3Projective varieties
III Positivity in Algebraic Geometry
3.3 Nef divisors
We can weaken the definition of an ample divisor to get an ample divisor.
Definition
(Numerically effective divisor)
.
Let
X
be a proper scheme, and
D
a
Cartier divisor. Then
D
is numerically effective (nef) if
D · C ≥
0 for all integral
curves C ⊆ X.
Similar to the case of ample divisors, we have
Proposition.
(i) D is nef iff D|
X
red
is nef.
(ii) D is nef iff D|
X
i
is nef for all irreducible components X
i
.
(iii) If V ⊆ X is a proper subscheme, and D is nef, then D|
V
is nef.
(iv)
If
f
:
X → Y
is a finite morphism of proper schemes, and
D
is nef on
Y
,
then f
∗
D is nef on X. The converse is true if f is surjective.
The last one follows from the analogue of Nakai’s criterion, called Kleinmann’s
criterion.
Theorem
(Kleinmann’s criterion)
.
Let
X
be a proper scheme, and
D
an
R
-
Cartier divisor. Then
D
is nef iff
D
dim V
[
V
]
≥
0 for all proper irreducible
subvarieties.
Corollary.
Let
X
be a projective scheme, and
D
be a nef
R
-divisor on
X
, and
H be a Cartier divisor on X.
(i) If H is ample, then D + εH is also ample for all ε > 0.
(ii) If D + εH is ample for all 0 < ε 1, then D is nef.
Proof.
(i)
We may assume
H
is very ample. By Nakai’s criterion this is equivalent
to requiring
(D + εH)
dim V
· [V ] =
X
dim V
p
ε
p
D
dim V −p
H
p
[V ] > 0.
Since any restriction of a nef divisor to any integral subscheme is also nef,
and multiplying with
H
is the same as restricting to a hyperplane cuts, we
know the terms that involve
D
are non-negative. The
H
p
term is positive.
So we are done.
(ii)
We know (
D
+
εH
)
· C >
0 for all positive
ε
sufficiently small. Taking
ε → 0, we know D · C ≥ 0.
Corollary. Nef(X) = Amp(X) and int(Nef(X)) = Amp(X).
Proof.
We know
Amp
(
X
)
⊆ Nef
(
X
) and
Amp
(
X
) is open. So this implies
Amp(X) ⊆ int(Nef(X)), and thus Amp(X) ⊆ Nef(X).
Conversely, if
D ∈ int
(
Nef
(
X
)), we fix
H
ample. Then
D − tH ∈ Nef
(
X
)
for small
t
, by definition of interior. Then
D
= (
D − tH
) +
tH
is ample. So
Amp(X) ⊇ int(Nef(X)).
Proof of Kleinmann’s criterion.
We may assume that
X
is an integral projective
scheme. The
⇐
direction is immediate. To prove the other direction, since the
criterion is a closed condition, we may assume
D ∈ Div
Q
(
X
). Moreover, by
induction, we may assume that
D
dim V
[
V
]
≥
0 for all
V
strictly contained in
X
,
and we have to show that D
dim X
≥ 0. Suppose not, and D
dim X
< 0.
Fix a very ample Cartier divisor H, and consider the polynomial
P (t) = (D + tH)
dim X
= D
dim X
+
dim X−1
X
i=1
t
i
dim X
i
H
i
D
dim X−i
+ t
dim X
H
dim X
.
The first term is negative; the last term is positive; and the other terms are
non-negative by induction since H is very ample.
Then on R
>0
, this polynomial is increasing. So there exists a unique t such
that P (t) = 0. Let
¯
t be the root. Then P (
¯
t) = 0. We can also write
P (t) = (D + tH) · (D + tH)
dim X−1
= R(t) + tQ(t),
where
R(t) = D · (D + tH)
dim X−1
, Q(t) = H · (D + tH)
dim X−1
.
We shall prove that R(
¯
t) ≥ 0 and Q(
¯
t) > 0, which is a contradiction.
We first look at Q(t), which is
Q(t) =
dim X−1
X
i=0
t
i
dim X − 1
i
H
i+1
D
dim X−i
,
which, as we argued, is a sum of non-negative terms and a positive term.
To understand R(t), we look at
R(t) = D · (D + tH)
dim X−1
.
Note that so far, we haven’t used the assumption that
D
is nef. If
t >
¯
t
, then
(
D
+
tH
)
dim X
>
0, and (
D
+
tH
)
dim V
[
V
]
>
0 for a proper integral subvariety,
by induction (and binomial expansion). Then by Nakai’s criterion,
D
+
tH
is
ample. So the intersection (
D
+
tH
)
dim X−1
is essentially a curve. So we are
done by definition of nef. Then take the limit t →
¯
t.
It is convenient to make the definition
Definition
(1-cycles)
.
For
K
=
Z, Q
or
R
, we define the space of 1-cycles to be
Z
1
(X)
K
=
n
X
a
i
C
i
: a
i
∈ K, C
i
⊆ X integral proper curves
o
.
We then have a pairing
N
1
K
(X) × Z
1
(X)
K
→ K
(D, E) 7→ D · E.
We know that
N
1
K
(
X
) is finite-dimensional, but
Z
1
(
X
)
K
is certainly uncountable.
So there is no hope that this intersection pairing is perfect
Definition
(Numerical equivalence)
.
Let
X
be a proper scheme, and
C
1
, C
2
∈
Z
1
(X)
K
. Then C
1
≡ C
2
iff
D · C
1
= D · C
2
for all D ∈ N
1
K
(X).
Definition
(
N
1
(
X
)
K
)
.
We define
N
1
(
X
)
K
to be the
K
-module of
K
1-cycles
modulo numerical equivalence.
Thus we get a pairing N
1
(X)
K
× N
1
(X)
K
→ K.
Definition (Effective curves). We define the cone of effective curves to be
NE(X) = {γ ∈ N
1
(X)
R
: γ ≡
X
[a
i
C
i
] : a
i
> 0, C
i
⊆ X integral curves}.
This cone detects nef divisors, since by definition, an
R
-divisor
D
is nef iff
D · γ ≥ 0 for all γ ∈ NE(X). In general, we can define
Definition
(Dual cone)
.
Let
V
be a finite-dimensional vector space over
R
and
V
∗
the dual of
V
. If
C ⊆ V
is a cone, then we define the dual cone
C
∨
⊆ V
∗
by
C
∨
= {f ∈ V
∗
: f(c) ≥ 0 for all c ∈ C}.
Then we see that
Proposition. Nef(X) = NE(X)
∨
.
It is also easy to see that C
∨∨
=
¯
C. So
Proposition. NE(X) = Nef(X)
∨
.
Theorem
(Kleinmann’s criterion)
.
If
X
is a projective scheme and
D ⊆
CaDiv
R
(X). Then the following are equivalent:
(i) D is ample
(ii) D|
NE(X)
> 0, i.e. D · γ > 0 for all γ ∈ NE(X).
(iii) S
1
∩ N E(X) ⊆ S
1
∩ D
>0
, where
S
1
⊆ N
1
(
X
)
R
is the unit sphere under
some choice of norm.
Proof.
– (1) ⇒ (2): Trivial.
– (2) ⇒ (1): If D|
NE(X)
> 0, then D ∈ int(Nef(X)).
– (2) ⇔ (3): Similar.
Proposition.
Let
X
be a projective scheme, and
D, H ∈ N
1
R
(
X
). Assume that
H is ample. Then D is ample iff there exists ε > 0 such that
D · C
H · C
≥ ε.
Proof. The statement in the lemma is the same as (D − εH) · C ≥ 0.
Example. Let X be a smooth projective surface. Then
N
1
(X)
R
= N
1
(X)
R
,
since
dim X
= 2 and
X
is smooth (so that we can identify Weil and Cartier
divisors). We certainly have
Nef(X) ⊆ NE(X).
This can be proper. For example, if
C ⊆ X
is such that
C
is irreducible with
C
2
< 0, then C is not nef.
What happens when we have such a curve? Suppose
γ
=
P
a
i
C
i
∈ NE
(
X
)
and
γ · C <
0. Then
C
must be one of the
C
i
, since the intersection with any
other curve is non-negative. So we can write
γ = a
C
C +
k
X
i=2
a
i
C
i
,
where a
C
, a
i
> 0 and C
i
6= C for all i ≥ 2.
So for any element in γ ∈ NE(X), we can write
γ = λC + γ
0
,
where λ > 0 and γ
0
= NE(X)
C≥0
, i.e. γ
0
· C ≥ 0. So we can write
NE(C) = R
+
[C] + NE(X)
C≥0
.
Pictorially, if we draw a cross section of
NE(C)
, then we get something like this:
γ · C = 0
C
To the left of the dahsed line is
NE(X)
C≥0
, which we can say nothing about,
and to the right of it is a “cone” to
C
, generated by interpolating
C
with the
elements of NE(X)
C≥0
.
Thus, [C] ∈ N
1
(X)
R
is an extremal ray of NE(X). In other words, if
λ[C] = µ
1
γ
1
+ µ
2
γ
2
where µ
i
> 0 and γ
i
∈ NE(X), then γ
i
is a multiple of [C].
In fact, in general, we have
Theorem
(Cone theorem)
.
Let
X
be a smooth projective variety over
C
. Then
there exists rational curves {C
i
}
i∈I
such that
NE(X) = NE
K
X
≥0
+
X
i∈I
R
+
[C
i
]
where
NE
K
X
≥0
=
{γ ∈ NE(X)
:
K
X
· γ ≥
0
}
. Further, we need at most
countably many C
i
’s, and the accumulation points are all at K
⊥
X
.