b) In (6.8) we showed that MP  .f  . 1// L0 . /. This implies
that
MP  .f  . 1// .1  'L.


// P


 .f  . 1// .1  'L./ / P
 .f  . 1// L0 ./:

Also, because D 7! 'L.D/ is a homomorphism, 'L. / D 'L./ , and so
MP  .f  . 1// .1  'L./ / P
 .f  1/ .1  'L. / / P
 .f  1/ L0 . /:
(c) The map on points J _ .k/ ! J.k/ defined by f _ is induced by

f  W Pic.J / ! Pic.C /.

(d) Lemma 6.7 can be generalized as follows. An effective canonical
divisor K defines a point on C .2g 2/ whose image in J will be denoted
g 2 / , and write a D
.
P Let a be a point of J such that a  is not in r.W
on C . Then W and .W g r /a intersect
i f .Pi / with P1 ; :::; Pg pointsP
r
g
r
properly, and W  .W
/a D .wi1 :::ir / where
wi1 :::ir D f .Pi1 / C    C f .Pir /
and the sum runs over the . gr / combinations obtained by taking r elements
from f1; 2; :::; gg. See Weil 1948, 39, Proposition 17.

7

WEIL’S CONSTRUCTION OF THE JACOBIAN VARIETY

24

S UMMARY 6.11. Between .C; P / and itself, there is a divisorial correspondence LP D L. fP g  C C  fP g/:
Between .C; P / and J there is the divisorial correspondence MP ; for
any divisorial correspondence L between .C; P / and a pointed k-scheme
.T; t /, there is a unique morphism of pointed k-schemes 'W T ! J such
that .1  '/ MP  LP . In particular, there is a unique map f P W C ! J
such that .1  f P / MP  LP and f .P / D 0.
Between J and J _ there is a canonical divisorial correspondence P
(the Poincaré sheaf); for any divisorial correspondence L between J and a
pointed k-scheme .T; t / there is a unique morphism of pointed k-schemes
W T ! J such that .1  / P  L.
Between J and J there is the divisorial correspondence L0 ./. The
unique morphism J ! J _ such that .1  / P  L0 ./ is 'L./ , which
is an isomorphism. Thus 'L./ is a principal polarization of J , called the
canonical polarization. There are the following formulas:
MP  .f  . 1// L0 ./  .f  1/ L0 ./ 1 :
Consequently,
LP  .f  f / L0 ./ 1 :
If f _ W J _ ! J is the morphism such that .f  1/ P  .1  f _ / MP ,
1
then f _ D 'L./
.
E XERCISE 6.12. It follows from (6.6) and the Riemann-Roch theorem
(AVs, 13.3) that . g / D gŠ. Prove this directly by studying the inverse
image of  (and its translates) by the map C g ! J . (Cf. AVs, 8.3, but note
that the map is not finite.) Hence deduce another proof of (6.6).

7 Weil’s construction of the Jacobian variety
As we saw in (5.6a), the Jacobian J of a curve C is the unique abelian
variety birationally equivalent to C .g/ . To construct J , Weil used the
Riemann-Roch theorem to define a rational law of composition on C .g/ and
then proved a general theorem that allowed him to construct an algebraic
group out of C .g/ and the rational law. Finally, he verified that the algebraic
group so obtained had the requisite properties to be called the Jacobian of
C . We give a sketch of this approach.
A birational group over k (or a nonsingular variety with a normal law
of composition in the terminology of Weil 1948, V) is a nonsingular variety
V together with a rational map mW V  V
V such that
(a) m is associative (that is, .ab/c D a.bc/ whenever both terms are
defined);
(b) the rational maps .a; b/ 7! .a; ab/ and .a; b/ 7! .b; ab/ from V  V
to V  V are both birational.
Assume that C has a k-rational point P .

7

WEIL’S CONSTRUCTION OF THE JACOBIAN VARIETY

25

L EMMA 7.1. (a) There exists an open subvariety U of C .g/  C .g/ such
that for all fields K containing k and all .D; D 0 / in U.K/,
h0 .D C D 0

gP / D 1:

(b) There exists an open subset V of C .g/  C .g/ such that for all fields
K containing k and all .D; D 0 / in V .K/,
h0 .D 0

D C gP / D 1:

P ROOF. (a) Let Dcan be the canonical relative effective divisor on C 
C .2g/ =C .2g/ constructed in 3. According to the Riemann-Roch theorem,
h0 .D gP /  1 for all divisors of degree 2g on C , and so (AVs, 4.2c) shows
that the subset U of C .2g/ of points t such that h0 ..Dcan /t gP / D 1 is
open. On the other hand, (5.2b) shows that there exist positive divisors D
of degree g such that h0 ..D C gP / gP / D 1, and so U is nonempty. Its
inverse image in C .g/  C .g/ is the required set.
(b) The proof is similar to that of (a): the Riemann-Roch theorem shows
that h0 .D 0 D C gP /  1 for all D and D 0 , we know that there exists a
D 0 such that h0 .D 0 gP C gP / D h0 .D 0 / D 1, and (AVs, 4.2) applied to
the appropriate invertible sheaf on C  C .g/  C .g/ gives the result.
2
P ROPOSITION 7.2. There exists a unique rational map
mW C .g/  C .g/

C .g/

whose domain of definition contains the subset U of (7.1a) and which is
such that for all fields K containing k and all .D; D 0 / in U.K/, m.D; D 0 / 
D C D 0 gP ; moreover m makes C .g/ into a birational group.
P ROOF. Let T be an integral k-scheme. If we identify C .g/ with the functor
it represents (see 3.13), then an element of U.T / is a pair of relative effective
divisors .D; D 0 / on C  T =T such that, for all t 2 T , h0 .Dt C Dt0 gP / D
1. Let L D L.D C D 0 g  P  T /. Then (AVs, 4.2d) shows that q .L/ is
an invertible sheaf on T . The canonical map q  q L ! L when tensored
with .q  q L/ 1 gives a canonical global section sW OT ! L ˝ .q  q L/ 1 ,
which determines a relative effective divisor m.D; D 0 / of degree g on
C  T =T (see 3.6). The construction is clearly functorial. Therefore we
have constructed a map mW U ! C .g/ as functors of integral schemes over
k, and this is represented by a map of varieties. On making the map
explicit in the case that K is the spectrum of a field, one sees easily that
m.D; D 0 /  D C D 0 gP in this case.
The uniqueness of the map is obvious. Also associativity is obvious
since it holds on an open subset of U.K/: m..D; D 0 /; D 00 / D m.D; .D 0 ; D 00 //
because each is an effective divisor on C linearly equivalent to D C D 0 C
D 00 2gP , and in general h0 .D C D 0 C D 00 2gP / D 1.
A similar argument using (7.1b) shows that there is a map rW V ! C .g/
such that .p; r/ is a birational inverse to
.a; b/ 7! .a; ab/W C .g/  C .g/

C .g/  C .g/ .

Because the law of composition is commutative, this shows that .a; b/ 7!
.b; ab/ is also birational. The proof is complete.
2

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WEIL’S CONSTRUCTION OF THE JACOBIAN VARIETY

26

T HEOREM 7.3. For any birational group V over k, there is a group variety
G over k and a birational map f W V
G such that f .ab/ D f .a/f .b/
whenever ab is defined; moreover, G is unique up to a unique isomorphism.
P ROOF. In the case that V .k/ is dense in V (for example, k is separably
closed), this is proved in Artin 1986, 2.3 (Briefly, one replaces V by
an open subset where m has better properties, and obtains G by patching
together copies of translates of U by elements of V .k/:/ From this it follows
that, in the general case, the theorem holds over a finite Galois extension k 0
of k. Let  2 Gal.k 0 =k/. Then f W Vk 0
 G is a birational map, and as
Vk 0 D Vk 0 , the uniqueness of G shows that there is a unique isomorphism
' W  G ! G such that ' ı f D f . For any ;  2 Gal.k 0 =k/;
.' ı ' / ı . f / D ' ı  .' ı f / D f D '  ı  f;
and so ' ı ' D '  . Descent theory (see 1.8) now shows that G is
defined over k.
2
Let J be the algebraic group associated by (7.3) with the rational group
defined in (7.2).
P ROPOSITION 7.4. The variety J is complete.
P ROOF. This can be proved using the valuative criterion of properness. (For
Weil’s original account, see Weil 1948, Théoreme 16, et seq.)
2
J is a morphism. If D
C OROLLARY 7.5. The rational map f W C .g/
and D 0 are linearly equivalent divisors on CK for some field K containing
k, then f .D/ D f .D 0 /.
P ROOF. The first statement follows from (AVs, 3.1). For the second, recall
that if D and D 0 are linearly equivalent, then they lie in a copy of projective
space contained in C .g/ (see 3.14). Consequently (AVs, 3.9) shows that
they map to the same point in J .
2
We now prove that J has the correct universal property.
T HEOREM 7.6. There is a canonical isomorphism of functors W PC0 ! J .
P ROOF. As in 4, it suffices to show that PCr is representable by J for
some r. In this case we take r D g. Let L be an invertible sheaf with
fibres of degree g on C  T . If dimk .Ct ; Lt / D 1 for some t, then this
holds for all points in an open neighbourhood Ut of t. As in the proof of
(7.2), we get a relative effective divisor sW OS ! L ˝ .q  q L/ 1 of degree
g on Ut . This family of Cartier divisors defines a map Ut ! C .g/ which,
when composed with f , gives a map L W Ut ! J . On the other hand,
if dimk .Ct ; Lt / > 1, then we choose an invertible sheaf L0 of degree
3 See also: Edixhoven, Bas; Romagny, Matthieu. Group schemes out of birational group

laws, Néron models. Autour des schémas en groupes. Vol. III, 15–38, Panor. Synthèses, 47,
Soc. Math. France, Paris, 2015.

8

GENERALIZATIONS

27

zero on C such that dim. .Ct ; Lt ˝ L0 // D 1, and define L W Ut ! C .g/
on a neighbourhood of t to be the composite of L˝p L0 with t a , where
a D f .D/ for D an effective divisor of degree g such that L.D gP /  L0 .
One checks that this map depends only on L, and that the maps for different
t agree on the overlaps of the neighbourhoods. They therefore define a map
T ! J.
2
R EMARK 7.7. Weil of course did not show that the Jacobian variety represented a functor on k-schemes. Rather, in the days before schemes, the
Jacobian variety was characterized by the universal property in (6.1) or
(6.4), and shown to have the property that Pic0 .C / ' J.k/. See Weil 1948
or Lang 1959.

8

Generalizations

It is possible to construct Jacobians for families of curves. Let W C ! S be
projective flat morphism whose fibres are integral curves. For any S-scheme
T of finite-type, define
PCr .T / D fL 2 Pic.C S T / j deg.Lt / D r all tg= 
where L  L0 if and only if L  L0 ˝ q  M for some invertible sheaf M on
T. (The degree of an invertible sheaf on a singular curve is defined as in the
nonsingular case: it is the leading coefficient of .C; Ln / as a polynomial
in n.) Note that PCr is a functor on the category of S -schemes of finite-type.
T HEOREM 8.1. Let W C ! S be as above; then there is a group scheme
J over S with connected fibres and a morphism of functors PC0 ! J such
that PC0 .T / ! J .T / is always injective and is an isomorphism whenever
C S T ! T has a section.
In the case that S is the spectrum of a field (but C may be singular),
the existence of J can be proved by Weil’s method (see Serre 1959, V).
When C is smooth over S , one can show as in 3 that C .r/ (quotient of
C S ::: S C by Sr / represents the functor DivrC=S sending an S-scheme
T to the set of relative effective Cartier divisors of degree r on C S T =T .
In general one can only show more abstractly that DivrC=S is represented
r
by a Hilbert scheme. There is a canonical map DivrC=S ! PC=S
and the
r
second part of the proof deduces the representability of PC=S from that of
DivrC=S . (The only reference for the proof in the general case seems to be
Grothendieck’s original rather succinct account in his Bourbaki Seminar,
#232;4 we sketch some of its ideas below.)
As in the case that the base scheme is the spectrum of a field, the
conditions of the theorem determine J uniquely; it is called the Jacobian
4 See also: Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure,
Nitin; Vistoli, Angelo. Fundamental algebraic geometry. Grothendieck’s FGA explained.
Mathematical Surveys and Monographs, 123. American Mathematical Society, Providence,
RI, 2005.

8

GENERALIZATIONS

28

scheme of C=S. Clearly J commutes with base change: the Jacobian of
C S T over T is J S T . In particular, if Ct is a smooth curve over k.t /,
then Jt is the Jacobian of Ct in the sense of 1. Therefore if C is smooth
over S , then J is an abelian scheme, and we may think of it as a family of
Jacobian varieties. If C is not smooth over S , then J need not be proper,
even in the case that S is the spectrum of a field.
E XAMPLE 8.2. Let C be complete smooth curve over an algebraically
closed
Pfield k. By a modulus for C one simply means an effective divisor
m D P nP P on C . Let m be such a modulus, and assume that deg.m/  2.
We shall associate with C and m a new curve Cm having a single singularity
at a point to be denoted by Q. The underlying topological space of Cm is
.C X S/ [ fQg, where S is the support of m. Let OQ D k C cQ , where
cQ D ff 2 k.C / j ord.f /  nP all P in Sg;
T
and define OCm to be the sheaf such that .U; OCm / D OP , where the
intersection is over the P in U . The Jacobian scheme Jm of Cm is an
algebraic group over k called the generalized Jacobian of C relative to
m. By definition, Jm .K/ is the group of isomorphism classes of invertible
sheaves on Cm of degree 0. It can also be described as the group of divisors
of degree 0 on C relatively prime to m, modulo the principal divisors defined
by elements congruent to 1 modulo m (an element of k.C / is congruent to
1 modulo if ordP .f 1/  nP for all P in S ). For each modulus m with
support on S there is a canonical map fm W C X S ! Jm , and these maps are
universal in the following sense: for any morphism f W C X S ! G from
C X S into an algebraic group, there is a modulus m and a homomorphism
'W Jm ! G such that f is the composite of fm ı ' with a translation. (For a
detailed account of this theory, see Serre 1959).
We now give a brief sketch of part of Grothendieck’s proof of (8.1).
First we need the notion of the Grassmann scheme.
Let E be a locally free sheaf of OS -modules of finite rank, and, for an
S -scheme T of finite-type, define GrassEn .T / to be the set of isomorphism
classes of pairs .V; h/, where V is a locally free OT -module of rank n
and h is an epimorphism OT ˝k E  V. For example, if E D OSm , then
GrassEn .T / can be identified with the set of isomorphism classes of pairs
.V; .e1 ; :::; em // where V is a locally free sheaf of rank n on T and the ei are
sections of V over T that generate V; two such pairs .V; .e1 ; :::; em // and
0 // are isomorphic if there is an isomorphism V 
.V 0 ; .e10 ; :::; em
! V 0 carryN C1
ing each ei to ei0 . In particular, GrassO
.T / D PN
1
S .T / (cf. Hartshorne
1977, II, 7.1).
P ROPOSITION 8.3. The functor T 7! GrassEn .T / is representable by a projective variety GnE over S.
P ROOF. The construction of GnE is scarcely more difficult than that of PN
S
(see EGA I,5 9.7).
2
5 Grothendieck, A., and Dieudonné, J, Eléments de géométrie algébrique I, Springer,

Heidelberg, (1971).

8

GENERALIZATIONS

29

Choose an r > 2g 2 and an m > 2g 2 C r. As in the case that S is
the spectrum of a field, we first need to construct the Jacobian under the
assumption that there is a section sW S ! C. Let E be the relative effective
divisor on C=S defined by s (see 3.9), and for any invertible sheaf L on
C S T , write L.m/ for L˝L.mE/. The first step is to define an embedding
of DivrC=S into a suitable Grassmann scheme.
Let D 2 DivrC=S .T /, and consider the exact sequence
0 ! L. D/ ! OCT ! OD ! 0
on C S T (we often drop the S from C S T /. This gives rise to an exact
sequence
0 ! L. D/.m/ ! OCT .m/ ! OD .m/ ! 0;
and on applying q we get an exact sequence
0 ! q L. D/.m/ ! q OC T .m/ ! q OD .m/ ! R1 q L. D/.m/ ! ::::
Note that, for all t in T , H 1 .Ct ; L. D/.m// is dual to H 0 .Ct ; L.K C D
mEt //, where Et is the divisor s.t / of degree one on Ct . Because of our
assumptions, this last group is zero, and so (see AVs, 4.2e) R1 q L. D/.m/
is zero, and we have an exact sequence
0 ! q L. D/.m/ ! q OCT .m/ ! q OD .m/ ! 0:
Moreover q OD .m/ is locally free of rank r, and
q .OCT .m// D q OC .m/ ˝ OT
q O .m/

(loc. cit.), and so we have constructed an element ˚.D/ of Grassr  C .T /.
On the other hand, suppose a D .q OCT .m/  V/ is an element
q O .m/
of Grassn C .T /. If K is the kernel of q  q OCT .m/  q  V, then
K. m/ is a subsheaf of q  q OCT , and its image under q  q OCT !
OCT is an ideal in OCT . Let .a/ be the subscheme defined by this ideal.
It is clear from the constructions that ˚.D/ D D for any relative divisor
of degree r. We have a diagram of natural transformations
˚

DivrC .T / ! Grassqr  OC .m/ .T / ! S.T /  DivrC .T /;

˚ D id;

where S.T / denotes the set of all closed subschemes of C S T . In particular, we see that ˚ is injective.
P ROPOSITION 8.4. The functor ˚ identifies DivrC with a closed subscheme
q O .m/
of Grassr  C .
P ROOF. See Grothendieck’s Bourbaki Seminar, p. 221-12 (or, under different hypotheses, Mumford 1966, Lecture 15).
2
r
Finally one shows that the fibres of the map DivrC=S ! PC=S
are represented by the projective space bundles associated with certain sheaves of
OS -modules (Grothendieck’s Bourbaki Seminars, p. 232-11; cf. 5.6c) and
r
deduces the representability of PC=S
(loc. cit.).

9

OBTAINING COVERINGS OF A CURVE FROM ITS JACOBIAN30

9 Obtaining coverings of a curve from its
Jacobian; application to Mordell’s conjecture
Let V be a variety over field k, and let W W ! V be a finite étale map. If
there is a finite group G acting freely on W by V -morphisms in such a way
that V D GnW , then .W; / is said to be a Galois covering6 of V with Galois group G. If G is abelian, then .W; / is said to be an abelian covering
of V . Fix a point P on V . Then the Galois coverings of V are classified
by the (étale) fundamental group 1 .V; P / and the abelian coverings by the
maximal abelian quotient 1 .V; P /ab of 1 .V; P /. For any finite abelian
group M , Hom.1 .V; P /; M / (set of continuous homomorphisms) is equal
to the set of isomorphism classes of Galois coverings of V with Galois
group M . If, for example, V is nonsingular and we take P to be the generic
point of V , then every finite connected étale covering of V is isomorphic
to the normalization of V in some finite extension K 0 of k.P / contained
in a fixed algebraic closure Kx of K; moreover, 1 .V; P / D Gal.K un =K/
where K un is the union of all finite extensions K 0 of k.P / in Kx such that
the normalization of V in K 0 is étale over V. The covering corresponding to
a continuous homomorphism ˛W Gal.K un =K/ ! M is the normalization of
V in KxKer.˛/ . (See 3 of my notes Lectures on Étale Cohomology or Milne
1980, I, 5, for a more detailed discussion of étale fundamental groups.)
Now let C be a complete nonsingular curve over a field k, and let
f D f P for some P in C.k/. From a finite étale covering J 0 ! J of J ,
we obtain an étale covering of C by pulling back relative to f :
J0
J

C0
f

C J J 0

C:

Because all finite étale coverings of J are abelian (cf. AVs, 15.3), we only
obtain abelian coverings of C in this way. The next proposition shows that
we obtain all such coverings.
Henceforth, k will be separably closed.
P ROPOSITION 9.1. If J 0 ! J is a connected étale covering of J , then
C 0 D C J J 0 ! C is a connected étale covering of C , and every connected abelian covering of C is obtained in this way. Equivalently, the map
1 .C; P /ab ! 1 .J; 0/ induced by f P is an isomorphism.
P ROOF. The equivalence of the two assertions follows from the interpretation of
Hom.1 .V; P /; M /
6 Some authors call a finite covering W ! V Galois if the field extension k.W /=k.V /
is Galois, i.e., if the covering is generically Galois, but this conflicts with Grothendieck’s
definition and is not the natural definition.

9

OBTAINING COVERINGS OF A CURVE FROM ITS JACOBIAN31

recalled above and the fact that 1 .J; 0/ is abelian. We shall prove the
second assertion. For this it suffices to show that for all integers n, the map
Hom.1 .J; 0/; Z=nZ/ ! Hom.1 .C; P /; Z=nZ/
induced by f P is an isomorphism. The next two lemmas take care of the
case that n is prime to the characteristic of k.
2
L EMMA 9.2. Let V be a complete nonsingular variety and let P be a point
of V ; then for all integers n prime to the characteristic of k,
Hom.1 .V; P /; Z=nZ/ ' Pic.V /n :
P ROOF. Let D be a (Weil) divisor on V such that nD D .g/ for some
g 2 k.V /, and let V 0 be the normalization of V in the Kummer extension
k.V /.g 1=n / of k.V /. A purity theorem (SGA 1, X, 3.1) shows that V 0 ! V
is étale if, for all prime divisors Z on V , the discrete valuation ring OZ
(local ring at the generic point of Z/ is unramified in k.V 0 /. But the
extension k.V 0 /=k.V / was constructed by extracting the nth root of an
element g such that ordZ .g/ D 0 if Z is not in the support of D and is
divisible by n otherwise, and it follows from this that OZ is unramified.
Conversely, let V 0 ! V be a Galois covering with Galois group Z=nZ.
Kummer theory shows that the k.V 0 /=k.V / is obtained by extracting the
nth root of an element g of k.V /. Let Z be a prime divisor on V . Because
OZ is unramified in k.V 0 /, ordZ .g/ must be divisible by n (or is zero), and
so .g/ D nD for some divisor D. Obviously D represents an element of
Pic.V /n . It is easy to see now that the correspondence we have defined
between coverings of V and elements of Pic.V /n is one-to-one. (For a
proof using étale cohomology, see Milne 1980, III, 4.11.)
2
L EMMA 9.3. The map Pic.J / ! Pic.C / defined by f induces an isomorphism Pic0 .J / ! Pic0 .C /.
P ROOF. This was noted in (6.10c).

2

In the case that n D p D char.k/, (9.2) and (9.3) must be replaced by
the following analogues.
L EMMA 9.4. For any complete nonsingular variety V and point P ,
Hom.1 .V; P /; Z=pZ/ ' Ker.1

F W H 1 .V; OV / ! H 1 .V; OV //;

where F is the map induced by a 7! ap W OV ! OV .
P ROOF. See (Milne 1980, p. 127) for a proof using étale cohomology as
well as for hints for an elementary proof.
2
L EMMA 9.5. The map f P W C ! J induces an isomorphism H 1 .J; OJ / !
H 1 .C; OC /.

9

OBTAINING COVERINGS OF A CURVE FROM ITS JACOBIAN32

P ROOF. See Serre 1959, VII, Théorème 9. (Alternatively, note that the same
argument as in the proof of (2.1) gives an isomorphism H 1 .J; OJ / !
T0 .J _ /, and we know that J ' J _ .)
2
To prove the case n D p m , one only has to replace OC and OJ by the
sheaves of Witt vectors of length m, Wm OC and Wm OJ . (It is also possible
to use a five-lemma argument starting from the case m D 1:)
C OROLLARY 9.6. For all primes `, the map of étale cohomology groups
H 1 .J; Z` / ! H 1 .C; Z` / induced by f is an isomorphism.
P ROOF. For any variety V , H 1 .Vet ; Z=nZ/ D Hom.1 .V; P /; Z=nZ/ (Milne
1980, III, 4). Therefore, there are isomorphisms
H 1 .J; Z=`m Z/ ! Hom.1 .J; P /; Z=`m Z/
! Hom.1 .C; P /; Z=`m Z/
! H 1 .C; Z=`m Z/;
and we obtained the required isomorphism by passing to the limit.

2

To obtain ramified coverings of C , one can use the generalized Jacobians.
P ROPOSITION 9.7. Let C 0 ! C be a finite abelian covering of C that is
unramified outside a finite set ˙. Then there is a modulus m with support on
˙ and an étale isogeny J 0 ! Jm whose pull-back by fm is C 0 X f 1 .˙ /.
P ROOF. See Serre 1959.

2

E XAMPLE 9.8. In the case that the curve is P1 and m D 0 C 1 , we have
Jm D P1 X f0; 1g, which is just the multiplicative group GL1 , and fm is
an isomorphism. For any n prime to the characteristic, there is a unique
unramified covering of P1 X f0; 1g of degree n, namely, multiplication
by n on P1 X f0; 1g. When k D C, this covering is the usual unramified
covering z 7! z n W C X f0g ! C X f0g.
P ROPOSITION 9.9. Let C be a curve of genus g over a number field k,
and let P be a k-rational point of C . Let S be a finite set of primes of k
containing all primes dividing 2 and such that C has good reduction outside
S . Then there exists a field k 0 of degree  22g over k and unramified over
2g
S , and a finite map fP W CP ! Ck 0 of degree  22 .g 1/C2gC1 , ramified
exactly over P , and such that CP has good reduction outside S .
S KETCH OF PROOF. Let C 0 be the pull-back of 2W J ! J ; it is an abelian
étale covering of C of degree 22g , and the Hurwitz genus formula (Hartshorne
1977, IV, 2.4) shows that the genus g 0 of C 0 satisfies
2g 0

2 D 22g .2g

2/;

so that g 0 D 22g .g 1/ C 1. Let D be the inverse image of P on C 0 . It is a
divisor of degree 22g on C 0 , and after an extension k 0 of k of degree  22g

10

ABELIAN VARIETIES ARE QUOTIENTS OF JACOBIAN VARIETIES33

unramified over S, some point P of D will be rational. Let m D D P ,
0
and let C 00 be the pull-back of the covering 2W Jm ! Jm (of degree  22g /
by C ˙ ! Jm , where ˙ D Supp.D/ fP g. Then C 00 is a curve over k 0 ,
and we take CP to be the associated complete nonsingular curve.
2
This result has a very striking consequence. Recall that a conjecture of
Shafarevich states the following:
9.10. For any number field k, integer g, and finite set S of primes of k,
there are only finitely many isomorphism classes of curves C of genus g
over k having good reduction at all primes outside S .
T HEOREM 9.11. Shafarevich’s conjecture (9.10) implies Mordell’s conjecture.
P ROOF. Let C be a curve of genus g  2 over k with good reduction
outside a set S containing all primes of k lying over 2. There is a finite field
extension K of k containing all extensions k 0 of k of degree  22g that are
unramified outside S . For each k-rational point P on C , Proposition 9.9
provides a map fP W CP ! CK of degree  a fixed bound B.g/ which is
ramified exactly over P ; moreover, CP has good reduction outside S. The
Hurwitz genus formula shows that
2g.CP /

2  B.g/.2g

2/ C B.g/

1:

Therefore Shafarevich’s conjecture implies that there can be only finitely
many curves CP . A classical result of de Franchis (Lang 1983, p.223) states
that for each CP , there are only finitely many maps CP ! C (this is where
it is used that g  2/. Therefore there can be only finitely many of k-rational
points on C , as predicted by Mordell.
2

10

Abelian varieties are quotients of Jacobian varieties

The main result in this section sometimes allows questions concerning
abelian varieties to be reduced to the special case of Jacobian varieties.
T HEOREM 10.1. For any abelian variety A over an infinite7 field k, there
is a Jacobian variety J and a surjective homomorpism J  A.
L EMMA 10.2. Let W W ! V be a finite morphism of complete varieties,
and let L be an invertible sheaf on V . If L is ample, then so also is   L.
P ROOF. We shall use the following criterion (Hartshorne 1977, III, 5.3):
an invertible sheaf L on a complete variety is ample if and only if, for all
7 The theorem is true also over finite fields (Gabber, O., On space filling curves and
Albanese varieties. Geom. Funct. Anal. 11 (2001), no. 6, 1192–1200; Poonen, Bjorn,
Bertini theorems over finite fields. Ann. of Math. (2) 160 (2004), no. 3, 1099–1127.)

10

ABELIAN VARIETIES ARE QUOTIENTS OF JACOBIAN VARIETIES34

coherent OV -modules F, H i .V; F ˝ Ln / D 0 for all i > 0 and sufficiently
large n. Also we shall need an elementary projection formula: if N and M
are coherent sheaves of modules on W and V respectively, then
 .N ˝   M/   N ˝ M:
(Locally, this says that if B is an A-algebra and N and M are modules over
B and A respectively, then N ˝B .B ˝A M /  N ˝A M as A-modules.)
Let F be a coherent OW -module. Because  is finite (hence affine), we
have by Hartshorne 1977, II, Ex.4.1, or Ex.8.2 that
H i .W; F ˝   Ln /  H i .V;  .F ˝   Ln //.
The projection formula shows that the second group equals H i .V;  F ˝
Ln /, which is zero for all i > 0 and sufficiently large n because L is ample
and  F is coherent (Hartshorne 1977, 4.1). The criterion now shows that
  L is ample.
2
L EMMA 10.3. Let V be a nonsingular projective variety of dimension  2
over a field k, and let Z be a hyperplane section of V relative to some
fixed embedding V ,! Pn . Then, for any finite map  from a nonsingular
variety W to V ,  1 .Z/ is geometrically connected (that is,  1 .Z/kx is
connected).
P ROOF. The hypotheses are stable under a change of the base field, and so
we can assume that k is algebraically closed. It then suffices to show that
 1 .Z/ is connected. Because Z is an ample divisor on V , the preceding
lemma shows that  1 .Z/ is the support of an ample divisor on W , which
implies that it is connected (Hartshorne 1977, III, 7.9).
2
We now prove the theorem. Since all elliptic curves are their own
Jacobians, we can assume that dim.A/ > 1. Fix an embedding A ,! Pn of
A into projective space. Then Bertini’s theorem Hartshorne 1977, II, 8.18,
shows that there exists an open dense subset U of the dual projective space
n
Pn_
x \ H is nonsingular
x of Pk
x such that, for all hyperplanes H in U , Ak
k
and connected. Because k is infinite, U.k/ is nonempty (consider a line L
in Pn_
x /, and so there exists such an H with coordinates in k. Then A \ H
k
is a (geometrically connected) nonsingular variety in Pn . On repeating the
argument dim.A/ 1 times, we arrive at a nonsingular curve C on A that
is the intersection of A with a linear subspace of Pn . Now (10.3) applied
several times shows that for any nonsingular variety W and finite map
W W ! A,  1 .C / is geometrically connected.
Consider the map J ! A arising from the inclusion of C into A, and
let A1 be the image of the map. It is an abelian subvariety of A, and if it is
not the whole of A, then there is an abelian subvariety A2 of A such that
A1  A2 ! A is an isogeny (see AVs, 12.1); in particular, A1 \ A2 is finite.
As C  A1 , this implies that C \ A2 is finite. Let W D A1  A2 and take
 to be the composite of the maps
A1  A2

1nA2

! A1  A2 ! A;

11

THE ZETA FUNCTION OF A CURVE

35

where n > 1 is an integer prime to the characteristic of k. Then  1 .C / is
not geometrically connected because q. 1 C / D nA12 .A2 \ C /. This is a
contradiction, and so A1 must equal A.
R EMARK 10.4. (a) Lemma 10.2 has the following useful restatement: let
V be a variety over a field k and let D be divisor on V such that the linear
system jDj is without base points; if the map V ! Pn defined by jDj is
finite, then D is ample.
(b) If some of the major theorems from étale cohomology are assumed,
then it is possible to give a very short proof of the theorem. They show
that, for any curve C on A constructed as in the above proof, the map
H 1 .A; Z` / ! H 1 .C; Z` / induced by the inclusion of C into A is injective
(see Milne 1980, VI, 5.6). But H 1 .A; Z` / is dual to T` A and H 1 .C; Z` / is
dual to T` J , and so this says that the map T` J ! T` A induced by J ! A
is surjective. Clearly this implies that J maps onto A.
O PEN Q UESTION 10.5. Let A be an abelian variety over an algebraically
closed field k. We have shown that there exists a surjection J  A with J
a Jacobian variety. Let A1 be the subvariety of J with support the identity
component of the kernel of this map. Then A1 is an abelian variety (AVs,
12), and so there is a surjection J1  A1 . Continuing in this way, we
obtain a sequence of abelian varieties A; A1 ; A2 ; :.. and a complex
::: ! J2 ! J1 ! A ! 0:
Is it possible to make the constructions in such a way that the sequence
terminates with 0? That is, does there exist a finite resolution (up to isogeny)
of an arbitrary abelian variety by Jacobian varieties?
A SIDE . In his 1928 thesis, Weil proved that, for the Jacobian J of a curve over a
number field K, the group J.K/ is finitely generated. Let A be an abelian variety
over a number field K. Then A is a quotient of a Jacobian J (10.1), and there is
an abelian subvariety B of J such that the homomorphism J ! A restricts to an
isogeny B ! A (AVs, 12.1). It follows that there exists an isogeny A ! B. As
B.K/ is finitely generated and the kernel of A.K/ ! B.K/ is finite, it follows that
A.K/ is finitely generated. This is the Mordell-Weil theorem. Everything used in
this proof was available in 1952 (see 14).

11

The zeta function of a curve

Let C be a complete nonsingular curve over a finite field k D Fq . The best
way to prove the Riemann hypothesis for C is to use intersection theory on
C  C (see Hartshorne 1977, V, Ex 1.10), but in this section we show how to
derive it from the corresponding result for the Jacobian of C . Recall (AVs,
19) that the characteristic polynomial of the Frobenius endomorphism
J of J acting on T` J is a polynomial P .X / of degree 2g with integral
coefficients whose roots ai have absolute value q 1=2 .
T HEOREM 11.1.
coordinates in k is
P The number N of points on C with
1=2
equal to 1
ai C q. Therefore, jN q 1j  2gq .

11

THE ZETA FUNCTION OF A CURVE

36

The proof will be based on the following analogue of the Lefschetz
trace formula. A map ˛W C ! C induces a unique endomorphism ˛ 0 of
x (cf. (6.1)). We define
J such that f P ˛ D ˛ 0 f P for any point P in C.k/
0
Tr.˛/ D Tr.˛ /.
P ROPOSITION 11.2. For any endomorphism ˛ of C ,
. ˛  / D 1

Tr.˛/ C deg.˛/:

Q
Recall (AVs, 12) that if P˛0 .X / D i .X ai /, then Tr.˛ 0 / D ai , and
that Tr.˛ 0 / D Tr.˛ 0 jT` J /.
We now show that the proposition implies the theorem. Let C W Ckx !
Ckx be the Frobenius endomorphism of C (see AVs, 19). Then . C  / D
N , the degree of C is q, and the map induced by C on J is J . Therefore
the formula in (11.2) immediately gives that in (11.1). Before proving (11.2)
we need a lemma.
L EMMA 11.3. Let A be an abelian variety of dimension g over a field k,
and let H be the class of an ample divisor in NS.A/. For an endomorphism
˛ of A, write DH .˛/ D .˛ C 1/ .H / ˛  .H / H . Then
Tr.˛/ D g

.H g 1  DH .˛//
:
.H g /

P ROOF. 8 The calculation in (AVs, 12.4) shows that
.˛ C n/ .H / D n.n

1/H C n.˛ C 1/ H

.n

1/˛  .H /

(because .2A / H D 4H inNS.A/), and so
.˛ C n/ H D n2 H C nDH .˛/ C ˛  .H /:
Now the required identity can be read off from the equation
P˛ . n/ D deg.˛ C n/ D

...˛ C n/ H /g /
.H g /

(see AVs, 8.3)

because P˛ . n/ D n2g C Tr.˛/n2g 1 C    .

2

We now prove (11.2). Consider the commutative diagram
f f

C C

J J



1˛ 0

J J


f

C

J

where f D f P for some rational point P of C . Consider the sheaf
L0 ./ D L.m 
def

8 See also Kleiman, Dix Exposes, p. 378.

 J

J  /

12

TORELLI’S THEOREM: STATEMENT AND APPLICATIONS 37

on J  J (see 6). Then
..1  ˛ 0 / ı .f  f // L0 ./ D ..f  f / ı .1  ˛// L0 ./
 .1  ˛/ .f  f / L0 ./
 .1  ˛/ .LP / 1
by a formula in (6.11). Now
 .1  ˛/ LP D L. ˛  .

P C

C  P //;

which has degree . ˛  / 1 deg.˛/. We next compute the sheaf by
going round the diagram the other way. As .1  ˛/ ı  D .1; ˛/, we have
..1  ˛/ ı / L.m /  .1 C ˛/ L./;
and
deg f  L..1 C ˛/ .// D deg f  .1 C ˛/ :
Similarly
deg f  ..1  ˛/ ı / L.  J / D deg f  
and
deg f  ..1  ˛/ ı / L.J  / D L.C  ˛  /;
and so we find that
1

. ˛  / C deg.˛/ D deg f  .D .˛//:

We know from (6.12) that . g / D gŠ, and it is possible to show that
f  .D .˛// D .f .C /  D .˛// is equal to .g 1/Š. g 1  D .˛// (see
Lang 1959, IV, 3). Therefore (11.3) completes the proof.
C OROLLARY 11.4. The zeta function Z.C; t / of C is equal to
.1

P .t /
:
t /.1 qt /

R EMARK 11.5. As we saw in (9.6),
H 1 .Cet ; Z` / ' H 1 .Jet ; Z` / ' .T` J /_ ;
and so (11.2) can be rewritten as
. ˛  / D

12

X2
i D0

. 1/i Tr.˛jH i .Cet ; Z` //:

Torelli’s theorem: statement and applications

Torelli’s theorem says that a curve C is uniquely determined by its canonically polarized Jacobian .J; /.

12

TORELLI’S THEOREM: STATEMENT AND APPLICATIONS 38

T HEOREM 12.1. Let C and C 0 be complete smooth curves over an algebraically closed field k, and let f W C ! J and f 0 W C 0 ! J 0 be the maps of
C and C 0 into their Jacobians defined by points P and P 0 on C and C 0 .
Let ˇW .J; / ! .J 0 ; 0 / be an isomorphism from the canonically polarized
Jacobian of C to that of C 0 .
(a) There exists an isomorphism ˛W C ! C 0 such that f 0 ı ˛ D ˙ˇ ı
f C c, for some c in J 0 .k/.
(b) Assume that C has genus  2. If C is not hyperelliptic, then the map
˛, the sign ˙; and c are uniquely determined by ˇ; P; P 0 . If C is
hyperelliptic, the sign can be chosen arbitrarily, and then ˛ and c are
uniquely determined.
P ROOF. (a) The proof involves complicated combinatorial arguments in
the W r — we defer it to the next section.
(b) Recall Hartshorne 1977, IV, 5, that a curve C is hyperelliptic if there
exists a finite map W C ! P1 of degree 2; the fibres of such a map form
a linear system on C of degree 2 and dimension 1, and this is the unique
such linear system on C . Conversely if C has a linear system of degree 2
and dimension 1, then the linear system defines a finite map W C ! P1 of
degree 2, and so C is hyperelliptic; the fibres of  are the members of the
linear system, and so the nontrivial automorphism  of C such that  ı  D 
preserves these individual members.
Now suppose that there exist ˛; ˛ 0 ; c, and c 0 such that
f 0 ı ˛ D Cˇ ı f C c
f 0 ı ˛ 0 D Cˇ ı f C c 0 :

(1)

Then f 0 .˛.Q// f 0 .˛ 0 .Q// D c c 0 for all Q 2 C.k/, which is a constant.
Since the fibres of the map Div0C .k/ ! J.k/ defined by f 0 are the linear
equivalence classes (see 2), this implies that for all Q and Q0 in C.k/;
˛.Q/

˛ 0 .Q0 /  ˛ 0 .Q/

˛.Q0 /;

i.e.,
˛.Q/ C ˛ 0 .Q0 /  ˛ 0 .Q/ C ˛.Q0 /:
Suppose ˛ ¤ ˛ 0 . Then ˛.Q0 / ¤ ˛ 0 .Q0 / for some Q0 2 C.k/ and, for
a suitable Q00 , ˛.Q0 / ¤ ˛.Q00 /. Therefore j˛.Q0 / C ˛ 0 .Q00 /j is a linear
system of dimension  1 (and degree 2) on C 0 . If C (hence C 0 / is nonhyperelliptic, there is no such system, and we have a contradiction. If C is
hyperelliptic, then there is a unique linear system of dimension 1 and degree
2, but it is obvious that by varying the points Q0 and Q00 we must get more
than one system. Again we have a contradiction. We conclude that ˛ D ˛ 0 ,
and this implies that c D c 0 .
On the other hand, suppose that the equations (1) hold with different
signs, say with a plus and a minus respectively. Then the same argument
shows that
˛.Q/ C ˛ 0 .Q/  ˛.Q0 / C ˛ 0 .Q0 /;

all Q; Q0 in C.k/:

12

TORELLI’S THEOREM: STATEMENT AND APPLICATIONS 39

Therefore f˛.Q/ C ˛ 0 .Q/ j Q 2 C.k/g is a linear system on C 0 of dimension  1, which is impossible if C is nonhyperelliptic. (In the case C is
hyperelliptic, there is an involution  of C 0 such that  ı ˛ D ˛ 0 :/
The case that the equations (1) hold with minus signs can be treated the
same way as the first case.
Finally let C 0 be hyperelliptic with an involution  such that jQ0 C Q0 j
is a linear system and f 0 .Q0 / C f 0 .Q0 / D constant. Then if f 0 ı ˛ D
ˇ ı f C c, we have f 0 ı ˛ D ˇ ı f C c 0 .
2
C OROLLARY 12.2. Let C and C 0 be curves of genus  2 over a perfect
field k. If the canonically polarized Jacobian varieties of C and C 0 are
isomorphic over k, then so also are C and C 0 .
P ROOF. Choose an isomorphism ˇW .J; / ! .J 0 ; 0 / defined over k. For
x and C 0 .k/,
x there is a
each choice of a pair of points P and P 0 in C.k/
0
unique isomorphism ˛W C ! C such that
0

f P ı ˛ D ˙ˇ ı f P C c

(2)

for some c in J 0 .k/ (in the case that C is hyperelliptic, we choose the
sign to be C/. Note that if the pair .P; P 0 / is replaced by .Q; Q0 /, then
0
0
f Q D f P C d and f Q D f P C e for some d 2 J. x
k/ and e 2 J 0 . x
k/, and
so
0

0

f Q ı˛ D f P ı˛Ce
D ˙ˇ ı f P C c C e
D ˙ˇ ı f Q x
Cˇ.d / C c C e.
In particular, we see that ˛ does not depend on the choice of the pair .P; P 0 /.
On applying  2 Gal. x
k=k/ to equation (2), we obtain an equation
0

f P ı  ˛ D ˙ˇ ı f P C c:
0

0

As f P D f P and f P D f P , we see that  ˛ D ˛, and so ˛ is defined
over k.
2
C OROLLARY 12.3. Let k be an algebraic number field, and let S be a finite
set of primes in k. The map C 7! .JC ; / sending a curve to its canonically
polarized Jacobian variety defines an injection from the set of isomorphism
classes of curves of genus  2 with good reduction outside S into the set of
isomorphism classes of principally polarized abelian varieties over k with
good reduction outside S .
P ROOF. Let R be the discrete valuation ring in k corresponding to a prime
of k not in S. Then C extends to a smooth proper curve C over Spec.R/,
and (see 8) the Jacobian J of C has generic fibre the Jacobian of C and
special fibre the Jacobian of the reduction of C. Therefore JC has good
reduction at the prime in question. The corollary is now obvious.
2

13

TORELLI’S THEOREM: THE PROOF

40

C OROLLARY 12.4. Suppose that for every number field k, every finite set
S primes of k, and every integer g, there are only finitely many principally
polarized abelian varieties of dimension g over k having good reduction
outside S. Then Mordell’s conjecture is true.
P ROOF. Combine the last corollary with (9.11).

2

R EMARK 12.5. Corollary (12.2) is false as stated without the condition
that the genus of C is greater than 1. It would say that all curves of genus
zero over k are isomorphic to P1 (but in general there exist conics defined
over k having no rational point in k/, and it would say that all curves of
genus 1 are isomorphic to their Jacobians (and, in particular, have a rational
point). However it is obviously true (without restriction on the genus) that
two curves over k having k-rational points are isomorphic over k if their
canonically polarized Jacobians are isomorphic over k.
A SIDE . The Torelli theorem holds over an arbitrary field k. If k is perfect, then
the uniqueness allows one to descend ˛, as in Corollary 12.2. The case of a perfect
field implies the general case because Hom.C; C 0 / does not acqure additional
elements when passing from k to kŒ,  p D 0. See mathoverflow.net, question
23848.

13

Torelli’s theorem: the proof

Throughout this section, C will be a complete nonsingular curve of genus
g  2 over an algebraically closed field k, and P will be a closed point
of C . The maps f P W C ! J and f .r/ W C .r/ ! J corresponding to P
will all be denoted by f . Therefore f .D C D 0 / D f .D/ C f .D 0 /, and if
f .D/ D f .D 0 /, then
D  D 0 C rP
where r D deg.D/ deg.D 0 /. As usual, the image of C .r/ in J is denoted
by W r . A canonical divisor K on C defines a point on C .2g 2/ whose
image in J will be denoted by . For any subvariety Z of J , Z  will denote
the image of Z under the map x 7!  x.
L EMMA 13.1. For all a in J.k/, .Wag 1 / D W ga 1 .
P ROOF. For any effective divisor D of degree g
h0 .K

D/ D h1 .K

1 on C ,

D/ D h0 .D/  1;

and so there exists an effective divisor D 0 such that K D  D 0 . Then
 f .D/ a D f .D 0 / — a, which shows that .Wag 1 /  W ga 1 . On
replacing a with a, we get that .W ga 1 /  Wag 1 , and so W ga 1 D
.W ga 1 /  .W ga 1 / .
2
L EMMA 13.2. For any r such that 0  r  g

1;

War  Wbg 1 ” a 2 Wbg 1 r :

13

TORELLI’S THEOREM: THE PROOF

41

P ROOF. (W If c D f .D/ C a with D an effective divisor of degree r,
and a D f .D 0 / C b with D 0 an effective divisor of degree g 1 r, then
c D f .D C D 0 / C b with D C D 0 an effective divisor of degree g 1.
)W As a 2 Wbg 1 , there is an effective divisor A of degree g 1 such
that a D f .A/ C b. Let D be effective of degree r. The hypothesis states
x effective of degree g 1, and so
that f .D/ C a D f . x
D/ C b for some D
x
f .D/ C f .A/ D f .D/ and
x C rP:
D CA  D
x 0 of degree g 1 such that A C A0 and
Choose effective divisors A0 and D
x CD
x 0 are linearly equivalent to K (cf. the proof of 13.1). Then
D
D CK

A0  K

x 0 C rP
D

and so
x 0  A0 C rP:
D CD
As the D’s form a family of dimension r, this shows that h0 .A0 C rP / 
r C 1. (In more detail, jA0 C rP j can be regarded as a closed subvariety of
C .rCg 1/ , and we have shown that it projects onto the whole of C .r/ .) It
follows from the Riemann-Roch theorem that h0 .K A0 rP /  1, and so
there is an effective divisor Ax of degree g 1 C r such that
A0 C Ax C rP  K:
x D f .A0 / and a D f . x
Therefore x
A/Cb 2
ACrP  K A0  A, and so f .A/
g 1 r
Wb
.
2
L EMMA 13.3. For any r such that 0  r  g 1;
\
W g 1 r D fW ga 1 j a 2 W r g
\
.W g 1 r / D fWag 1 j a 2 W r g:
P ROOF. Clearly, for a fixed a in J.k/;
W g 1 r  W ga 1 ” Wag 1 r  W g 1 ;
and (13.2) shows that both hold if a 2 W r . Therefore
\
W g 1 r  fW ga 1 j a 2 W r g:
Conversely, c 2 W ga 1 ” a 2 W gc 1 , and so if c 2 W ga 1 for all a 2 W r ,
then W r  W gc 1 and Wcr  W g 1 . According to (13.2), this implies that
c 2 W g 1 r , which completes the proof the first equality. The second
follows from the first and the equation
\
\
fWag 1 j a 2 W r g D f.W ga 1 / j a 2 W r g
\
D . fW ga 1 j a 2 W r g/ :
2

13

TORELLI’S THEOREM: THE PROOF

42

L EMMA 13.4. Let r be such that 0  r  g 2, and let a and b be points of
J.k/ related by an equation a C xP
D b C y with x 2 W 1 and y 2 W g 1 r .
g 1
r
rC1
rC1
If Wa
6 W
, then Wa \ Wbg 1 D WaCx
[ S with S D WarC1 \
P g 2  b
.Wy a / .
P ROOF. Write x D f .X / and y D f .Y / with X and Y effective divisors
of degree 1 and g 1 r. If Y  X, then, because f .X / C a D f .Y / C
b, we will have a D f .Y X / C b with Y X an effective divisor of
degree g 2 r. Therefore a 2 Wbg 2 r , and so WarC1  Wbg 1 (by 13.2).
Consequently, we may assume that X is not a point of Y .
Let c 2 WarC1 \ Wbg 1 . Then c D f .D/ C a D f .D 0 / C b for some
effective divisors D and D 0 of degree r C 1 and g 1. Note that
f .D/ C y D f .D/ C a C x

b D f .D 0 / C x;

and so D C Y  D 0 C X.
If D C Y D D 0 C X , then D  X , and so
c D f .D/ C a D f .D

X / C x C aI

r
in this case c 2 WaCx
.
0
If D C Y ¤ D C X, then h0 .D C Y /  2, and so for any point Q of
x of degree
C.k/, h0 .D C Y Q/  1, and there is an effective divisor Q
x
g 1 such that D C Y  Q C Q. Then

c D f .D/ C a D f . x
Q/ C a
and so c 2

y C f .Q/;

T g 1
fWa yCd j d 2 W 1 g D .W g 2 /a y (by 13.3). As .W g 2 /a y D

.Wyg a2 / and c is in WarC1 by assumption, this completes the proof that
r
WarC1 \ Wbg 1  WaCx
[ S.
The reverse inclusion follows from the obvious inclusions:
r
WaCx
 WarC1 I
r
r
WaCx
D WbCy
 Wbg 1 I

.Wyg a2 /  .Wyg a1 x / D Wbg 1 :

2

L EMMA 13.5. Let a 2 J.k/ be such that W 1 6 Wag 1 ; then there is a
unique effective divisor D.a/ of degree g on C such that
f .D.a// D a C 

(3)

and W 1  Wag 1 , when regarded as a divisor on C , equals D.a/.
P ROOF. We use the notations of 6; in particular,  D W g 1 . For a D 0,
(13.1) says that . / D . Therefore, on applying (6.8), we find that
W 1  Wag 1 D f .C /  . /aC D f
def

1

.. /aC / D D;

where D is a divisor of degree g on C such that f .g/ .D/ D a C . This is
the required result.
2

13

TORELLI’S THEOREM: THE PROOF

43

We are now ready to prove (12.1a). We use ˇ to identify J with J 0 , and
write V r for the images of C 0.r/ in J . As W g 1 and V g 1 define the same
polarization of J , they give the same element of NS.J / (see AVs, 12), and
therefore one is a translate of the other, say W g 1 D Vcg 1 , c 2 J.k/. To
prove (12.1a), we shall show that V 1 is a translate of W 1 or of .W 1 / .
Let r be the smallest integer such that V 1 is contained in a translate of
W rC1 or .W rC1 / . The theorem will be proved if we can show that r D 0.
(Clearly, r < g 1:/ Assume on the contrary that r > 0. We may suppose
(after possibly replacing ˇ by ˇ/ that V 1  WarC1 . Choose an x in W 1
and a y in W g 1 r , and set b D a C x y. Then, unless WarC1  Wbg 1 ,
we have (with the notations of 13.4)
V 1 \ Wbg 1 D V 1 \ WarC1 \ Wbg 1
r
D .V 1 \ WaCx
/ [ .V 1 \ S /:
r
Note that, for a fixed a, WaCx
depends only on x and S depends only on y.
Fix an x; we shall show that for almost all y, V 1 6 Wbg 1 , which
implies that WarC1 6 Wbg 1 for the same y. As y runs over W g 1 r , b
1 r
1 r
runs over W g.aCx/
. Now, if V 1  Wbg 1 for all b in W g.aCx/
, then V 1 
r
WaCx (by 13.3). This contradicts the definition of r, and so there exist b for
g 1
which V 1 6 Wbg 1 . Note that V 1  Wbg 1 .D VbCc
/ ” b 2 Vcg 2
1 r
(by 13.2). Therefore Vcg 2 6 W g.aCx/
, and so the intersection of these

1 r
sets is a lower dimensional subset of W g.aCx/
whose points are the b for

which V 1  Wbg 1 .
We now returnTto the consideration
of the intersection V 1 \ Wbg 1 ,
T
r
1
1
which equals .V
WaCx / [ .V
S / for almost all y. We first show
r
1
that V \ WaCx contains at most one point. If not, then as b runs over
1 r
almost all points of W g.aCx/
(for a fixed x/, the element D 0 .b/ D f 0 1 .V 0 
def

r

Wbg 1 / (cf. 13.5) will contain at least two fixed points (because WaCx
g 1
g 1
0
g
WaCx y D Wb /, and hence f .D .b// will lie in a translate of V 2 .
As f 0 .D 0 .b// D b C  0 , we would then have .W g 1 r / contained in a
translate of V g 2 , say Vdg 2 , and so
\
\
fVcg u1 j u 2 Vdg 2 g  fW gu 1 j u 2 .W g 1 r / g:

On applying (13.3) to each side, we then get an inclusion of V in a translate
of .W r / , contradicting the definition of r.
r
Keeping y fixed and varying x, we see from (3) that V 1 \ WaCx
must
contain at least one point, and hence it contains exactly one point; according
to the preceding argument, the point occurs in D 0 .b/ with multiplicity one
for almost all choices of y.
It is now easily seen that we can find x; x 0 in W 1 and y in W g 1 r
such that
x
.D 0 .b/ D/D 0 .a C x y/ D Q C D
and
.D 0 .b 0 / D/D 0 .a C x 0

x
y/ D Q0 C D

13

TORELLI’S THEOREM: THE PROOF

44

x is an effective divisor of degree g 1 on C 0
where Q; Q0 are in C 0 and D
not containing Q or Q0 . By equation (3), f .Q/ f .Q0 / D x x 0 , and
hence W 1 has two distinct points in common with some translate of V 1 .
g 2 
Now, if x; x 0 are in W 1 , then W gx 1 \ W gx 0 1 D W g 2 [ .WxCx
(by
0/
13.4). According to (13.3), we now get an inclusion of some translate of
V g 2 in W g 2 or .W g 2 / . Finally (13.3) shows that
\
V 1 D fV e j e 2 V g 2 g
which is contained in a translate of W 1 or W 1 according as V g 2 is
contained in a translate of W g 2 or .W g 2 / . This completes the proof.
A SIDE . When I posted this article on my website, I complained that I found the
above proof unilluminating, although short and elementary, and asked for advice
on the many other proofs of the theorem.
Roy Smith responded as follows: “you ask on your website for advice on
conceptual proofs of Torelli. ... here goes. There are many, and the one you give
there is the least conceptual one, due I believe to Martens. Of course you also
wanted short, ....well maybe these are not all so short.
The one due to Weil is based on the fact that certain self intersections of
a jacobian theta divisor are reducible, and is sketched in Mumford’s Michigan
notes.9 Indeed about four proofs are sketched there. The most geometric one, due
to Andreotti–Mayer and Green is to intersect at the origin of the jacobian, those
quadric hypersurfaces occurring as tangent cones to the theta divisor at double
points, thus recovering the canonical model of the curve as their base locus, with
some few exceptions. To show that this works, one can appeal to the deformation
theoretic results of Kempf. That is, since the Italians proved that a canonical curve
is cut out by quadrics most of the time, one needs to know that the ideal of all
quadrics containing the canonical curve is generated by the ones coming as tangent
cones to theta. The ones that do arise that way cut out the directions in moduli
of abelian varieties where theta remains singular in codimension three. But these
equisingular deformations of theta embed into the deformations of the resolution
of theta by the symmetric product of the curve, which Kempf showed are equal
to the deformations of the curve itself. Hence every equisingular deformation of
theta(C) comes from a deformation of C, and these are cut out by the equations in
moduli of abelian varieties defined by quadratic hypersurfaces containing C. Hence
the tangent cones to theta determine C. This version of Green’s result is in a paper
of Smith and Varley10 .
Perhaps the shortest geometric proof is due to Andreotti, who computed
the branch locus of the canonical map on the theta divisor, and showed quite
directly that it equals the dual variety of the canonical curve. This is explained in
Andreotti’s paper from about 1958,11 and quite nicely too, with some small errata,
in the book by Arbarello, Cornalba, Griffiths and Harris.12 I recommend this proof
for conceptualness and completeness in a reasonably short argument.
There are other short proofs that Torelli holds for general curves, simply from
the fact that the quadrics containing the canonical curve occur as the kernel of the
9 Mumford, David, Curves and their Jacobians. The University of Michigan Press, Ann
Arbor, Mich., 1975.
10 Smith, Roy; Varley, Robert. Deformations of theta divisors and the rank 4 quadrics
problem. Compositio Math. 76 (1990), no. 3, 367–398.
11 Andreotti, Aldo. On a theorem of Torelli. Amer. J. Math. 80 1958 801–828.
12 Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J. Geometry of algebraic curves.
Vol. I. 1985, Vol II. 2011, Springer-Verlag.

14

BIBLIOGRAPHIC NOTES

45

dual of the derivative of the Torelli map from moduli of curves to moduli of abelian
varieties. This is described in the article on the Prym Torelli problem by Smith
and Varley.13 There is also a special argument there for genus 4, essentially using
Zariski’s main theorem on the map from moduli of curves to moduli of jacobians.
There are also inductive arguments, based on the fact that the boundary of
moduli of curves of genus g contains singular curves of genus g 1, and allowing
one to use lower genus Torelli results to deduce degree Torelli for later genera.
Then of course there is Matsusaka’s proof, derived from Torelli’s original proof
that given an isomorphism of polarized jacobians, the theta divisor defines the
graph of an isomorphism between their curves.”

14

Bibliographic notes for “Abelian Varieties” and “Jacobian Varieties”

The theory of abelian varieties over C is very old. On the other hand,
the “abstract” theory over arbitrary fields, can be said to have begun with
Weil’s famous announcement of the proof of the Riemann hypothesis for
function fields (Sur les fonctions algébriques a corps de constantes fini,
C.R. 210 (1940) 592–594). Parts of the projected proof (for example, the
key “lemme important”) can best be understood in terms of intersection
theory on the Jacobian variety of the curve, and Weil was to spend the next
six years developing the foundational material necessary for making his
proof rigorous. Unable in 1941 to construct the Jacobian as a projective
variety, Weil was led to introduce the notion of an abstract variety (that is,
a variety that is not quasi-projective). He then had to develop the theory
of such varieties, and he was forced to develop his intersection theory by
local methods rather than the projective methods used by van der Waerden
(Einfuhring in die algebraische Geometrie, Springer, 1939). In 1944 Weil
completed his book, Foundations of algebraic geometry, AMS Coll., XXIX,
1946, which laid the necessary foundations in algebraic geometry, and in
1946 he completed his two books Sur les courbes algébriques et les variétés
qui s’en déduisent, Hermann, 1948, and Weil 1948, which developed the
basic theory of Abelian varieties and Jacobian varieties and gave a detailed
account of his proof of the Riemann hypothesis. In the last work, abelian
varieties are defined much as we defined them and Jacobian varieties are
constructed, but it was not shown that the Jacobian could be defined over
the same field as the curve.
Chow (Algebraic systems of positive cycles in an algebraic variety,
Amer. J. Math. 72 (1950) 247-283 and Chow 1954) gave a construction of
the Jacobian variety which realized it as a projective variety defined over
the same ground field as the original curve. Matsusaka (On the algebraic
construction of the Picard variety, Japan J. Math 21 (1951) 217–235 and
22 (1952) 51–62) gave the first algebraic construction of the Picard and
Albanese varieties and demonstrated also that they were projective and had
13 Smith, R.; Varley, R. The Prym Torelli problem: an update and a reformulation as a

question in birational geometry. Symposium in Honor of C. H. Clemens (Salt Lake City,
UT, 2000), 235–264, Contemp. Math., 312, Amer. Math. Soc., Providence, RI, 2002.

14

BIBLIOGRAPHIC NOTES

46

the same field of definition as the original varieties. Weil showed that his
construction of a group variety starting from a birational group could also be
carried out without making an extension of the ground field (On algebraic
groups of transformations, Amer. J. Math., 77 (1955) 355–391), and in (The
field of definition of a variety, Amer. J. Math., 78 (1956) 509–524) he further
developed his methods of descending the field of definition of a variety.
Finally Barsotti (A note on abelian varieties, Rend. Circ. Mat. di Palermo, 2
(1953) 236–257), Matsusaka (Some theorems on abelian varieties, Nat. Sci.
Report Ochanomizu Univ. 4 (1953) 22–35), and Weil (On the projective
embedding of abelian varieties, in Algebraic geometry and topology, A
symposium in honor of S. Lefschetz, Princeton, 1957, pp. 177–181) showed
that all abelian varieties are projective. In a course at the University of
Chicago, 1954-55, Weil made substantial improvements to the theory of
abelian varieties (the seesaw principle and the theorem of the cube, for
example), and these and the results mentioned above together with Chow’s
theory of the “k-image” and “k-trace” (Abelian varieties over function fields,
Trans. AMS, 78 (1955) 253-275) were incorporated by Lang in his book
Lang 1959. The main lacuna at this time (1958/1959) was a satisfactory
theory of isogenies of degree p and their kernels in characteristic p; for
example, it was not known that the canonical map from an abelian variety to
the dual of its dual was an isomorphism (its degree might have been divisible
by p/. Cartier (Isogenies and duality of abelian varieties, Ann of Math. 71
(1960) 315-351) and Nishi (The Frobenius theorem and the duality theorem
on an abelian variety, Mem. Coll. Sc. Kyoto (A), 32 (1959) 333-350)
settled this particular point, but the full understanding of the p-structure of
abelian varieties required the development of the theories of finite group
schemes and Barsotti-Tate groups. The book of Mumford (Mumford 1970)
represents a substantial contribution to the subject of abelian varieties: it
uses modern methods to give a comprehensive account of abelian varieties
including the p-theory in characteristic p, and avoids the crutch of using
Jacobians to prove results about general abelian varieties. (It has been a
significant loss to the mathematical community that Mumford did not go on
to write a second volume on topics suggested in the introduction: Jacobians;
Abelian schemes: deformation theory and moduli; The ring of modular
forms and the global structure of the moduli space; The Dieudonné theory
of the “fine” characteristic p structure; Arithmetic theory: abelian schemes
over local, global fields. We still lack satisfactory accounts of some of these
topics.)
Much of the present two articles has been based on these sources; we
now give some other sources and references. “Abelian Varieties” will be
abbreviated by AVs and “Jacobian Varieties” by JVs.
The proof that abelian varieties are projective in (AVs 7) is Weil’s 1957
proof. The term “isogeny” was invented by Weil: previously, “isomorphism”
had frequently been used in the same situation. The fact that the kernel of
mA has m2g elements when m is prime to the characteristic was one of the
main results that Weil had to check in order to give substance to his proof
of the Riemann hypothesis. Proposition 11.3 of AVs is mentioned briefly
by Weil in Variétés abéliennes, Colloque d’algebre et theorie des nombres,

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BIBLIOGRAPHIC NOTES

47

Paris, 1949, 125-128, and is treated in detail by Barsotti (Structure theorems
for group varieties, Annali di Mat. 38 (1955) 77-119). Theorem 14.1 is
folklore: it was used by Tate in Endomorphisms of abelian varieties over
finite fields, Invent. math., 2 (1966) 134-144, which was one of the starting
points for the work that led to Faltings’s proof of Mordell’s conjecture in
1983. The étale cohomology of an abelian variety is known to everyone
who knows étale cohomology, but I was surprised not to be able to find an
adequate reference for its calculation: in Kleiman, Algebraic cycles and the
Weil conjectures, in Dix exposés sur la cohomologie des schémas, NorthHolland, 1968, pp 359–386, Jacobians are used, and it was unaccountably
omitted from Milne 1980. In his 1940 announcement, Weil gives a definition
of the em -pairing (in our terminology, exm -pairing) for divisor classes of
degree zero and order m on a curve which is analogous to the explicit
description at the start of 16 of AVs. The results of that section mainly
go back to Weil’s 1948 monograph, Weil 1948, but they were reworked
and extended to the p-part in Mumford’s book. The observation (see 16.12
of AVs) that .A  A_ /4 is always principally polarized is due to Zarhin
(A finiteness theorem for unpolarized Abelian varieties over number fields
with prescribed places of bad reduction, Invent. math. 79 (1985) 309-321).
Theorem 18.1 of AVs was proved by Narasimhan and Nori (Polarizations on
an abelian variety, in Geometry and Analysis, Springer, (1981), p. 125–128).
Proposition 20.1 of AVs is due to Grothendieck (cf. Mumford, Geometric
Invariant Theory, Springer, 1965, 6.1), and (20.5) of AVs (defining the K=ktrace) is due to Chow (reference above). The Mordell-Weil Theorem was
proved by Mordell (On the rational solutions of the indeterminate equations
of the third and fourth degrees, Proc. Cambridge Phil. Soc. 21 (1922)
179-192) (the same paper in which he stated his famous conjecture) for an
elliptic curve over the rational numbers and by Weil (L’arithmétique sur
les courbes algébriques, Acta Math. 52 (1928) 281-315) for the Jacobian
variety of a curve over a number field. (Weil, of course, stated the result in
terms of divisors on a curve.)
The first seven sections of JVs were pieced together from two disparate
sources, Lang’s book, Lang 1959, and Grothendieck’s Bourbaki talks14 ,
with some help from Serre 1959, Mumford 1966, and the first section of
Katz and Mazur 1985 (Arithmetic Moduli of Elliptic Surfaces, Princeton,
1985).
Rosenlicht (Generalized Jacobian varieties, Ann. of Math.,59 (1954)
505-530, and A universal mapping property of generalized Jacobians, ibid,
(1957), 80-88), was the first to construct the generalized Jacobian of a curve
relative to a modulus. The proof that all abelian coverings of a curve can
be obtained from isogenies of its generalized Jacobians (Theorem 9.7 of
JVs) is due to Lang (Sur les séries L d’une variété algébrique, Bull. SMF,
84 (1956) 555-563). Results close to Theorem 8.1 of JVs were obtained
by Igusa (Fibre systems of Jacobian varieties I, II, III, Amer. J. Math., 78
(1956) p. 171–199, p. 745-760, and 81 (1959) p. 453–476). Theorem 9.11
14 Grothendieck, A, Technique de descente et théoremes d’existence en géometrie
algébrique, I–VI. Séminaire Bourbaki 190, 195, 212, 221, 232, 236 (1959/62).

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48

is due to Parshin (Algebraic curves over function fields, I, Math. USSR—
Izvestija, 2 (1968) 1145-1169). Matsusaka (On a generating curve of an
abelian variety, Nat Sc. Rep. Ochanomizu Univ. 3 (1952) 1-4) showed
that every abelian variety over an algebraically closed field is generated by
a curve (cf. 10.1 of JVs). Regarding (11.2) of JVs, Hurwitz (Math. Ann.
28 (1886)) was the first to show the relation between the number of fixed
points of a correspondence on a Rieman surface C and the trace of a matrix
describing its action on the homology of the surface (equivalently that of its
Jacobian). This result of Hurwitz inspired both Lefschetz in his proof of his
trace formula and Weil in his proof of the Riemann hypothesis for curves.
Proofs of Torelli’s theorem can be found in Andreotti (On a theorem
of Torelli, Amer. J. Math., 80 (1958) 801–821), Matsusaka (On a theorem
of Torelli, Amer. J. Math., 80 (1958) 784–800), Weil (Zum Beweis des
Torellischen Satzes, Gott. Nachr. 2 (1957) 33-53), and Ciliberto (On a proof
of Torelli’s theorem, in Algebraic geometry — open problems, Lecture
notes in math. 997, Springer, 1983 pp. 113–223). The proof in 13 of JVs
is taken from Martens (A new proof of Torelli’s theorem, Ann. Math. 78
(1963) 107–111). Torelli’s original paper is Sulle varieta di Jacobi, Rend.
R. Acad. Sci. Torino, 50 (1914–15) 439–455. Torelli’s theorem shows that
the map from the moduli space of curves into that of principally polarized
abelian varieties is injective on geometric points; a finer discussion of the
map can be found in a paper by Oort and Steenbrink (The local Torelli
problem for algebraic curves, in Algebraic Geometry Angers 1979, Sijthoff
& Noordhoff, 1980, pp. 157-204).
Finally, we mention that Mumford’s notes Curves and their Jacobians
(footnote 9) provide a useful survey of the topics in its title, and that the
commentaries by Weil in his Collected Papers (Springer, 1979) give a
fascinating insight into the origins of parts of the subject of arithmetic
geometry.

BIBLIOGRAPHY

49

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Index
abelian covering, 30
birational group, 24
canonical polarization, 24
degree of a relative divisor, 12
descent datum, 4
DivrC , 13
divisorial correspondence, 3
effective Cartier divisor, 10
Galois covering, 30
Galois group, 30
generalized Jacobian, 28
Jacobian scheme, 28
Jacobian variety, 3
modulus, 28
period of a curve, 4
pointed scheme, 3
relative effective Cartier divisor, 10
split divisor, 13
symmetric morphism, 9
symmetric power of a variety, 9

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