1Divisors

III Positivity in Algebraic Geometry


1.2 Weil divisors
If
X
is a sufficiently nice Noetherian scheme,
L
is a line bundle over
X
, and
s ∈ H
0
(
X, L
), then the vanishing locus of
s
is a codimension-1 subscheme. More
generally, if
s
is a rational section, then the zeroes and poles form codimension 1
subschemes. Thus, to understand line bundles, a good place to start would be
to understand these subschemes. We will see that for suitably nice schemes, we
can recover L completely from this information.
For the theory to work out nicely, we need to assume the following technical
condition:
Definition
(Regular in codimension 1)
.
Let
X
be a Noetherian scheme. We say
X is regular in codimension 1 if every local ring O
x
of dimension 1 is regular.
The key property of such schemes we will make use of is that if
Y ⊆ X
is
a codimension 1 integral subscheme, then the local ring
O
X,Y
is a DVR. Then
there is a valuation
val
Y
: O
X,Y
→ Z,
which tells us the order of vanishing/poles of our function along Y .
Definition
(Weil divisor)
.
Let
X
be a Noetherian scheme, regular in codimension
1. A prime divisor is a codimension 1 integral subscheme of
X
. A Weil divisor
is a formal sum
D =
X
a
i
Y
i
,
where the
a
i
∈ Z
and
Y
i
are prime divisors. We write
WDiv
(
X
) for the group of
Weil divisors of X.
For K a field, a Weil K-divisor is the same where we allow a
i
∈ K.
Definition
(Effective divisor)
.
We say a Weil divisor is effective if
a
i
≥
0 for
all i. We write D ≥ 0.
Definition
(Principal divisor)
.
If
f ∈ K
(
X
), then we define the principal
divisor
div(f) =
X
Y
val
Y
(f) · Y.
One can show that
val
Y
(
f
) is non-zero for only finitely many
Y
’s, so this is
a genuine divisor. To see this, there is always a Zariski open
U
such that
f|
U
is invertible, So
val
Y
(
f
)
6
= 0 implies
Y ⊆ X \ U
. Since
X
is Noetherian,
X \ U
can only contain finitely many codimension 1 subscheme.
Definition (Support). The support of D =
P
a
i
Y
i
is
supp(D) =
[
Y
i
.
Observe that
div
(
fg
) =
div
(
f
) +
div
(
g
). So the principal divisors form a
subgroup of the Weil divisors.
Definition
(Class group)
.
The class group of
X
,
Cl
(
X
), is the group of Weil
divisors quotiented out by the principal divisors.
We say Weil divisors
D, D
0
are linearly equivalent if
D − D
0
is principal, and
we write D ∼ D
0
.
Example. Take X = A
1
K
, and f =
x
3
x+1
∈ K(X). Then
div(f) = 3[0] − [−1].
A useful result for the future is the following:
Theorem
(Hartog’s lemma)
.
Let
X
be normal, and
f ∈ O
(
X \ V
) for some
V ≥ 2. Then f ∈ O
X
. Thus, div(f ) = 0 implies f ∈ O
×
X
.