1Divisors
III Positivity in Algebraic Geometry
1.5 Linear systems
We end by collecting some useful results about divisors and linear systems.
Proposition.
Let
X
be a smooth projective variety over an algebraically closed
field. Let D
0
be a divisor on X.
(i)
For all
s ∈ H
0
(
X, O
X
(
D
)),
div
(
s
) is an effective divisor linearly equivalent
to D.
(ii)
If
D ∼ D
0
and
D ≥
0, then there is
s ∈ H
0
(
O
X
(
D
0
)) such that
div
(
s
) =
D
(iii)
If
s, s
0
∈ H
0
(
O
X
(
D
0
)) and
div
(
s
) =
div
(
s
0
), then
s
0
=
λs
for some
λ ∈ K
∗
.
Proof.
(i) Done last time.
(ii)
If
D ∼ D
0
, then
D − D
0
=
div
(
f
) for some
f ∈ K
(
X
). Then (
f
) +
D
0
≥
0.
So f induces a section s ∈ H
0
(O
X
(D
0
)). Then div(s) = D.
(iii) We have
s
0
s
∈ K(X)
∗
. So div
s
0
s
. So
s
0
s
∈ H
0
(O
∗
) = K
∗
.
Definition
(Complete linear system)
.
A complete linear system is the set of all
effective divisors linearly equivalent to a given divisor D
0
, written |D
0
|.
Thus, |D
0
| is the projectivization of the vector space H
0
(X, O(D
0
)).
We also define
Definition
(Linear system)
.
A linear system is a linear subspace of the projective
space structure on |D
0
|.
Given a linear system
|V |
of
L
on
X
, we have previously seen that we get a
rational map
X 99K P
(
V
). For this to be a morphism, we need
V
to generate
L
.
In general, the O
X
action on L induces a map
V ⊗ O
X
→ L.
Tensoring with L
−1
gives a map
V ⊗ L
−1
→ O
X
.
The image is an
O
X
-submodule, hence an ideal sheaf. This ideal
b
|V |
is called
the base ideal of
|V |
on
X
, and the corresponding closed subscheme is the base
locus. Then the map
X 99K P
(
V
) is a morphism iff
b
|V |
=
O
X
. In this case, we
say |V | is free.
Accordingly, we can define
Definition
((Very) ample divisor)
.
We say a Cartier divisor
D
is (very) ample
when O
X
(D) is.
By Serre’s theorem, if
A
is ample and
X
is projective, and
L
is any line
bundle, then for n sufficiently large, we have
h
0
(X, L ⊗ A
n
) =
X
(−1)
i
h
i
(X, L ⊗ A
n
) = χ(X, L ⊗ A
n
).
In the case of curves, Riemann–Roch lets us understand this number well:
Theorem (Riemann–Roch theorem). If C is a smooth projective curve, then
χ(L) = deg(L) + 1 − g(C).
It is often easier to reason about very ample divisors than general Cartier
divisors. First observe that by definition, every projective normal scheme has a
very ample divisor, say
H
. If
D
is any divisor, then by Serre’s theorem,
D
+
nH
is globally generated for large
n
. Moreover, one can check that the sum of a
globally generated divisor and a very ample divisor is still very ample, e.g. by
checking it separates points and tangent vectors. Hence
D
+
nH
is very ample
for large n. Thus, we deduce that
Proposition.
Let
D
be a Cartier divisor on a projective normal scheme .Then
D ∼ H
1
− H
2
for some very ample divisors
H
i
. We can in fact take
H
i
to be
effective, and if
X
is smooth, then we can take
H
i
to be smooth and intersecting
transversely.
The last part is a consequence of Bertini’s theorem.
Theorem
(Bertini)
.
Let
X
be a smooth projective variety over an algebraically
closed field
K
, and
D
a very ample divisor. Then there exists a Zariski open set
U ⊆ |D|
such that for all
H ∈ U
,
H
is smooth on
X
and if
H
1
6
=
H
2
, then
H
1
and H
2
intersect transversely.