3Projective varieties
III Positivity in Algebraic Geometry
3.1 The intersection product
First, we have to define
NS
(
X
), and to do so, we need to introduce the intersection
product. Thus, given a projective variety
X
over an algebraically closed field
K
with
dim X
=
n
, we seek a multilinear symmetric function
CaDiv
(
X
)
n
→ Z
,
denoted
(D
1
, . . . , D
n
) 7→ (D
1
· . . . · D
n
).
This should satisfy the following properties:
(i)
The intersection depends only on the linear equivalence class, i.e. it factors
through Pic(X)
n
.
(ii)
If
D
1
, . . . , D
n
are smooth hypersurfaces that all intersect transversely, then
D
1
· · · · · D
n
= |D
1
∩ · · · ∩ D
n
|.
(iii)
If
V ⊆ X
is an integral subvariety of codimension 1, and
D
1
, . . . , D
n−1
are
Cartier divisors, then
D
1
· . . . · D
n−1
· [V ] = (D
1
|
V
· . . . · D
n−1
|
V
).
In general, if V ⊆ X is an integral subvariety, we write
D
1
· . . . · D
dim V
· [V ] = (D
1
|
V
· . . . · D
dim V
|
V
).
There are two ways we can define the intersection product:
(i)
For each
D
i
, write it as
D
i
∼ H
i,1
− H
i,2
with
H
i,j
very ample. Then
multilinearity tells us how we should compute the intersection product,
namely as
X
I⊆{1,...,n}
(−1)
|I|
Y
i∈I
H
i,2
·
Y
j6∈I
H
j,1
.
By Bertini, we can assume the
H
i,j
intersect transversely, and then we just
compute these by counting the number of points of intersection. We then
have to show that this is well-defined.
(ii)
We can write down the definition of the intersection product in a more
cohomological way. Recall that for n = 2, we had
D · C = χ(O
X
) + χ(O
X
(−C − D)) − χ(O
X
(−C)) − χ(O
X
(−D)).
For general
n
, let
H
1
, . . . , H
n
be very ample divisors. Then the last property
implies we have
H
1
· · · · · H
n
= (H
2
|
H
1
· . . . · H
n
|
H
1
).
So inductively, we have
H
1
· · · · · H
n
=
X
I⊆{2,...,n}
(−1)
|I|
χ
H
1
, O
H
1
−
X
i∈I
H
i
!!
But we also have the sequence
0 → O
X
(−H
1
) → O
X
→ O
H
1
→ 0.
Twisting this sequence shows that
H
1
· · · · · H
n
=
X
I⊆{2,...,n}
(−1)
|I|
χ
O
X
−
X
i∈I
H
i
− χ
O
X
−H
1
−
X
i∈I
H
i
=
X
J⊆{1,...,n}
(−1)
|J|
χ
O
X
−
X
i∈J
H
i
We can then adopt this as the definition of the intersection product
D
1
· · · · · D
n
=
X
J⊆{1,...,n}
(−1)
|J|
χ
O
X
−
X
i∈J
D
i
!!
.
Reversing the above procedure shows that property (iii) is satisfied.
As before we can define
Definition
(Numerical equivalence)
.
Let
D, D
0
∈ CaDiv
(
X
). We say
D
and
D
0
are numerically equivalent, written
D ≡ D
0
, if
D ·
[
C
] =
D
0
·
[
C
] for all integral
curves C ⊆ X.
In the case of surfaces, we showed that being numerically equivalent to zero
is the same as being in the kernel of the intersection product.
Lemma. Assume D ≡ 0. Then for all D
2
, . . . , D
dim X
, we have
D · D
2
· . . . · D
dim X
= 0.
Proof.
We induct on
n
. As usual, we can assume that the
D
i
are very ample.
Then
D · D
2
· · · · D
dim X
= D|
D
2
· D
3
|
D
2
· · · · · D
dim X
|
D
2
.
Now if D ≡ 0 on X, then D|
D
2
≡ 0 on D
2
. So we are done.
Definition
(Neron–Severi group)
.
The Neron–Severi group of
X
, written
NS(X) = N
1
(X), is
N
1
(X) =
CaDiv(X)
Num
0
(X)
=
CaDiv(X)
{D | D ≡ 0}
.
As in the case of surfaces, we have
Theorem
(Severi)
.
Let
X
be a projective variety over an algebraically closed
field. Then
N
1
(
X
) is a finitely-generated torsion free abelian group, hence of
the form Z
n
.
Definition (Picard number). The Picard number of X is the rank of N
1
(X).
We will not prove Severi’s theorem, but when working over
C
, we can indicate
a proof strategy. Over
C
, in the analytic topology, we have the exponential
sequence
0 → Z → O
an
X
→ O
∗,an
X
→ 0.
This gives a long exact sequence
H
1
(X, Z) → H
1
(X, O
an
X
) → H
1
(X, O
∗,an
X
) → H
2
(X, Z).
It is not hard to see that
H
i
(
X, Z
) is in fact the same as singular cohomology,
and
H
1
(
O
∗,an
X
) is the complex analytic Picard group. By Serre’s GAGA theorem,
this is the same as the algebraic
Pic
(
X
). Moreover, the map
Pic
(
X
)
→ H
2
(
X, Z
)
is just the first Chern class.
Next check that the intersection product [
C
]
·D
is the same as the topological
intersection product [
C
]
·c
1
(
D
). So at least modulo torsion, we have an embedding
of the Neron–Severi group into H
2
(X, Z), which we know is finite dimensional.
In the case of surfaces, we had the Riemann–Roch theorem. We do not have
a similarly precise statement for arbitrary varieties, but an “asymptotic” version
is good enough for what we want.
Theorem
(Asymptotic Riemann–Roch)
.
Let
X
be a projective normal variety
over
K
=
¯
K
. Let
D
be a Cartier divisor, and
E
a Weil divisor on
X
. Then
χ
(
X, O
X
(
mD
+
E
)) is a numerical polynomial in
m
(i.e. a polynomial with
rational coefficients that only takes integral values) of degree at most
n
=
dim X
,
and
χ(X, O
X
(mD + E)) =
D
n
n!
m
n
+ lower order terms.
Proof.
By induction on
dim X
, we can assume the theorem holds for normal
projective varieties of dimension
< n
. We fix
H
on
X
very ample such that
H
+
D
is very ample. Let
H
0
∈ |H|
and
G ∈ |H
+
D|
be sufficiently general. We
then have short exact sequences
0 → O
X
(mD + E) → O
X
(mD + E + H) → O
H
0
((mD + E + H)|
H
0
) → 0
0 → O
X
((m − 1)D + E) → O
X
(mD + E + H) → O
G
((mD + E + H)|
G
) → 0.
Note that the middle term appears in both equations. So we find that
χ(X, O
X
(mD + E)) + χ(H
0
, O
H
0
((mD + E + H)|
H
))
= χ(X, O
X
((m − 1)D + E)) + χ(H
0
, O
G
((mD + E + H)|
G
))
Rearranging, we get
χ(O
X
(mD − E)) − χ(O
X
((m − 1)D − E))
= χ(G, O
G
(mD + E + H)) − χ(H
0
, O
H
0
(mD + E + H)).
By induction (see proposition below) the right-hand side is a numerical polyno-
mial of degree at most n − 1 with leading term
D
n−1
· G − D
n−1
· H
(n − 1)!
m
n−1
+ lower order terms,
since
D
n−1
· G
is just the (
n −
1) self-intersection of
D
along
G
. But
G − H
=
D
.
So the LHS is
D
n
(n − 1)!
m
n−1
+ lower order terms,
so we are done.
Proposition.
Let
X
be a normal projective variety, and
|H|
a very ample
linear system. Then for a general element
G ∈ |H|
,
G
is a normal projective
variety.
Some immediate consequences include
Proposition. Let X be a normal projective variety.
(i) If H is a very ample Cartier divisor, then
h
0
(X, mH) =
H
n
n!
m
n
+ lower order terms for m 0.
(ii) If D is any Cartier divisor, then there is some C ∈ R
>0
such that
h
0
(mD) ≤ C · m
n
for m 0.
Proof.
(i)
By Serre’s theorem,
H
i
(
O
x
(
mH
)) = 0 for
i >
0 and
m
0. So we apply
asymptotic Riemann Roch.
(ii)
There exists a very ample Cartier divisor
H
0
on
X
such that
H
0
+
D
is
also very ample. Then
h
0
(mD) ≤ h
0
(m(H
0
+ D)).