1Divisors
III Positivity in Algebraic Geometry
1.3 Cartier divisors
Weil divisors do not behave too well on weirder schemes. A better alternative is
Cartier divisors. Let
L
be a line bundle, and
s
a rational section of
L
, i.e.
s
is a
section of L|
U
for some open U ⊆ X. Given such an s, we can define
div(s) =
X
Y
val
Y
(s) · Y.
To make sense of
val
Y
(
s
), for a fixed codimension 1 subscheme
Y
, pick
W
such
that
W ∩ U 6
=
∅
, and
L|
W
is trivial. Then we can make sense of
val
Y
(
s
) using
the trivialization. It is clear from Hartog’s lemma that
Proposition. If X is normal, then
div : {rational sections of L} → WDiv(X).
is well-defined, and two sections have the same image iff they differ by an element
of O
∗
X
.
Corollary. If X is normal and proper, then there is a map
div{rational sections of L}/K
∗
→ WDiv(X).
Proof. Properness implies O
∗
X
= K
∗
.
Example.
Take
X
=
P
1
K
, and
s
=
X
2
X+Y
∈ H
0
(
O
(1)), where
X, Y
are our
homogeneous coordinates. Then
div(s) = 2[0 : 1] − [1 : −1].
This lets us go from line bundles to divisors. To go the other direction, let
X
be a normal Noetherian scheme. Fix a Weil divisor
X
. We define the sheaf
O
X
(D) by setting, for all U ⊆ X open,
O
U
(D) = {f ∈ K(X) : div(f) + D|
U
≥ 0}.
Proposition. O
X
(D) is a rank 1 quasicoherent O
X
-module. .
In general,
O
X
(
D
) need not be a line bundle. If we want to prove that it is,
then we very quickly see that we need the following condition:
Definition
(Locally principal)
.
Let
D
be a Weil divisor on
X
. Fix
x ∈ X
. Then
D
is locally principal at
x
if there exists an open set
U ⊆ X
containing
x
such
that D|
U
= div(f)|
U
for some f ∈ K(X).
Proposition.
If
D
is locally principal at every point
x
, then
O
X
(
D
) is an
invertible sheaf.
Proof. If U ⊆ X is such that D|
U
= div(f)|
U
, then there is an isomorphism
O
X
|
U
→ O
X
(D)|
U
g 7→ g/f.
Definition
(Cartier divisor)
.
A Cartier divisor is a locally principal Weil divisor.
By checking locally, we see that
Proposition. If D
1
, D
2
are Cartier divisors, then
(i) O
X
(D
1
+ D
2
) = O
X
(D
1
) ⊗ O
X
(D
2
).
(ii) O
X
(−D)
∼
=
O
X
(D)
∨
.
(iii) If f ∈ K(X), then O
X
(div(f))
∼
=
O
X
.
Proposition.
Let
X
be a Noetherian, normal, integral scheme. Assume that
X
is factorial, i.e. every local ring
O
X,x
is a UFD. Then any Weil divisor is Cartier.
Note that smooth schemes are always factorial.
Proof.
It is enough to prove the proposition when
D
is prime and effective. So
D ⊆ X is a codimension 1 irreducible subvariety. For x ∈ D
– If x 6∈ D, then 1 is a divisor equivalent to D near x.
–
If
x ∈ D
, then
I
D,x
⊆ O
X,x
is a height 1 prime ideal. So
I
D,x
= (
f
) for
f ∈ m
X,x
. Then f is the local equation for D.
Recall the Class group
Cl
(
X
) was the group of Weil divisors modulo principal
equivalence.
Definition
(Picard group)
.
We define the Picard group of
X
to be the group
of Cartier divisors modulo principal equivalence.
Then if X is factorial, then Cl(X) = Pic(X).
Recall that if
L
is an invertible sheaf, and
s
is a rational section of
L
, then
we can define div(s).
Theorem.
Let
X
be normal and
L
an invertible sheaf,
s
a rational section of
L. Then O
X
(div(s)) is invertible, and there is an isomorphism
O
X
(div(s)) → L.
Moreover, sending L to divs gives a map
Pic(X) → Cl(X),
which is an isomorphism if X is factorial (and Noetherian and integral).
So we can think of
Pic
(
X
) as the group of invertible sheaves modulo isomor-
phism, namely H
1
(X, O
∗
X
).
Proof.
Given
f ∈ H
0
(
U, O
X
(
div
(
s
))), map it to
f · s ∈ H
0
(
U, L
). This gives the
desired isomorphism.
If we have to sections
s
0
6
=
s
, then
f
=
s
0
/s ∈ K
(
X
). So
div
(
s
) =
div
(
s
0
) +
div
(
f
), and
div
(
f
) is principal. So this gives a well-defined map
Pic
(
X
)
→
Cl(X).