III Positivity in Algebraic Geometry

1Divisors

1.1 Projective embeddings

Our results here will be stated for rather general schemes, since we can, but at

the end, we are only interested in concrete algebraic varieties.

We are interested in the following problem — given a scheme

, can we

embed it in projective space? The first observation that

comes with a very

special line bundle

(1), and every embedding

X → P

gives pulls back to a

corresponding sheaf L on X.

The key property of

(1) is that it has global sections

, . . . , x

which

generate O(1) as an O

-module.

Definition

(Generating section)

Let

be a scheme, and

a sheaf of

modules. Let

, . . . , s

∈ H

(

X, F

) be sections. We say the sections generate

F if the natural map

n+1

i=0

→ F

induced by the s

is a surjective map of O

-modules.

The generating sections are preserved under pullbacks. So if

is embedded

into

, then it should have a corresponding line bundle generated by

+ 1

global sections. More generally, if there is any map to

at all, we can pull

back a corresponding bundle. Indeed, we have the following theorem:

Theorem. Let A be any ring, and X a scheme over A.

(i)

X → P

is a morphism over

, then

∗

(1) is an invertible sheaf

on X, generated by the sections ϕ

∗

, . . . , ϕ

∗

∈ H

(X, ϕ

∗

(1)).

(ii)

is an invertible sheaf on

, and if

, . . . , s

∈ H

(

X, L

) which generate

, then there exists a unique morphism

X → P

such that

∗

(1)

∼

and ϕ

∗

= s

This theorem highlights the importance of studying line bundles and their

sections, and in some sense, understanding these is the whole focus of the course.

Proof.

(i)

The pullback of an invertible sheaf is an invertible, and the pullbacks of

, . . . , x

generate ϕ

∗

(1).

(ii) In short, we map x ∈ X to [s

(x) : · · · : s

(x)] ∈ P

In more detail, define

= {p ∈ X : s

6∈ m

This is a Zariski open set, and

is invertible on

. Thus there is a dual

section

∨

∈ L

∨

such that

⊗ s

∨

∈ L ⊗ L

∨

∼

is equal to 1. Define

the map X

→ A

by the map

K[A

] → H

, O

)

7→ s

⊗ s

∨

Since the

generate, they cannot simultaneously vanish on a point. So

. Identifying

as the chart of

where

= 0, this defines

the desired map X → P

The theorem tells us we get a map from X to P

. However, it says nothing

about how nice these maps are. In particular, it says nothing about whether or

not we get an embedding.

Definition

(Very ample sheaf)

Let

be an algebraic variety over

, and

be an invertible sheaf. We say that

is very ample if there is a closed immersion

ϕ : X → P

such that ϕ

∗

(1)

∼

It would be convenient if we had a good way of identifying when

is very

ample. In this section, we will prove some formal criteria for very ampleness,

which are convenient for proving things but not very convenient for actually

checking if a line bundle is very ample. In specific cases, such as for curves, we

can obtain some rather more concrete and usable criterion.

So how can we understand when a map

X → P

is an embedding? If it were

an embedding, then it in particular is injective. So given any two points in

there is a hyperplane in

that passes through one but not the other. But being

an embedding means something more. We want the differential of the map to

be injective as well. This boils down to the following conditions:

Proposition.

Let

, and

a projective variety over

. Let

be an

invertible sheaf on

, and

, . . . , s

∈ H

(

X, L

) generating sections. Write

, . . . , s

for the linear span. Then the associated map

X → P

is a

closed embedding iff

(i)

For every distinct closed points

p 6

q ∈ X

, there exists

p,q

∈ V

such that

p,q

∈ m

but s

p,q

6∈ m

(ii)

For every closed point

p ∈ X

, the set

{s ∈ V | s ∈ m

}

spans the vector

space m

Definition

(Separate points and tangent vectors)

With the hypothesis of the

proposition, we say that

– elements of V separate points if V satisfies (i).

– elements of V separate tangent vectors if V satisfies (ii).

Proof.

(⇒)

Suppose

is a closed immersion. Then it is injective on points. So suppose

p 6

are (closed) points. Then there is some hyperplane

p,q

passing through

but not

. The hyperplane

p,q

is the vanishing locus

of a global section of

(1). Let

p,q

∈ V ⊆ H

(

X, L

) be the pullback of

this section. Then s

p,q

∈ m

and s

p,q

6∈ m

. So (i) is satisfied.

To see (ii), we restrict to the affine patch containing

, and

is a closed

subvariety of

. The result is then clear since

is exactly the

span of

, . . . , s

. We used

to know what the closed points of

look like.

(⇐)

We first show that

is injective on closed points. For any

p 6

q ∈ X

write the given s

p,q

∗

= ϕ

∗

for some

∈ K

. So we can take

p,q

to be given by the vanishing set

, and so it is injective on closed point. It follows that it is also

injective on schematic points. Since

is proper, so is

, and in particular

ϕ is a homeomorphism onto the image.

To show that

is in fact a closed immersion, we need to show that

→ ϕ

∗

is surjective. As before, it is enough to prove that it holds

at the level of stalks over closed points. To show this, we observe that

is trivial, so

∼

(unnaturally). We then apply the

following lemma:

Lemma. Let f : A → B be a local morphism of local rings such that

◦ A/m

→ B/m

is an isomorphism;

◦ m

→ m

is surjective; and

◦ B is a finitely-generated A-module.

Then f is surjective.

To check the first condition, note that we have

p,P

∼

p,X

∼

Now since

p,P

is generated by

, . . . , x

, the second condition is the

same as saying

p,P

→

p,X

is surjective. The last part is immediate.

Unsurprisingly, this is not a very pleasant hypothesis to check, since it requires

us to really understand the structure of

. In general, given an invertible sheaf

, it is unlikely that we can concretely understand the space of sections. It

would be nice if there is some simpler criterion to check if sheaves are ample.

One convenient place to start is the following theorem of Serre:

Theorem

(Serre)

Let

be a projective scheme over a Noetherian ring

a very ample invertible sheaf, and

a coherent

-module. Then there exists

a positive integer

(

F, L

) such that for all

n ≥ n

, the twist

F ⊗ L

generated by global sections.

The proof of this is straightforward and will be omitted. The idea is that

tensoring with

lets us clear denominators, and once we have cleared all the

denominators of the (finitely many) generators of

, the resulting sheaf

F ⊗ L

will be generated by global sections.

We can weaken the condition of very ampleness to only require the condition

of this theorem to hold.

Definition

(Ample sheaf)

Let

be a Noetherian scheme over

, and

invertible sheaf over

. We say

is ample iff for any coherent

-module

there is an

such that for all

n ≥ n

, the sheaf

F ⊗ L

is generated by global

sections.

While this seems like a rather weak condition, it is actually not too bad.

First of all, by taking

to be

, we can find some

that is generated by

global sections. So at least it gives some map to

. In fact, another theorem of

Serre tells us this gives us an embedding.

Theorem

(Serre)

Let

be a scheme of finite type over a Noetherian ring

and

an invertible sheaf on

. Then

is ample iff there exists

m >

0 such

that L

is very ample.

Proof.

(⇐)

Let

be very ample, and

a coherent sheaf. By Serre’s theorem, there

exists n

such that for all j ≥ j

, the sheafs

F ⊗ L

, (F ⊗ L) ⊗ L

, . . . , (F ⊗ L

m−1

) ⊗ L

are all globally generated. So F ⊗ L

is globally generated for n ≥ mj

(⇒)

Suppose

is ample. Then

is globally generated for

sufficiently large.

We claim that there exists t

, . . . , t

∈ H

(X, L

) such that L|

are all

trivial (i.e. isomorphic to O

), and X =

By compactness, it suffices to show that for each

p ∈ X

, there is some

t ∈ H

(

X, L

) (for some

) such that

p ∈ X

and

is trivial on

. First

of all, since

is locally free by definition, we can find an open affine

containing p such that L|

is trivial.

Thus, it suffices to produce a section

that vanishes on

X − U

but

not at

. Then

p ∈ X

⊆ U

and hence

is trivial on

. Vanishing on

is the same as belonging to the ideal sheaf

. Since

is coherent,

ampleness implies there is some large

such that

⊗ L

is generated by

global sections. In particular, since

⊗ L

doesn’t vanish at

, we can

find some

t ∈

Γ(

X, I

⊗ L

) such that

t 6∈ m

(

⊗ L

)

. Since

is a

subsheaf of O

, we can view t as a section of L

, and this t works.

Now given the

, for each fixed

, we let

}

generate

as an

-algebra. Then for large

extends to a global section

∈

Γ(

X, L

) (by clearing denominators). We can pick an

large enough

to work for all

. Then we use

, c

}

as our generating sections

to construct a morphism to

, and let

, x

}

be the corresponding

coordinates. Observe that

implies the

already generate

Now each

gets mapped to

⊆ P

, the vanishing set of

. The map

→ ϕ

∗

corresponds to the map

A[y

, y

] → O

where

is mapped to

. So by assumption, this is surjective,

and so we have a closed embedding.

From this, we also see that

Proposition.

Let

be a sheaf over

(which is itself a projective variety over

K). Then the following are equivalent:

(i) L is ample.

(ii) L

is ample for all m > 0.

(iii) L

is ample for some m > 0.

We will also frequently make use of the following theorem:

Theorem

(Serre)

Let

be a projective scheme over a Noetherian ring

, and

L is very ample on X. Let F be a coherent sheaf. Then

(i) For all i ≥ 0 and n ∈ N, H

(F ⊗ L

) is a finitely-generated A-module.

(ii)

There exists

∈ N

such that for all

n ≥ n

(

F ⊗ L

) = 0 for all

i > 0.

The proof is exactly the same as the case of O(1) on P

As before, this theorem still holds for ample sheaves, and in fact characterizes

them.

Theorem.

Let

be a proper scheme over a Noetherian ring

, and

invertible sheaf. Then the following are equivalent:

(i) L is ample.

(ii) For all coherent F on X, there exists n

∈ N such that for all n ≥ n

, we

have H

(F ⊗ L

) = 0.

Proof. Proving (i) ⇒ (ii) is the same as the first part of the theorem last time.

To prove (ii) ⇒ (i), fix a point x ∈ X, and consider the sequence

0 → m

F → F → F

→ 0.

We twist by

, where

is sufficiently big, and take cohomology. Then we have

a long exact sequence

0 → H

F(n)) → H

(F(n)) → H

(n)) → H

F(n)) = 0.

In particular, the map

(

))

→ H

(

)) is surjective. This mean at

(

) is globally generated. Then by compactness, there is a single

large

enough such that F(n) is globally generated everywhere.