50
CHAPTER 2. SCHEMES
(Week 14)
Now we consider a special situation when X → Pn is an immersion.
Definition 2.7.4. An invertible sheaf L on a noetherian scheme X is ample if for every coherent
sheaf F on X, there is n0 > 0 such that for every n ≥ n0 , F ⊗ L n is generated by global sections.
We usually see another equivalent definition.
Theorem 2.7.5. Let X be a scheme of finite type over a neotherian ring A and L is invertible.
Then L is ample ⇔ L m is very ample over Spec A for some m > 0.
Proof. Serre’s theorem shows ⇐. Please read [Hartshorne, Thm 7.6] for the beautiful proof of
⇒.
Example 2.7.6. (a) Let X = Pnk . Then O(d) is very ample iff d > 0.
(b) Let X = P1 × P1 . Invertible sheaves are of the form O(a) £ O(b). It is ample iff a, b > 0.
(c) Consider a divisor on a curve of genus g. If deg D ≥ 2g + 1, then D is very ample. Hence
D is ample if deg D > 0. This can be proved by Riemann-Roch.
Linear systems of divisors
Before invertible sheaves were introduced, people study linear systems of divisors. It has the
advantage of been ‘more’ geometric.
Assume for the rest of the section that X is a non-singular over an algebraically closed field k.
For an invertible sheaf L on X, a nonzero section s ∈ Γ(X, L ) determines an effective divisor (s).
It carries the following property:
(a) If D is a divisor and s 6= 0 ∈ Γ(X, L (D)), then the effective divisor (s) ∼ D. Indeed, (s) is
obviously effective. Regard s as a rational function f ∈ K(X), Let D locally be defined as (Ui , fi ).
The section s is locally defined by f /(fi−1 ) = f fi . Then (s) = (f fi ) = (f ) + D, (s) ∼ D.
(b) Every effective divisor linearly equivalent to D equals some (s).
(c) Suppose X is projective. Two sections s, s0 have the same zero iff s0 = λs for some λ ∈ k ∗ .
So we can add two sections, and quotient by the k ∗ equivalence relation.
Definition 2.7.7. (a) A complete linear system on a variety is the set of all effective divisors
linearly equivalent to a given divisor D, denoted by |D|.
(b) A linear system is a linear subspace of a complete linear system.
Definition 2.7.8. A point P is a base point of a linear system d iff P is contained in the support
of every divisor in d. A linear system is called base-point-free if there is no base point.
Simple observation: Let d corresponds to the subspace V ⊆ Γ(X, L ). P is a base point iff
sP ∈ mP LP for all s ∈ V . Therefore d is base-point-free iff L is generated by global sections in V .
2.7. PROJECTIVE MORPHISMS
51
Example 2.7.9. (a) Consider the projective space Pnk . Take the divisor D = (x0 = 0). Then the
complete linear system |D| is the linear space Γ(Pnk , L (D)). It is spanned by x0 , x1 , . . . , xn , so is
n + 1 dimensional. But unfortunately we will call the linear system |D| of dimension n, which tell
us the dimension of the target projective space induced by the linear system.
in
Since the the sections x0 , . . . , xn has common zero only at x1 = · · · xn = 0 which is not a point
so |D| is base-point-free.
Pnk ,
(b) Consider a subspace V spanned by x1 , . . . , xn . It has base point at (1 : 0 : · · · : 0). Which
defines a rational map Pn 99K Pn−1 projecting from the point (1 : 0 : · · · : 0).
Proposition 2.7.10. Let f : X → Pn be a morphism determined by the linear system d. Then f
is a closed immersion iff
(1) d separates points, i.e., for any two closed points P, Q ∈ X there is a divisor D ∈ d, P ∈ D
but Q ∈
/ D.
(2) d separates tangent directions, i.e., given a closed point P and a tangent vector t ∈ TP :=
/ TP (D) = (mP,D /m2P,D )∨ . (If a morphism sends
(mP /m2P )∨ , there is a D ∈ d that P ∈ D and t ∈
two nonzero tangent vector to the same image, then it must send a non-zero vector to 0.)
Example 2.7.11. Linear system of cubics.(Taken from [Beauville] complex algebraic surfaces, prop
4.9)
Suppose p1 , . . . , pr (r ≤ 6) are points in P2k in general position, i.e. any 3 are not on a line
and any 6 are not on a conic. Let d be the linear system of cubics passing through p1 , . . . , pr . Let
π : Pr → P2 be the blow-up of p1 , . . . , pr . Let d = 9 − r. Then d defines a closed embedding
f : Pr → Pd . The image is a surface of degree d in Pd , called the del Pezzo surface of degree d.
(The degree is 9 − r since each blow-up decrease the number of intersection of two divisors by 1.)
Now we show for r = 6, f : P6 → P3 is indeed a closed embedding. Denote by Qxij the conic
passing through pk (k 6= i, j) and x.
(1) It separates points. Given x 6= y ∈ P6 . We need to find a cubic curve passing through
pi (1 ≤ i ≤ 6) and x but missing y. The union of a line with a conic is a cubic. So we can choose
a line passing through two points pi , pj and missing y, and let the conic pass through pk (k 6= i, j)
and x, and hope this conic will both miss y. It turns out, by changing the line if necessary, we can
always make this happen.
(2) It separates tangents. For a point x ∈ P2 − {p1 , . . . , p6 } and a tangent v, we can choose a
line through x that is in different direction than v and a conic. For a point x in the exceptional
divisor E1 , take the two conics Qx23 and Qx24 and they intersect at p1 , p5 , p6 and the intersection at
p1 is at least of multiplicity 2. But the total intersection should be 2 × 2 = 4, so the intersection
multiplicity at p1 is exactly 2. Then they have different tangent after blown up, so one of them is
not v.
Proj (E) and blow up
Definition 2.7.12. Let F be a quasi-coherent sheaf of graded OX algebra on a scheme X and
assume F0 = OX , F1 is a coherent OX -module, and F is locally generated by F1 as an OX -algebra.
We define the scheme Proj F as follows.
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CHAPTER 2. SCHEMES
Take an affine covering {Ui } of X and we have the natural morphism Vi = Proj SF (Ui ) → U .
It can be shown that these Vi can be glued together, and the resulting scheme is denoted by Proj F.
Definition 2.7.13. (General definition of blow-ups)
Let X be a scheme and I an ideal sheaf. Define the Rees algebra
S=
∞
M
(I t)d
d=0
as a sheaf of graded rings such that deg t = 1.
We call Proj S the blow-up of X along I , denoted by BlI X.
Proposition 2.7.14. (Universal property of blow-up) Let X be a noetherian scheme, I a coherent
sheaf of ideals, π : X̃ → X the blow-up along I . If f : Z → X is a morphism such that f −1 I · OZ
is an invertible sheaf on ideals on Z, then there exists a unique morphism g : Z → X̃.
Theorem 2.7.15. Let X be a quasi-projective variety over k. If Z is a variety and f : Z → X is a
birational morphism, then there exists a coherent sheaf of ideals I on X such that Z is isomorphic
to the blow-up of X along I .
Here is a useful tool to compute the blow-up.
Proposition 2.7.16. Let I = (f1 , . . . , fk ) be a finitely generated ideal of a ring A. Let S =
d
⊕∞
d=0 (It) . Let X = Spec A and X̃ = BlI X = Proj S.
(a) X̃ has an affine covering by Ui := Spec S(fi ) for 1 ≤ i ≤ k.
(b) S(fi t) ∼
− ui fi , . . . , fn − ui fi )/fi∞ -torson.
= A[u1 , . . . , ubi , . . . , un ]/(f1 − u1 fi , . . . , fi\
Proof. Part (a) follows from the standard fact on affine coverings of Proj S.
Part (b). Without loss of generality assume i = 1. We have
na¯
o±
¯
d
S(f1 t) =
a
∈
I
,
d
∈
Z
∼
¯
≥0
f1d
where
a
f1d
∼ 0 iff af1` = 0 for some ` ≥ 0, i.e. a is an f1∞ -torsion.
Since a ∈ I d , we can write a =
P
ai1 ···ik f1i1 · · · fkik , ai1 ···ik ∈ A and i1 + · · · + ik = d. Then
³ f ´ik
³ f ´i2
X
a
2
k
·
·
·
.
=
a
i1 ···ik
d
f
f
fi
1
1
Let ui = fi /f1 , then the above is in A[u2 , . . . , uk ]/(f2 − u1 f1 , . . . , fk − uk f1 ). Finally, remember to
quotient out the equivalence relation ∼.
e 2.
Example 2.7.17. Blow up (x, y 2 ) in C2 , denoted by C
By previous proposition, the blow-up is covered by
U1 = C[x, y, u]/(y 2 − ux)/(y 2 )∞ -torsion = C[x, y, u]/(y 2 − ux)
2.7. PROJECTIVE MORPHISMS
53
and
U2 = C[x, y, v]/(x − vy 2 )/x∞ -torsion = C[y, v].
So A2 has a A1 singularity in chart U1 . See Figure 2.1. The green curve is the exceptional divisor
e 2 → C2 . On the other hand, C
e 2 can be obtained by blowing up the point (0:0:1:0) on the
of C
2
3
projective cone y − ux in P , then cutting out the strict transform of the infinite curve defined by
y 2 − ux.
Figure 2.1: Blow up (x, y 2 ) in A2 .