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correspondence L on C × T (some pointed variety T ) such that Lt is of degree 0 for
all t, there is a regular map Õ : T ’! J sending the distinguished point of T to 0 and
such that (1 × Õ)"MH"L.
Proof. See JV 1.2.
Remark 17.4. (a) The Jacobian variety is defined even when C(k) ="; however,
it then doesn t (quite) represent the functor P (because the functor is not repre-
sentable). See JV p168.
(b) The Jacobian variety commutes with extension of scalars, i.e., Jac(Ck ) =
(Jac(C))k for any field k ƒ" k.
(c) Let M be the sheaf in (17.3); as x runs through the elements of J(k), Mx runs
through a set of representatives for the isomorphism classes of invertible sheaves of
degree 0 on C.
(d) Fix a point P0 in J(k). There is a regular map ÕP : C ’! J such that, on
points, ÕP sends P to [P - P0]; in particular, ÕP sends P0 to 0. The map ÕQ differs
0 0 0
from ÕP by translation by [P0 - Q0] (regarded as a point on J).
(e) The dimension of J is the genus of C. If C has genus zero, then Jac(C) =0
(this is obvious, because Pic0(C) = 0, even when one goes to the algebraic closure).
If C has genus 1, then Jac(C) =C (provided C has a rational point; otherwise it
differs from C because Jac(C) always has a point).
66 J.S. MILNE
Construction of the Jacobian variety. Fix a nonsingular projective curve over k.
For simplicity, assume k = kal. We want to construct a variety such that J(k) is the
group of divisor classes of degree zero on C. As a first step, we construct a variety
whose points are the effective divisors of degree r, some r >0. Let Cr = C ×C ×...×C
(r copies). A point on Cr is an ordered r-tuple of points on C. The symmetric group
on r letters, Sr, acts on Cr by permuting the factors, and the points on the quotient
df
variety C(r) = Cr/Sr are the unordered r-tuples of points on C. But an unordered
r-tuple is just an effective divisor of degree r, £Pi. Thus
df
C(r) = Divr(C) = {effective divisors of degree r on C}.
Write À for the quotient map Cr ’! C(r), (P1, ..., Pr) ’! Pi.
Lemma 17.5. The variety C(r) is nonsingular.
Proof. In general, when a finite group acts freely on a nonsingular variety, the
quotient will be nonsingular. In our case, there are points on Cr whose stabilizer sub-
group is nontrivial, namely the points (P1, ..., Pr) in which two (or more) Pi coincide,
and we have to show that they don t give singularities on the quotient variety. The
worst case is a point Q =(P, ..., P ), and here one can show that
OQ
=
the power series ring in the elementary symmetric functions Ã1, ..., Ãr in the Xi, and
this is a regular ring. See JV 3.2.
Let Picr(C) be the set of divisor classes of degree r. For a fixed point P0 on C, the
map
[D] ’! [D + rP0]: Pic0(C) ’! Picr(C)
is a bijection (both Pic0(C) and Picr(C) are fibres of the map deg: Pic(C) ’! Z).
This remains true when we regard Pic0(C) and Picr(C) as functors of varieties over
k (see above), and so it suffices to find a variety representing the Picr(C).
For a divisor of degree r, the Riemann-Roch theorem says that
(D) =r +1- g + (K - D)
where K is the canonical divisor. Since deg(K) = 2g - 2, deg(K - D)
(K - D) =0 when deg(D) > 2g - 2. Thus,
(D) =r +1- g >0, if r =deg(D) > 2g - 2.
In particular, every divisor class of degree r contains an effective divisor, and so the
map
Õ : {effective divisors of degree r} ’!Picr(C), D ’! [D]
is surjective when r >2g - 2. We can regard this as a morphism of functors
Õ : C(r) Picr(C).
Suppose that we could find a section s to Õ, i.e., a morphism of functors
s : Picr(C) ’! C(r) such that Õ æ% s = id. Then s æ% Õ is a morphism of functors
ABELIAN VARIETIES 67
C(r) ’! C(r) and hence by (17.1) a regular map, and we can form the fibre product:
C(r)
(1,sæ%Õ)
(" ("
"
C(r) × C(r)
Then
b ’!Õ(b)
J (k) ={(a, b) " C(r) × C(r) | a = b, b = s æ% Õ(a)} ’! Picr(C)
is an isomorphism. Thus we will have constructed the Jacobian variety; in fact J
will be a closed subvariety of C(r). Unfortunately, it is not possible to find such a
section: the Riemann-Roch theorem tells us that, for r >2g - 2, each divisor class
of degree r is represented by an (r - g)-dimensional family of effective divisors, and
there is no nice functorial way of choosing a representative. However, it is possible
to do this locally , and so construct J as a union of varieties, each of which is a
closed subvariety of an open subvariety of C(r). For the details, see JV §4.
18. Abel and Jacobi
Abel 1802-29.
Jacobi 1804-1851.
Let f(X, Y ) " R[X, Y ]. We can regard the equation f(X, Y ) = 0 as defining Y
(implicitly) as a multivalued function of X. An integral of the form,
g(Y )dX
with g(Y ) a rational function, is called an abelian integral, after Abel who made a
2
profound study of them. For example, if f(X, Y ) =Y - X3 - aX - b, then
dX dX
=
1
Y
2
(X3 + aX + b)
is an example of an abelian integral in this case it is a elliptic integral, which had
been studied in the eighteenth century. The difficulty with these integrals is that,
unless the curve f(X, Y ) = 0 has genus 0, they can t be evaluated in terms of the
elementary functions.
Today, rather than integrals of multivalued functions, we prefer to think of differ-
entials on a Riemann surface, e.g., the compact Riemann surface (i.e., curve over C)
defined by f(X, Y ) =0.
Let C be a compact Riemann surface. Recall10 that C is covered by coordinate
neigbourhoods (U, z) where U can be identified with an open subset of C and z is the
complex variable; if (U1, z1) is a second open set, then z = u(z1) and z1 = v(z) with
u and v holomorphic functions on U )" U1. To give a differential form É on C, one has
to give an expression f(z)dz on each (U, z) such that, on U )" U1,
f(u(z1)) · u (z1) · dz1 = f1(z1) · dz1.
10
See Cartan, H., Elementary Theory of Analytic Functions of One or Several Complex Variables,
Addison Wesley, 1963, especially VI.4.
68 J.S. MILNE
A differential form is holomorphic if each of the functions f(z) is holomorphic (rather
than meromorphic). Let É be a differential on C and let ³ be a path in U )" U1; then
f(z) æ% dz = f1(z1) æ% dz1.
³ ³
Thus, it makes sense to integrate É along any path in C.
Theorem 18.1. The set of holomorphic differentials on C forms a g-dimensional
vector space where g is the genus of C.
We denote this vector space by “(C, &!1). If É1, ..., Ég is a basis for the space,
then every holomorphic differential is a linear combination, É = £ aiÉi, of the
Éi, and É = a Éi; therefore it suffices to understand the finite set of integrals
³ ³
{ É1, . . . , Ég}.
³ ³
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