The Jacobian of a Curve
Saúl Quispe
Departamento de Matemática
Universidad de Concepción
7 de junio de 2012
1 / 18
The Jacobian of a Curve
Jacobi inversion theorem
Lemma 1. A holomorphic map F : X → Y between compact connected complex manifolds of the same dimension is surjective if the
Jacobian matrix of the map F has nonzero determinant at some
point of X.
Proof. It is known that F (X) is a subvariety of Y . Since JF is
nonsingular at some point, then F (X) contains an open set in Y .
Thus dim(Y ) = dim(F (X)). This implies that F is surjective since
X is compact and Y is Hausdorff and connected.
2 / 18
The Jacobian of a Curve
Jacobi inversion theorem
Theorem 1. The restriction of the Abel-Jacobi map to X (g)
µ : X (g) → J(X),
g
X
k=1
"
pk 7→
g Z
X
k=1
pk
p0
ω1 , · · · ,
g Z
X
k=1
pk
ωg
#
.
p0
is surjective.
P
Proof. Let D = k pk be a point of X (g) with all the pk ’s distinct,
zk be a local coordinate on X near the point pk and (z1 , · · · , zg ) be
the corresponding coordinates on X (g) near D.
3 / 18
The Jacobian of a Curve
Jacobi inversion theorem
P
Let D0 = gk=1 qk near D and z̄k be the coordinate of qk .
By calculus
Z zk
∂ Z zk
∂
∂µ
ω1
ωg 0
ω1 ), · · · ,
ωg ) (D0 ) =
(D0 ) =
(
(
,··· ,
(D )
∂zk
∂zk z0
∂zk z0
dzk
dzk
where ωi = fik (zk ) dzk (i = 1, · · · , g) near pk , with fik (zk ) holomorphic.
Thus the Jacobian matrix of the map µ near D is given by


Jµ(D0 ) = 
ω1
dz1 (q1 )
···
..
.
ωg
(q
dz1 1 ) · · ·
ω1
dzg (qg )


..
.
.
ωg
dzg (qg )
We note that changing the local coordinate zi multiplies the ith
column by a nonzero factor but does not affect the rank of Jµ.
4 / 18
The Jacobian of a Curve
Jacobi inversion theorem
I
We may choose q1 , so that ω1 (q1 ) 6= 0, and then, subtracting
a multiple of ω1 from ω2 , · · · , ωg , we may arrange that
ω2 (q1 ) = · · · = ωg (q1 ) = 0.
I
Next, we may choose q2 so that ω2 (q2 ) 6= 0, and then arrange
as before that ω3 (q2 ) = · · · = ωg (q2 ) = 0.
P
Continuing in this way, the Jacobian matrix at D0 = k qk
will be triangular with zeros below the diagonal and nonzero
on the diagonal.
I
So µ has maximal rank at D0 , i.e. there is a point D0 at which
the Jacobian matrix of µ is nonsingular, and by Lemma 1 the AbelJacobi map must be surjective.
5 / 18
The Jacobian of a Curve
Picard group
The subgroup of Div X consisting of the principal divisors is denoted
by Prin X.
Definition 1. Two divisors D1 , D2 ∈ Div X are linearly equivalent,
written D1 ∼ D2 , if their difference D1 − D2 is principal.
Definition 2. The Picard group of X is the quotient group
Pic X =
Div X
.
Prin X
Pic0 X =
Div0 X
,
Prin X
We can also define
which is a subgroup of Pic X.
6 / 18
The Jacobian of a Curve
Consequences
Corollary 1. If X is a curve of genus g ≥ 1, then the Abel map
induces an isomorphism
µ̄ : Pic0 X → J(X).
Proof. We consider the sequence of homomorphisms
M∗ (X)
div
/ Div0 X
µ
/ J(X)
/ 0.
Then
I Abel Theorem implies Im(div) = ker(µ).
I Fixing p0 on X, Jacobi inversion Theorem implies that µ must
be surjective on X (g) − gp0 ⊂ Div0 (X). Thus
Pic0 X = Div0 X/Im(div) ∼
= J(X).
7 / 18
The Jacobian of a Curve
Consequences
Lemma 2. Let X be a curve of genus g ≥ 1. Then there is no point
where all holomorphic 1-forms on X vanish.
Proof. Suppose there exists p ∈ X where all holomorphic 1-form
vanish. Then
dim L(1) (−p) = g,
and by the Riemann-Roch theorem
dim L(p) = 2.
i.e. there exists a non-constant meromorphic function f with a
simple pole at p. The associated holomorphic map F : X → C∞ is
an isomorphism, which implies g = 0. Contradiction to g ≥ 1.
8 / 18
The Jacobian of a Curve
Consequences
Corollary 2. Let X be a curve of genus g ≥ 1. Then the map
µ : X → J(X)
is an embedding.
Proof.
I
We consider the Jacobian of the Abel-Jacobi map in p ∈ X,
dµ(p) = (ω1 (p), · · · , ωg (p))T 6= 0.
Then µ is an immersion.
I
Assume that µ(p) = µ(q), with p 6= q. Then µ(p − q) = 0 in
J(X). By Abel Theorem there exists f ∈ M(X) such that
div(f ) = p − q. The associated holomorphic map
F : X → C∞ would then be an isomorphism. Since g ≥ 1,
this is a contradiction.
9 / 18
The Jacobian of a Curve
Consequences
Remark. Let X be a compact Riemann surface of genus one. Then
I
J(X) is itself a complex torus of dimension one.
I
Given any point p0 ∈ X there is an isomorphism µ of X with
the complex torus J(X) such that µ(p0 ) = 0.
I
X is an abelian group, the group law being induced by the
Abel-Jacobi isomorphism.
10 / 18
The Jacobian of a Curve
Consequences
Let X be a compact Riemann surface of genus g, and D be a divisor
on X. Then:
I
The complete linear system of D, is the set
|D| = {E ∈ Div X : E ∼ D and E ≥ 0}.
I
The map S : P(L(D)) → |D|, is a 1-1 correspondence.
I
If D ≥ 0, then
dim |D| = dim L(D) − 1.
11 / 18
The Jacobian of a Curve
Consequences
Theorem 2. The fiber of µ : X (d) → J(X) is a projective space.
More precisely, for any D ∈ X (d) , we have
µ−1 (µ(D)) = |D|.
Proof.
I
We first prove that µ−1 (µ(D)) ⊂ |D|. Suppose E ∈ X (d)
such that
µ(E) = µ(D).
Then µ(E − D) = 0, and by sufficiency part of the Abel
Theorem there exist f ∈ M(X) such that
div(f ) = E − D.
However, E ∈ X (d) implies that E ≥ 0, thus E ∈ |D|.
12 / 18
The Jacobian of a Curve
Consequences
I
Conversely, suppose E ∈ |D|, i.e., there exists f ∈ L(D) such
that E = div(f ) + D. By necessity part of the Abel Theorem,
we have
µ(E) = µ(div(f )) + µ(D) = 0 + µ(D) = µ(D),
whence E ∈ µ−1 (µ(D)).
13 / 18
The Jacobian of a Curve
Consequences
Let X be a compact Riemann surface of genus g ≥ 2, and K be a
canonical divisor on X. Then:
I If we consider the canonical map
φK : X → Pg−1 , p 7→ [ω1 (p), · · · , ωg (p)],
I
then φK (X) is a non-degenerate curve in Pg−1 , where
ωi (i = 1, · · · , g) is a basis of Ω1 (X).
P
If D = dk=1 pk ∈ X (d) , then
φK (D) = φK (p1 ), · · · , φK (pd ),
I
is the projective subespace in Pg−1 spanned by
φK (p1 ), · · · , φK (pd ).
The
dim L(K − D) = g − 1 − dim φK (D).
(1)
14 / 18
The Jacobian of a Curve
Consequences (Riemann-Roch Theorem)
Theorem 3. Let X be a curve of genus g ≥ 2, and D an effective
divisor. Then
dim |D| = deg(D) − 1 − dim φK (D).
Pd
(d) with all the p ’s
Proof. Let D =
k
k=1 pk be a point of X
distinct, zk be a local coordinate on X near the point pk and
(z1 , · · · , zd ) be the corresponding coordinates on X (d) near D.
Let r := dim |D| and let λ1 , · · · , λr be local coordinates in the
projective space |D| near D; we shall write
Dλ = p1 (λ) + · · · + pd (λ)
to denote the point of |D| corresponding to the value λ of the
coordinates. Then, write zk (pk (λ)) = zk (λ). Suppose ωi (i =
1, · · · , g) is a basis of Ω1 (X), and near pk
ωi = fik (zk ) dzk ,
where fik (zk ) is holomorphic.
15 / 18
The Jacobian of a Curve
Consequences (Riemann-Roch Theorem)
Since the points pk (λ) move with r degrees of freedom, for general
λ we have that
rank
∂(z1 , · · · , zd )
(λ) = r.
∂(λ1 , · · · , λr )
(2)
Then, by Theorem 2, we have
X Z zk (λ)
fik (zk ) dzk ≡ constant (modulo periods),
k
p0
and differentiating with respect to λν we obtain
X
∂zk
fik (zk (λ))
(λ) = 0.
∂λν
(3)
k
This relation, together with (2), tells us that
dim φK (D) ≤ d − 1 − r.
(4)
16 / 18
The Jacobian of a Curve
Consequences (Riemann-Roch Theorem)
To show that, in this relation, we have an equality, we proceed
by contradiction. Suppose (4) is a strict inequality. Then, given
D0 ∈ |K − D|, by (1) we would have
dim |D0 | > g − d + r − 1.
Now, applying (4) to D0 we would get
dim φK (D0 ) < 2g − 2 − d − 1 − (g − d + r − 1) = g − 2 − r. (5)
Then, applying (1) to D0 , together with (5), tells us that
dim L(D) = dim L(K − D0 ) > g − 1 − (g − 2 − r) = 1 + r.
So
dim |D| > r,
which is absurd.
17 / 18
The Jacobian of a Curve
Reference
E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris
Geometry of Algebraic Curves Vol. 1,
Springer-Verlag, New York Berlin Heidelberg Tokyo, 1985.
P. Griffiths
Introduction to Algebraic Curves ,
Translations of Mathematical Monographs Vol. 76. American
Mathematical Society, 1989.
R. Miranda
Algebraic Curves and Riemann Surfaces ,
Graduate Studies in Mathematics Vol. 5. American
Mathematical Society, 1995.
18 / 18