Ample and Nef cones
Giulio Orecchia
8 October 2014
1
Notation
In what follows, a
variety
nite type over a eld
will always denote a separated, geometrically integral scheme of
k.
Throughout the lecture, a
surface
will mean a variety of dimension 2, while a
mean a prime divisor (i.e. a closed subvariety of dimension
2
curve
on it will
1).
Line bundles
Pic(X) was dened for X a smooth projective variety over a eld
Div(X)/ PDiv(X). It turns out that the group Pic(X) is isomorphic to the
group of isomorphism classes of line bundles on X , with operation given by the tensor product
−1
and with inverse L
:= Hom OX (L, OX ). See Hartshorne's book for a detailed explanation.
Therefore we can dene an intersection form on isomorphism classes of line bundles on X by
(L1 .L2 ) := (D1 .D2 ) for [D1 ], [D2 ] the divisor classes corresponding to L1 and L2 respectively.
In the last lecture, the group
k
as the quotient
3
Ample line bundles
Denition 3.1. Let Y be a scheme and X a scheme over Y . We say that a line bundle L
on X is very ample relative to Y if for some r ∈ Z≥0 there exists an immersion (i.e. an open
immersion followed by a closed immersion) of
Remark 3.2.
X
relative to
Let
Y
Y
L∼
= i∗ (O(1)).
X proper over Y and L a very ample line bundle on
i∗ O(1) with i : X → PrY an immersion. Then i is a closed
r
II.4.4.], the image of the immersion i is closed in PY .
Y ∼
= Spec A, saying that L is very ample is equivalent to asking
global sections s0 , . . . , sr such that the corresponding Y -morphism
In the case of
L is generated by
X → PrA is an immersion.
that
Denition 3.4.
Let
X
for every coherent sheaf
F ⊗ Lm
such that
be noetherian,
isomorphic to
immersion. Indeed, by [1, Ex.
Remark 3.3.
Y -schemes i : X → PrY
This is an immediate consequence of [1, Theorem II.7.1.]
be a noetherian scheme.
F
on
X
A line bundle
there is an integer
n
L
on
X
such that for every
is called
m≥n
ample
if
the sheaf
is generated by its global sections.
Remark 3.5.
If
X
is ane, every invertible sheaf is ample, since every coherent sheaf is the
sheaf associated to a nitely generated
Γ(X, OX )-module.
1
Theorem 3.6.
bundle on
X.
X be a scheme of nite type over a noetherian ring A and let L be a line
L is ample if and only if there exists an integer n0 > 0 such that Lm is
Spec A for all m ≥ n0 .
Let
Then
very ample over
Proof on [1, Theorem II.7.6.] plus [1, Ex. II.7.5.]
Let's see some facts about line bundles. Let
type over a noetherian ring
i) If
L
is ample and
M
ii) If
L
and
M
X
in lexicographic order, is an immersion.
be a geometrically integral proper scheme over a eld
are ample, then
Lm
both
L ⊗ M. This follows from
Pr × Ps → P(r+1)(s+1)−1 , (a0 , . . . , ar ), (b0 , . . . , bs ) 7→
are two (very) ample line bundles, then so is
(. . . , ai bj , . . .)
L−1
is ample for suciently large n. This
L is ample and M is generated by global sections,
is ample.
the fact that the Segre embedding
iii) Let
of nite
Ln ⊗ M
is arbitrary, then
follows immediately by the fact that if
L⊗M
L and M be line bundles on a scheme X
A.
and
Lm
L
is torsion in
Pic(X)
and
k.
X = Spec k .
If both
L
and its dual
Indeed, for some
are very ample and hence generated by global sections.
m>0
Therefore
there are non-zero morphisms
ϕ : OX → Lm
global section of
Lm
and
ψ : Lm → OX
global section of
L−m .
Lm → Lm
and OX → OX which are both
X . Then they are isomorphisms since HomOX (Lm , Lm ) =
HomOX (OX , OX ) = Γ(X, OX ) = k . Hence ϕ is an isomorphism. Now, if OX is very
0
ample, X embeds into Pk = Spec k . So X = Spec k since Γ(X, O X ) = k .
By composing we get two morphisms
non-zero by integrality of
Let
X
be a smooth projective variety over a eld
Denition 3.7.
line bundle
We say that a divisor
O(D)
D
on
X
k.
is ample, resp. very ample, if the associated
is so.
Denition 3.8. A divisor D = ni Di on X is eective if ni ≥ 0 for all i.
Remark 3.9. Every very ample divisor is linearly equivalent to an eective one.
P
Indeed, if D
O(D) is the corresponding very ample line bundle, then Γ(X, O(D)) 6= 0.
0
If s ∈ Γ(X, O(D)) is a non-zero global section, we can associate to it a divisor D in the
following way: for every prime divisor Z , we let sZ be a generator of O(D)ηZ , where ηZ is
the generic point of Z . Then there is a unique a ∈ O ηZ such that asZ = s. If we let nZ be
P
0
the valuation of a, then D =
nZ Z is eective and linearly equivalent to D.
0
One can also recover D as the schematic support of the cokernel of the map
is very ample and
O → O(D)
given by the global section
s.
X a smooth projective surface over a eld, there exists a bilinear symmetric
( . ) on Div(X) which extends uniquely to Pic(X). The following theorem plays a central
Recall that for
form
role in this lecture.
Theorem 3.10 (Nakai-Moishezon-Kleiman).
surface
X
over a eld
k
is ample if and only if
C ⊂ X.
2
L on a smooth projective
(L. O(C)) > 0 for all curves
A line bundle
(L)2 > 0
and
For a proof see [1, Theorem V.1.10.]
As a consequence, the notion of ampleness is invariant under numerical equivalence. Therefore
we can speak about ample classes in the groups
0
Pic(X)/ Pic (X)
(since
0
Denition 3.11.
A
(L. O(C)) ≥ 0
4
nef
Num(X)
NS(X) =
another class of line bundles.
line bundle on a smooth projective surface is a line bundle
for all curves
L
such
C ⊂ X.
Riemann-Roch on surfaces
Theorem 4.1 (Riemann-Roch).
L
and
Pic (X) ⊂ Pic (X)).
It is also natural to dene in
that
Num(X) = Pic(X)/ Picn (X)
n
be a line bundle on
X.
Let
X
be a smooth projective surface over a eld
k.
Let
Then
−1
χ(X, L)) = 1/2(L.L + L.ωX/k
) + χ(X, OX ).
5
Ample and nef cone
A subset
Let
X
space
C⊂V
of a real vector space
V
is a
cone
be a smooth projective surface. The form
if R≥0 · C = C .
( . ) extends to the ρ-dimensional real vector
NS(X)R .
Denition 5.1.
The
positive cone
C+
X ⊂ NSR (X)
is the connected component of the set
C X := {α ∈ NS(X)|α2 > 0}
that contains ample
classes.
Remark 5.2.
i) Of course, by Nakai-Moishezon-Kleiman Theorem, all ample classes are contained in
C X . Notice that if H and H 0 are two ample classes, then for all α, β > 0 we have
(αH + βH 0 )2 > 0, since H 2 > 0, H 02 > 0 and H · H 0 ≥ 0. Indeed, for some m > 0,
0
the classes mH and mH contain very ample line bundles, hence eective divisors, and
0
2
H · H = 1/m (mH · mH 0 ) > 0. Therefore H and H 0 belong to the same connected
component of the aforementioned set C X .
ii) The cone
CX
−1.
e0 , e1 , . . . , eρPbe a orthonormal basis of N S(X)R
i > 0. If α =
xi ei , then α2 > 0 implies x20 > 0.
has exactly two connected components, switched by multiplication by
By Hodge Index Theorem, we can let
for which
e20 = 1
and
e2i = −1
for
Then
C+
X = {α ∈ C X : x0 > 0}.
iii) If
x ∈ C+X
is non-zero and
y ∈ CX ,
then
y ∈ C+
X
if and only if
consider the continuous map
ϕ : C X → R, y → (x.y)
3
(x.y) > 0.
To see this,
ϕ(y) = 0.
P
i>0 xi yi .
Assume
x0 y0 =
x = (x0 , . . . , xn ) for aPbasis as in ii), andPy = (y0 , . . . , yn ), then
2
2
2
x20 ≥
i>0 xi and y0 >
i>0 yi .. By Cauchy-
If
By assumption,
Schwarz, we get
X
X
X
(x0 y0 )2 = (
xi yi )2 ≤ (
x2i )(
yi2 ) < x20 y02
i>0
which is a contradiction. Hence
ϕ
respectively. Hence
Denition 5.3.
The
i>0
i>0
ϕ(C X ) ⊂ R \ {0}. Then e0 , −e0 map to R>0
C + (X) and negative on − C + (X).
and
R<0
is positive on
ample cone
Amp(X) ⊂ NS(X)R
is the set of all nite sums
The
nef cone
P
ai Li
with
Li
ample and
ai > 0.
Nef(X) ⊂ NS(X)R
is the set of all classes
α ∈ NS(X)R
with
(α.C) ≥ 0
for all curves
C ⊂ X.
Remark 5.4.
i) Clearly the ample cone is contained in the positive cone.
ii) It is not true in general that the nef cone is spanned by nef classes of line bundles;
iii) It is true that the two inequalities
Proposition 5.5.
for all curves
C
(α.α) > 0
and
(α.C) > 0
X be a smooth projective surface.
α2 ≥ 0. Hence
Let
then
If
describe the ample cone.
α ∈ NS(X) is such that (α.C) ≥ 0
Nef(X) ⊂ C + (X).
Proof.
α2 < 0. By the Hodge Index Theorem,the hyperplane
+
α cuts the positive cone C X into two parts, according to the sign of (L.α). Take L ∈ C +
X
n
such that (L.α) < 0. Then by HAG, Corollary V.1.8, L is eective for some n > 0. Since α
n
is nef, n(α.L) = (α.L ) ≥ 0, which is a contradiction.
Suppose by contradiction that
⊥
Proposition 5.6.
Let
X
be a smooth projective surface over a eld
k.
Then
Amp(X) = Int Nef(X) ⊂ Nef(X) = Amp(X)
where Int stands for interior
Proof.
Amp(X) = Int Nef(X).
H is an ample class, then (H.H) ≥ 1 and (H.C) ≥ 1 for
any curve C . Obviously Amp(X) ⊂ Nef(X), therefore Amp(X) ⊂ Int Nef(X). On the other
hand, if α ∈ Int Nef(X), then α − H is still in the nef cone for some > 0 not too big. Then
α = α − H + H is a sum of a nef and an ample class, which is easily checked to be ample
The nef cone is closed by denition, so it is enough to show that
The ample cone is open. Indeed, if
(a consequence of Proposition 5.5 and Remark 5.4, iii))
We consider now the case where
X
is a
K3
surface over a eld
k.
Then the adjunction formula
reads
(C.C) = 2g(C) − 2
for all curves
C ⊂ X.
Therefore either
(C.C) = −2,
which happens when
(C.C) ≥ 0.
The following is a corollary of Nakai-Moishezan-Kleiman Theorem.
4
C ∼
= P1k ,
or
Corollary 5.7.
A line bundle
L
on a
K3
surface
i)
(L)2 > 0,
ii)
(L.C) > 0
for every smooth rational curve
iii)
(L.H) > 0
for one ample line bundle
Proof.
H
X
over a eld
P1 ∼
= C ⊂ X,
k
is ample if and only if
and
(or, equivalently, for all of them).
L is ample, then i), ii), iii) are satised. On the other hand, for every non-rational
+
curve on X , (C.C) ≥ 0. Moreover, (C.H) > 0 for any H ample, hence C ∈ C X . By i) and
+
iii) it also follows that L ∈ C X . Now, by Remark 5.2 iii) it follows that (L.C) > 0. Therefore
(L.C) > 0 for all curves C ⊂ X , and we conclude by Nakai-Moishezon-Kleiman Theorem.
If
Corollary 5.8.
For a
K3
surface over a eld
k
one has
Amp(X) = {α ∈ C +
X : (α.C) > 0
6
for all
P1 ∼
= C ⊂ X}.
Chambers and walls
X be a K3 surface over a eld k . We have seen that for a curve C ⊂ X , (C.C) ≥ −2 and
(C.C) = −2 if and only if C is smooth and rational. We dene the subset of −2-classes of the
Neron-Severi group NS(X) (which, for a K3 surface is isomorphic to Num(X) and Pic(X)):
Let
∆ := {δ ∈ NS(X) : δ 2 = −2}
Let
L
be the class of a line bundle in
∆.
By Riemann-Roch theorem we have
−1
χ(L) = h0 (L) − h1 (L) + h2 (L) = χ(OX ) + (L)2 /2 + (L.ωX/k
)/2 = 2 + (L)2 /2.
Serre duality and triviality of
ωX/k
imply that
h0 (L) + h0 (L−1 ) ≥ 2 + (L)2 /2 = 1.
L and its dual is eective. Notice also that,
NS(X) = Pic(X), eective divisor classes are well dened in NS(X).
+
−
Then we let ∆ = ∆ ∪ ∆
where
Hence exactly one of
∆+ := {δ ∈ NS(X) : δ 2 = −2
We let
δ ⊥ ⊂ NS(X)R
Denition 6.1.
and
δ
since for a
is an eective divisor class},
be the hyperplane orthogonal to
K3-surface
∆− := −∆+ .
δ ∈ ∆+ .
The connected components of
[
C+
X\
δ⊥
δ∈∆+
are called the
chambers
Remark 6.2.
locally nite in
of
C+
X,
while the
δ⊥
are the
The union of the walls is closed in
NS(X)R .
Proposition 6.3.
walls
of the chambers.
NS(X)R ,
and the collection of the walls is
[2, Chapt. 8, Remark 2.2.]
The ample cone
Amp(X)
is a chamber of
5
C+
X.
Proof.
Let
ℵ := {α ∈ C +
X : (α.δ) > 0
for all
δ ∈ ∆+ }.
α in Amp(X) are given by the conditions α ∈ C +
X and
(α.C) > 0 for all curves in X with C.C = −2. Hence ℵ ⊂ Amp(X). On the other hand,
+
if H ∈ Amp(X), then (H.δ) > 0 for all δ ∈ ∆
since δ is an eective divisor class. So
+
Amp(X) = ℵ, which is a chamber of C X .
This is a chamber of
Now, for all
δ∈∆
C+
X.
Classes
consider the isometry on
NS(X)R
given by
sδ (x) = x + (x.δ)δ.
2
sδ (δ) = −δ and sδ = id on δ ⊥ . If x ∈ C +
X , then sδ (x) > 0.
+
hence sδ (x) ∈ C X . Therefore the sδ 's preserve the positive cone.
Then
Denition 6.4.
δ ∈ ∆}
The subgroup
W
of the orthogonal group
Moreover
O(NS(X)R )
(sδ (x).x) > 0,
generated by
{sδ :
is called Weyl group.
The Weyl group
W (∆) = ∆,
and
of chambers of
S
W preserves the union of the walls δ∈∆+ δ ⊥ . Indeed one
⊥
⊥
0
if x ∈ δ , then sδ 0 (x) ∈ sδ 0 (δ)
for all δ ∈ ∆. Therefore W
acts on the set
C+
X.
Proposition 6.5.
The cone
can check that
The Weyl group acts simply transitively on the set of chambers of
+
Nef(X) ∩ C (X)
is a fundamental domain for the action of the Weyl group on
C + (X).
Proof on [2, Chapt. 8, Prop. 2.6. and Corollary 2.11]
References
[1] Robin Hartshorne,
Algebraic Geometry
Berlin, New York: Springer-Verlag, 1977
[2] Daniel Huybrechts
C+
X.
Lectures on K3 surfaces
6