2. Preliminaries
2.1. Divisors and line bundles. Let X be an irreducible complex
variety of dimension n. The group of k-cycles on X is
Zk (X ) = fZ linear combinations of subvarieties of dimension kg:
We will often denote Zk (X ) by Z n k (X ). The group of Weil divisors
is just
WDiv(X ) = Zn 1(X ):
The group of Cartier divisors is
Div(X ) := (X; KX =OX ):
Here KX denotes the sheaf of non-zero rational functions on X and
OX is the sgeaf of nonvanishing regular functions. Therefore, a Cartier
divisor D can be given by specifying an open cover Ui of X and sections
fi 2 (Ui ; KX (Ui)) such that fi=fj 2 (Ui \ Uj ; OX (Ui \ Uj )). Given
divisors D; D0 2 Div(X ), then by denition D + D0 = fUi ; fifi0g. There
is a natural homomorphism of groups
Div(X ) ! WDiv(X )
P
given by D ! [D] = ordV (D) [V ] where ordV (D) is the order of
vanishing of the fi along the codimension 1 subvariety. This homomorphism is injective when X is normal and an isomorphism when X is
smooth.
Exercise 2.1. Let X = Spec k[x; y; z]=(xy z2 ) be the quadric cone
given by the equation xy z2 . Show that WDiv(X ) 6= Div(X ). (In fact
WDiv(X ) = Z=2Z and Div(X ) = 0.
There is a short exact sequence
0 ! OX ! KX ! KX =OX ! 0:
Therefore, there is a natural homomorphism
Div(X ) = Pic(X )
= H 0(X; KX =OX ) ! H 1(X; OX ) which sends a divisor D = fUi; fig to the line bundle OX (D) = fUi \
Uj ; fi=fj g. The kernel of this map is the group of principal divisors
Princ(X ) Div(X ):
Two divisors D; D0 are linearly equivalent D D0 if D D0 2 Princ(X ).
We denote by
jDj := fD0jD0 Dg
4
the complete linear series of D. Notice that D = PH 0(X; OX (D)).
Given any subvectorspace V H 0(X; OX (D)) there is a rational map
V : X 99K PV = jV j
given by
x ! [s0 (x) : s1(x) : ::: : sn(x)]
where s0; :::; sn is a basis of V . This map is only dened away from the
base locus of V and it sends x to the hyperplane of divisors containing
x.
We say that D is very ample if D is an embedding. D is ample
if there is a positive integer m such that mD is ample.
2.2. Rules of coherent cohomology. Let X be a variety (over an
algebraically closed eld k), and F a quasi-coherent sheaf1, then there
exist k-vector spaces H i(X; F ) such that:
(1) For any homomorphism of OX -modules a : F ! G , there is
an induced k-linear map
a : H i(X; F ) ! H i(X; G );
(2) H 0(X; F ) = (X; F );
(3) If 0 ! F 0 ! F ! F 00 ! 0 is a short exact sequence, of
quasi-coherent sheaves on X , then there is a coboundary map
d : H i(X; F 00) ! H i+1(X; F 0) shuch that the sequence
H i(X; F 0) ! H i(X; F ) ! H i(X; F 00) ! H i+1(X; F 0) is exact;
(4) H i(X; F ) = 0 for all i > dim X ;
(5) If F is coherent2, X is proper then H i(X; F ) is nite dimensional and we denote its dimension by hi (X; F ).
(6) hi(Pr ; OPr (n)) = 0 for
1;
all n 2 Z and all 1 r ni r
r
+
n
0
r
r
r
h (P ; OP (n)) = n if n 0 and 0 if n < 0; h (P ; OP ( n
r 1)) = h0 (Pn; OPr (n)). (cf. [Hartshorne] III.5.1.
Recall that the Euler characteristic of a cohrent sheaf F is
X
(X; F ) = hi(X; F ):
i0
1i.e. it is locally isomorphic to the cokernel of a homomorphism of locally free
sheaves
2i.e. it is locally isomorphic to the cokernel of a homomorphism of locally free
sheaves of nite rank
5
Theorem 2.2 (Asymptotic Riemann-Roch). Let X be an irreducible
projective variety of dimension n, and D 2 Div(X ). Then (X; F OX (mD)) is a polynomial of degree at most n in m such that
n
(X; F OX (mD)) = rank(F ) Dn mn + O(mn 1):
Proof. See Hirzebruch-Riemann-Roch:
(X; F OX (mD)) = ch(F OX (mD)) Td(X ):
˜
2.3. Serre's Theorems. Recall that a coherent sheaf F is globally
generated if the homomorphism
H 0(X; F ) OX ! F
is surjective.
Theorem 2.3 (Serre). Let X be a projective scheme and D be a very
ample line bundle and F a coherent sheaf. Then there is an integer
n0 > 0 such that for all n n0 , F OX (nD) is globally generated.
Proof. [Hartshorne] II x5. The idea is that we may assume that X =
PN . Since F is coherent, it is locally generated by nitely many sections
of O. Each local section is the restriction of some global section of
OPN (n) for n 0. By compactness, we only need nitely many such
sections.
˜
Theorem 2.4 (Serre Vanishing). Let X be a projective scheme and D
be a very ample line bundle and F a coherent sheaf. Then there is an
integer n0 > 0 such that for all n n0 and all i > 0,
H i(X; F OX (nD)) = 0:
Proof. (cf. [Hartshorne] III.5.2) We may assume that X = PN (replace
F by DF ). The Theorem is clear if F is a nite direct sum of sheaves
of the form OPN (q). We can nd a short exact sequence of coherent
sheaves
0 ! K ! OPN (qi) ! F ! 0
(eg. use (2.3)). We now consider the exact sequence
H i(OPN (qi + n)) ! H i(F OPN (n)) ! H i+1(K OPN (n)) and proceed by descending induction on i so that we may assume that
hi+1(K OPN (n)) = 0 for n 0 and i 0. Since hi(OPN (qi + n)) = 0
for n 0 and i 0, we have hi(F OPN (n)) = 0 as required for
n 0 and i 0.
˜
Recall that we also have the following
6
Proposition 2.5. The following are equivalent
(1) D is ample;
(2) mD is ample for all m > 0;
(3) mD is ample for some m > 0;
(4) for any coherent sheaf F , there exists an integer m > 0 such
that F OX (nD) is globally generated.
Proof. Excercise. (See [Hartshorne] II x7).
˜
2.4. The cone of eective 1-cycles. For any D 2 Div(X ) and C =
P
ni Ci 2 Z1(X ) we dene
X
D C = ni (D Ci)
by considering the normalization i : C~i ! Ci and letting
D Ci = deg(i OX (D)):
Two Cartier divisors D; D0 are numerically equivalent D D0 if
D C = D0 C 8 C irreducible curves on X:
Two curves C; C 0 are numerically equivalent C C 0 if
C D = C 0 D 8 D 2 Div(X ):
Remark 2.6. Recall that there is a short exact sequence
0 ! Z ! OX ! OX ! 0
inducing a short exact sequence
Pic0(X ) ! Pic(X ) ! NS(X ) ! 0:
here Pic0(X ) = H 1(X; OX )=H 1(X; Z) is the group of topologically trivial line bundles and the Neron Severi group is
NS(X ) = Imm(H 1(X; OX ) ! H 2(X; Z)):
Since the group Div(X )= is a quotient of NS(X ), it is a free abelian
group of nite rank which we denote by (X ) the Picard number of
X.
We will adopt the following notation:
N1(X ) = (Z1(X )= ) R;
N 1 (X ) = (Z 1(X )= ) R;
so that (X ) = dimR N1 (X ) = dimR N 1 (X ): We dene the cone of
eective 1-cycles to be
NE (X ) N1(X )
7
P
to be the cone generated by f ni Ci s: t: ni 0g. Given a morphism
f : X ! Y and an irreducible curve C X , we let f(C ) = df (C )
where d = deg(C ! f (C )). If f (C ) is a point, then we set fC = 0.
One sees that
f D C = D fC 8D 2 Div(Y ):
Extending by linearity, we get an injective homomorphism
f : N 1 (Y ) ! N 1 (X )
and surjective homomorphisms
f : N1 (X ) ! N1 (Y ); NE (X ) ! NE (Y ):
Let W be a subvariety of X of codimension i, and D 2 Div(X ) then we
may also dene the intersection numbrs Di W 2 Z. Suppose that this
has been done for all j > i, then pick a very ample divisor H such that
D + H is also very ample. Then (D + H )i W is just the degree of the
image of W via H 0(X;OX (D+H )) and Dj H i j W have been previously
dened (as the codimension of H i j W is j > i). Hence Di W is also
dened. We have the following important results:
Theorem 2.7. [Nakai-Moishezon criterion] Let D be a Cartier divisor
on a projective scheme X , then D is ample if and only if for any 0 i n 1 and any subvariety W of codimension i, one has Di W > 0.
It is enough to assume that X is proper.
Proof. (cf. [KM98] 1.37) We may assume that X is irreducible. Clearly,
if D is ample, then Di W > 0. For the converse implication, we
proceed by induction on n = dim X . When dim X = 1, the Theorem
is obvious. So we may assume that DjZ is ample for all for all proper
closed subschemes Z ( X .
Claim 1. h0(X; OX (D)) > 0 for some k > 0 (actually (D) = n).
We choose a very ample divisor B such that D + B is very ample. Let
A 2 jD + B j be a general member. Consider the short exact sequence
0 ! OX (kD B ) ! OX (kD) ! OB (kD) ! 0:
Since OB (kD) is ample, for all k 0 we have that hi(B; OB (kD)) = 0
for all i > 0, and so
hi(OX (kD B )) = hi(OX (kD)) for i 2 and k 0:
The same argument applied to the short exact sequence
0 ! OX (kD B ) ! OX ((k + 1)D) ! OA((k + 1)D) ! 0
shows that
hi(OX (kD B )) = hi (OX ((k + 1)D)) for i 2 and k 0:
8
Putting this toghether, we see that for i 2 and k 0, one has
hi(OX (kD)) = hi (OX ((k + 1)D)) and so the number hi(OX (kD)) is
constant. But then for k 0
h0 (X; OX (kD)) h0 (X; OX (kD)) h1(X; OX (kD))
= (X; OX (kD)) + (constant) = Dn=n kn + O(k 1):
Claim 2. OX (kD) is generated by global sections for some k > 0.
Fix a non-zero section s 2 H 0(X; OX (mD)). Let S be the divisor
dened by s. Consider the short exact sequence
0 ! OX ((k 1)mD) ! OX (kmD) ! OS (kmD) ! 0:
By induction OS (kmD) is generated by global sections and so it sufces to show that for k 0, the homomorphism H 0(X; OX (kmD)) !
H 0(S; OS (kmD)) is surjective. Arguing as in Claim 1, one sees that
h1(X; OX (kmD)) is a decreasing sequence and is hence eventually constant as required.
Conclusion of the proof. kD : X ! PN is a nite morphism
for k 0. If infact C is a curve contracted by kD , then kD C =
OPN (1) kD (C ) = 0 which contradicts C D > 0. The Theorem now
follows as kD = kD OPN (1) is ample (as it is the pullback of an ample
˜
line bundle via a nite map).
We have the following useful corollary:
Theorem 2.8. [Kleiman's Theorem] Let X be a proper variety, D a
nef divisor. Then Ddim Z Z 0 for all irreducible subvarieties Z X .
Proof. (cf. [Lazarsfeld05] 1.4.9) We assume that X is projective (Chow's
Lemma) and irreducible. When dim X = 1, the Theorem is clear. By
induction on n = dim X , we may assume that
Ddim Z Z 0 8Z X irreducible of dim Z < n;
and we must show that Dn 0. Fix H an ample divisor and consider
the polynomial
P (t) := (D + tH )n 2 Q[t]:
We must show that P (0) 0. For 1 k n, the coecient of tk is
D n k H k 0:
Assume that P (0) < 0, then one sees that P (t) has a unique real root
t0 > 0.
For any rational number t > t0, one sees that
(D + tH )dim Z Z > 0 8Z X irreducible of dim Z n;
9
and so by (2.7), D + tH is ample. We write
P (t) = Q(t) + R(t) = D (D + tH )n 1 + tH (D + tH )n 1:
As D + tH is ample for t > t0 , one has that (D + tH )n 1 is an eective 1cycle, so Q(t) > 0 for all rational numbers t > t0 and so Q(t0 ) 0. One
sees that all the coecients of R(t) are non-negative and the coecient
of tn is H n > 0. It follows that R(t0 ) > 0 and so P (t0) > 0 which is
˜
the required contradiction.
Exercise 2.9. Let X be a projective variety, H an ample divisor on
X . A divisor D on X is nef if and only if D + H is ample for all
rational numbers > 0.
Exercise 2.10. Let X be a projective variety, H an ample divisor on
X . A divisor D on X is ample if and only if there exists an > 0 such
that
DC H C
for all irreducible curves C X .
Theorem 2.11. [Nakai's Criterion] Let D be a divisor on a projective
scheme X . D is ample if and only if D Z > 0 for any Z 2 NE (X )
f0g.
˜
Proof. Exercise.
Theorem 2.12. [Seshadri's Criterion] Let D be a divisor on a projective scheme X . D is ample if and only if there exists an > 0 such
that
C D multx(C )
for all x 2 C X .
Proof. Assume that D is ample, then there exists an integer n > 0 such
that nD is very ample, but then
nD C multx(C )
for all x 2 C X .
For the reverse implication, we proceed by induction. Therefore, we
may assume that for any irreducible subvariety Z ( X , the divisor DjZ
is ample and so Ddim Z Z > 0. By (2.7), it is enough to show that
Dn > 0. Let
: X0 ! X
be the blow up of X at a smooth point. Then D E is nef. In fact,
for any curve C 0 X 0 we either have C = (C 0) is a curve and then
(D E ) C 0 = D C multx C 0;
10
or (C 0) = x and then since C 0 E = Pn 1 and OE (E ) = OPn 1 ( 1)
(D E ) C 0 = deg C 0 > 0:
But then, by (2.8)
(D E )n = Dn n 0
and this completes the proof.
˜
11