Complex Algebraic Surfaces Class 7

COMPLEX ALGEBRAIC SURFACES CLASS 7

RAVI VAKIL

CONTENTS

1. How basic aspects of surfaces change under blow-up 1 2. Rational maps of surfaces, linear systems, and elimination of indeterminacy 3 3. The universal property of blowing up 4 3.1. Applications of the universal property of blowing up 4

Recap of last time. Last time we began discussing blow-ups:

¼ ¼

Ô ¾ Ë Ë = BÐ Ë : Ë ! Ë

Given , there is a surface Ô and a morphism , unique up to

´Ë fÔgµ

isomorphism, such that (i) the restriction of to is an isomorphism onto

½ ½ ½

fÔg ´Ôµ È ´Ôµ Ô Ë , and (ii) is isomorphic to . is called the exceptional divisor , and is called the exceptional divisor.

A key example, and indeed the analytic-, formal-, or etale-local situation, was given by

= A blowing up Ë at the origin, which I’ll describe again soon when it comes up in a proof.

For the deﬁnition, complex analytically, you can take the same construction. Then you

need to think a little bit about uniqueness. There is a more intrinsic deﬁnition that works

¼ d

Á Ë = ÈÖÓ j ¨ Á

¼ algebraically, let be the ideal sheaf of the point. Then d .

1. HOW BASIC ASPECTS OF SURFACES CHANGE UNDER BLOW-UP

×ØÖicØ

Ë C C

Deﬁnition. If C is a curve on , deﬁne the strict transform of to the the closure of

ÔÖÓÔ eÖ

´C \ Ë Ôµ Ë Ô j C

the pullback on , i.e. ¼ . The proper transform is given by the Ë E

£ ¼

Ç ´C µ=Ç ´C µ Ë

pullback of the deﬁning equation, so for example Ë .

×ØÖicØ ÔÖÓÔ eÖ ×ØÖicØ £

C · ÑE C Ñ Ô C = C · ÑE C

Lemma. If has multiplicity at , then , i.e. = . Ñ

Proof. The multiplicity of C being means that in local coordinates, the deﬁning equation

Ñ Ñ

has terms of degree Ñ, but not lower. (Better: the deﬁning equation lies in but not

Ñ·½

Ü Ý Ñ .) Analytically, this means that the leading term in and has degree m.

Date: Wednesday, October 23.

Í =

We do this by local calculation which will be useful in general. (Draw picture.) ¼

f´´Ü ;Ý µ; [½; Ú ]µ : Ý = Ü Ú g = ËÔ ec k [Ü ;Ý ;Ú]=Ý = Ü Ú = ËÔ ec k [Ü ;Ú]

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

¼ . The exceptional

E Ý =¼

divisor is given by ¼ (after morphism).

Í = f´´Ü ;Ý µ; [Ù; ½]µ : Ü = Ý Ùg = ËÔ ec k [Ü ;Ý ;Ù]=Ü = Ý Ù = ËÔ ec k [Ý ;Ù]

½ ½ ½ ½ ½ ½ ½ ½ ½

½ . The

E Ý =¼

exceptional divisor is given by ½ .

´Ü; Ý µ = ËÔ ec k [Ü; Ý ] ´Ü ;Ý ;Úµ 7! ´Ü ;Ý µ ´Ü ;Ý ;Úµ 7! ´Ü ;Ý µ

¼ ¼ ¼ ½ ½ ½ ½

Map down to . ¼ , .

f ´Ü; Ý µ=¼ Í f ´Ü; Ý µ= f ´Ü; Ý µ · higheÖ = f ´Ü ; Ü Úµ ·

Ñ Ñ ¼ ¼

Given a function . Pull it back to ¼ : ¡¡¡ higheÖ · .

Exercise. To see if you understood that, do the same calculation on patch 2.

¼ ¼

: Ë ! Ë Ë Ô E Ë

Theorem. Suppose is a blow up of at , with exceptional curve . Let

¼ £ £ ¼ ¼ £ ¾

D Ë D ¡ D = D ¡ D E ¡ D =¼ E = ½

D and be divisors on . Then , , .

Remark. A curve on a smooth surface that is isomorphic to È and has self-intersection

½µ is called a ´ -curve.

Proof. The ﬁrst we did yesterday. The second: by Serre’s moving lemma, we can move

Ô C Ô

D away from , then pull back. For the third: choose a curve passing through with

×ØÖicØ

C ¡ E = ½

multiplicity 1. (How to do this: hyperplane section of Ë .) Then . Also

ÔÖÓÔ eÖ ×ØÖicØ ÔÖÓÔ eÖ

¡ E =¼ C · E = C

C .as , we’re done.

¼ £

Ë ¨ Z ! Èic Ë ´D; Òµ 7! D · ÒE

Theorem. (a) There is an isomorphism Èic deﬁned by . ÆË (b) The same with Èic replaced by .

Proof. The arguments are the same for both parts, so I’ll do (a). It is surjective: the divisors E

upstairs are either E or strict transforms (which are proper transforms plus ’s). It is

D · ÒE = ¼ E Ò = ¼

injective: if , then intersect with to see that ; then apply £ to see =¼

that D .

= Ã · E Ã Ë

Theorem. Ë .

Ã Ñ E Ã = = Ã · ÑE

Ë E

Proof. Clearly Ë for some . By the adjunction formula for ,

´E µj Ã E

Ë . Taking degrees:

¾= ´ Ã · ÑE · E µ ¡ E = Ñ ½: Ë

Exercise/Remark. If you want practice with the canonical bundle in local coordinates,

Ã Ô

take a meromorphic section of Ë that has neither zero nor pole at (possible by Serre’s

f ´Ü; Ý µdÜ ^ dÝ Í

moving lemma), write it as , and pull it back to the open set ½ to see that

f ´Ü ;Ü Ú µdÜ ^ d´Ü Ú µ= f ´Ü ;Ü Ú µÜ dÜ ^ dÚ

¼ ¼ ¼ ¼ ¼ ¼ ¼ you get ¼ .

2 2. RATIONAL MAPS OF SURFACES, LINEAR SYSTEMS, AND ELIMINATION OF

INDETERMINACY

99 ÃX X

A rational map Ë , where is a variety, means a morphism from an dense open C set of Ë . Recall that a rational map from a curve to a projective variety can always be

extended to a morphism. Similarly, a rational map from a surface Ë to a projective variety

can be extended over most points; the set of indeterminacy is a ﬁnite set of points. More

: Ë 99 ÃÈ Ò ·½ precisely, given a map . This is given by sections of some line bundle. It

makes sense except where the sections are all zero. This will be in codimension 2.

´Ë F µ Ë ´Ë µ

Let F be this ﬁnite set. We’ll denote the image of , and denote it . (I’m

Ë ´C F µ

not sure we need to take the closure.) If C is a curve on , then we’ll denote the

´C µ

image of C , and denote it . Here we deﬁnitely need to take the closure.

Ë Î Ò

Now suppose you have a divisor D on . Given a subspace of dimension of

¼ £

´Ë; Ç ´D µµ ÈÎ À , we might hope to get a map to projective space . (This is called a linear

system of dimension Ò; I should have introduced this notation earlier.) If it is base point free, we do.

If it has base points, the locus could have components of dimension 1. Such a compo- Î

nent is called a ﬁxed component of the linear system Î . The ﬁxed part of is the biggest

F D F divisor contained in every element of Î . So if this ﬁxed part is , then has no ﬁxed components.

(I’m not happy with how I explained the previous paragraphs in class. I hope this is clearer.)

Lemma. If the linear systems has no fixed part, then it has only a finite number of fixed points.

Proof. Take two general sections, and look at their two zero-sets. Where do they intersect? ¾

At a bunch of points. Hence we get at most D ?

We’ve basically shown that there is a bijection between:

: Ë 99 ÃÈ ´Ë µ g

(i) f rational maps such that is contained in no hyperplane

Ë Ò g (ii) f linear systems on without ﬁxed part and of dimension

(Explain the correspondence.)

: Ë 99 Ã X

Theorem (Elimination of indeterminacy). Let be a rational map from a

¼ ¼

: Ë ! Ë

surface to a projective variety. Then there exists a surface Ë , a morphism

: Ë ! X which is the composite of a ﬁnite number of blow-ups, and a morphism f such

that the diagram

º ²

Ë 99 Ã X

is commutative. ¾

Proof. Idea: blow up ﬁxed points, show that D decreases.

È ´Ë µ

We immediately reduce to the case where X is , and isn’t contained in any

Î jD j Ò Ë hyperplane of È . Then corresponds to a linear system of dimension on ,

with no ﬁxed component. If Î has no base point, then we’re done.

Ü Ë ! Ë Ü

Otherwise, we blow up a base point , and consider ½ at (and hence a rational

Ë 99 ÃË

map ½ ). The exceptional curve is now in the ﬁxed part of the linear system, with

½ kE

some multiplicity k . So we subtract to get rid of the ﬁxed part, i.e. get a new linear

Î j D kE j : Ë 99 ÃË D = D kE

½ ½ ½ system ½ , to get the same rational map , given by . If this is a morphism, we win, otherwise we keep going.

At some point, this process must stop (and hence we win in the long run). We prove

= i i i = ¼

this is the case when D , by induction on . Base case, : the number of ﬁxed

= ¼ i > ¼

points is bounded by D , so there aren’t any. Inductive step: Now . Then

D =

we blow-up once, and we get a new surface with divisor class. On this surface, ½

¾ ¾ ¾

D kE µ´D kE µ = D k < D ´ . So by the inductive hypothesis, the process will terminate on this new surface, completing the induction.

3. THE UNIVERSAL PROPERTY OF BLOWING UP

: X ! Ë

Theorem (Universal property of blowing up). Let f be a birational morphism

Ô Ë f of surfaces, and suppose that the rational map f is undeﬁned at a point of . Then

factorizes as

f : X ! Ë := BÐ Ë ! Ë

Ô where g is a birational morphism and is the blow-up at .

Proof: next day.

3.1. Applications of the universal property of blowing up. Two theorems.

f : Ë ! Ë

Theorem (all birational morphisms factor into blow-ups). Let ¼ be a birational

: Ë ! Ë k ½;:::;Ò

k k ½

morphism of surfaces. Then there is a sequence of blow-ups k ( )

Ù : Ë ! Ë f = Æ ¡¡¡ Æ Æ Ù

½ Ò

and an isomorphism Ò such that .

f Ô Ë f

Proof. If is an isomorphism, we’re done. Otherwise, there is a point of ¼ such that

Ô Ë ! Ë =BÐ Ë

Ô ¼

is undeﬁned At , and we can factor through ½ . We can repeat this.

Ò´f µ Ò´f µ < Ò´f µ E

k k ½

If k is the number of contracted curves of :if is the exceptional

: Ë ! Ë E Ë

k k ½ divisor of k , then the preimage of in contains a curve which is contracted

f f

½ k by k but not . As the number of contracted curves can’t be negative, the process must

terminate.

: Ë 99 Ã Ë

Theorem (all birational maps can be factored into blow-ups). Let be a ¼¼

birational map of surfaces. Then there is a surface Ë and a commutative diagram

¼¼

f g

º ²

Ë 99 Ã Ë g where the morphisms f and are composites of blow-ups.

Proof. By the theorem of elimination of indeterminacy, we can ﬁnd such a diagram such g that f is a composition of blow-ups. By the Theorem above, must then be a composition of blow-ups too.

We’ve now proved some powerful stuff, so let’s take a step back and see what we now know, and how it relates to classiﬁcation.

Two surfaces are birational iff they can be be related by sequences of blow-ups. We’ll

be interested in birational classiﬁcation, but biregular classiﬁcation is very close.

f : Ë ! Ë Ò ÆË´Ë µ

If is birational which is the composition of blow-ups, then =

¼ Ò

ÆË´Ë µ ¨ Z ,so Ò is independent of the choice of blow-ups. Exercise: Use this to show

that every birational morphism from Ë to itself is an isomorphism.

i ¼

: Ë ! Ë

Fact. In a blow-up, À of the structure sheaf is preserved, i.e. if is a blow-up,

£ i i

: À ´Ç µ ! À ´Ç µ Ë then Ë is an isomorphism.

The algebraic way of proving this fact comes from the Leray spectral sequence, and the

¼ ¼

Ç Ê Ç i > ¼ = Ç = ¼

£ Ë £ Ë fact that Ë and for . This in turn requires some inﬁnitesimal analysis, in the form of “formal function theorems”. I suspect that there should be a relatively straightforward analytic proof.

In particular, by these numbers are birational invariants.

So look at what this means for the Hodge diamond. When you blow up, you add 1 to the central entry (the rank of the Neron-Severi group). Everything else is constant.

Next day: More consequences of these powerful theorems. Proof of the universal property of blowing up. Castelnuovo’s criterion for blowing down curves.