Dave Bayer and David Eisenbud

Home , David Eisenbud

RIBBONS AND THEIR CANONICAL EMBEDDINGS

Abstract. We study double structures on the pro jective line and on certain other

varieties, with a view to having a nice family of degenerations of curves and K3

surfaces of given genus and Cli ord index. Our main interest is in the canonical

emb eddings of these ob jects, with a view toward Green's Conjecture on the free

resolutions of canonical curves. We give the canonical embeddings explicitly, and

exhibit an approach to determining a minimal free resolution.

Contents

Ribb ons and their morphisms

(1) General ribb ons and their morphisms

(2) Rational ribb ons 1: the restricted cotangent sequence

(3) Rational ribb ons 2: gluing

(4) Classi cation of line bundles

Canonically emb edded ribb ons

(5) The canonical emb edding of a ribb on

(6) Canonical ribb ons are smooth points of Hilb

(7) Ribb ons on surfaces

(8) Free resolutions

(9) App endix: Osculating bundles of the rational normal curve

Both authors are grateful to the NSF for partial supp ort during the preparation of this pap er. 1

2 DAVE BAYER AND DAVID EISENBUD

Intro duction

What is the limit of the canonical mo del of a smo oth curve as the curve degen-

erates to a hyp erelliptic curve? \A ribb on" | more precisely \a ribb on on P " |

may b e de ned as the answer to this riddle. A ribb on on P is a double structure on

the pro jective line. Such ribb ons represent a little-studied degeneration of smo oth

curves that shows promise esp ecially for dealing with questions ab out the Cli ord

indices of curves.

The theory of ribb ons is in some resp ects remarkably close to that of smo oth

curves, but ribb ons are much easier to construct and work with. In this pap er we

discuss the classi cation of ribb ons and their maps. In particular, we construct

the \holomorphic di erentials" | sections of the canonical bundle { of a ribb on,

and study prop erties of the canonical emb edding. Aside from the genus, the main

invariant of a ribb on is a number we call the \Cli ord index", although the de nition

for it that we give is completely di erent from the de nition for smo oth curves. This

name is partially justi ed here, and but much more so by two subsequent works:

In the pap er of Fong [1993] a strong smo othing result for ribb ons is proved. In the

pap er of Eisenbud-Green [1994] it is shown that the Cli ord index of a ribb on may

be re-expressed in terms of a certain notion of generalized linear series, and the

semicontinuity of the Cli ord index as a smo oth curve degenerates to a ribb on is

established. Together, these results imply that any ribb on may be deformed to a

smo oth curve of the same Cli ord index.

Our original motivation for studying ribb ons came from an attack on a conjecture

of Mark Green concerning the free resolution of a canonical curve. Before stating

the conjecture, we intro duce two notions. If

I S = k [x ;:::;x ]

0 g 1

is the homogeneous ideal of a canonically emb edded curve C of arithmetic genus g ,

then the free resolution of S=I is known to have the form

b a

g 3 g 3

(g +1) ! (g +2) S 0 ! S (g 1) ! S

b a

i i

(i 2) ! ::: (i 1) S ::: ! S

a b

1 1

! S (2) S (3) ! S ! S=I ! 0

with a = b for all i. In this situation the free mo dules notated S (i

g 2i i

1) ab ove form a sub complex, which we think of as the \two-linear part" of the

resolution, since it b egins with the quadrics in the ideal of the canonical curve and

continues with matrices of linear forms. Because of Green's conjecture, which we

are ab out to state, we will de ne the resolution Clifford index of C to be

the length of the 2-linear part of this resolution | that is, the largest i for which

a 6=0. i

RIBBONS AND THEIR CANONICAL EMBEDDINGS 3

By contrast, the usual Clifford index of a smo oth curve C of genus g 3 is

de ned as the maximum, over all line bundles L on C such that h (L) > 1 and

h (L) > 1 of the quantity

def

CliffL = degree L 2(h (L) 1)

0 1

= g +1 h (L) h (L):

With this terminology, Green's Conjecture on canonical curves is the

assertion that the Cli ord index and the resolution Cli ord index agree for smo oth

curves over an algebraically closed eld of characteristic 0. In terms of the new

Cli ord index we de ne for a ribb on, we make the

Canonical ribb on conjecture. The resolution Cli ord index and the Cli ord

index agree for ribbons over a eld of characteristic 0.

Because of the smo othing results of Fong [1993] and Eisenbud-Green [1994 Prop o-

sition 2.3], a pro of of our conjecture for some ribb on of eachgenus and Cli ord index

would imply Green's conjecture for a generic curve of each Cli ord index (that is,

for a generic curve in some comp onent of the lo cus of curves of each Cli ord index).

The restriction to characteristic 0 is really necessary in b oth cases, since examples

of Schreyer for smo oth curves and examples given below for ribb ons show that the

conjectures sometimes fail in nite characteristic.

Perhaps the most imp ortant di erence between the case of ribb ons and in the

case of smo oth curves is that two smo oth curves of the same Cli ord index and

genus may have di erent graded b etti numb ers, but the graded betti numbers of a

ribb on are completely determined by the genus and Cli ord index. This is b ecause

all ribb ons of given genus and Cli ord index are hyp erplane sections of a particular

"K3 carp et" { a double structure on a 2-dimensional rational normal scroll (at

least over an algebraically closed eld of characteristic 0, this is the unique double

structure on the scroll with trivial canonical bundle that can be emb edded in the

ambient space of the usual emb edding of the scroll { see Hulek and Van de Ven,

[1985]). K3 carp ets are the sub ject of a planned pap er by Eisenbud and Schreyer.

Aside from this, what we know ab out the canonical ribb on conjecture is rather

similar to what we know of Green's conjecture itself. We can prove the analogues of

No ether's and Petri's Theorems, which deal with the cases of Cli ord index 1 and

2. We can also prove, by machine computation, that the conjecture is true in all

cases up to genus 12 (see the table of results at the end of section 8). The pro ofs of

these sp ecial cases are quite di erent in the case of ribb ons, and are in a sense more

direct and algebraic, than in the case of smo oth curves, so that we are hop eful that

the study of ribb ons will be useful in further work on Green's conjecture.

We next discuss the material of this pap er in more detail: The rst section b elow

is devoted to the general theory of ribb ons. Here we work with double structures

on a more general reduced scheme D . First we classify the ribb ons on D by certain

extensions of the sheaf of di erentials of D (this familiar idea go es back at least

4 DAVE BAYER AND DAVID EISENBUD

to Lichtenbaum-Schlessinger [1967]). Next we describ e morphisms: given a mor-

phism from D to another scheme X ,we explain what data is necessary to describ e

morphisms from the ribb on to X . From this description we show how to tell when

the extended morphism is a closed immersion. In case X is a pro jective space, we

further explain how to tell whether the image of the ribb on is \arithmetically of

depth 2", the analogue of the condition \pro jectivly normal" for smo oth vari-

eties. Finally we show that the morphisms from the ribb on to another ribb on on D

which induce the identity on D are precisely those obtained by blowing up closed

subschemes of D .

For most of the rest of the pap er we sp ecialize to the case where the underlying

scheme D is the pro jective line over an algebraically closed eld k (although many

of our results could be generalized at least to the case where D is a nonsingular

curve over an arbitrary eld). We will call such a ribb on a rational ribbon, or

simply a ribb on when the context is clear.

In section 2 we sp ecialize the classi cation theory to rational ribb ons. Here the

two fundamental invariants are the (arithmetic) genus and the Cli ord index. The

latter is de ned in terms of the restricted cotangent sequence. An alternate, and

p erhaps the quickest de nition is the following: We say that a ribb on is split, or

1 2

\hyp erelliptic" if it is isomorphic to P Sp ec k []=( ) and we de ne the Cli ord

index of a ribb on C as the minimum number of blow-ups of C at reduced p oints

of C necessary to obtain a split ribb on. Some preliminary evidence is given that

red

this notion of Cli ord index is the \right" one, in the sense that it parallels the

prop erties of the Cli ord index for smo oth curves. For example it is shown that

a ribb on of Cli ord index a has a \generalized linear series" of dimension 1 and

degree a +2 in a suitable sense. Much further justi cation may be found in the

rest of the pap er. Thus the theory of ribb ons gives one direct access to curves of

arbitrary genus and Cli ord index.

In section 3 we present a di erent view of the construction and classi cation

of ribb ons, this time by gluing together ribb ons on the ane line (such a ribb on

is necessarily of the form Sp ec k [s; ]=() .) We give the translation between the

necessary \gluing data" and the data of the restricted cotangent sequence of the

ribb on. This version of the classi cation, though somewhat computational, is nec-

essary for our treatment of line bundles, and in particular for our computation of

the canonical emb edding of the ribb on.

In section 4 we discuss line bundles and their global sections on a ribb on. The

Picard group of line bundles on a ribb on of genus g is simply k Z, where the

second factor is given by the degree of the restriction of the line bundle to P (equal

to half the degree of the line bundle itself.) The line bundles of degree 0 form a

formally principle homogeneous space under the group H (O (g 1)) k , and

this accounts for the rst factor. The global sections of line bundles are computed

in terms of an exact sequence coming from restriction to P .

After these preliminaries, we turn to the main concern of the pap er, the canonical

emb edding of a ribb on. The canonical bundle is discussed in section 5, using the

RIBBONS AND THEIR CANONICAL EMBEDDINGS 5

theory develop ed in sections 3 and 4 to identify the global sections. We prove

\No ether's Theorem for Ribb ons": the canonical series provides an emb edding

of any non-hyp erelliptic ribb on, and the canonical image is arithmetically Cohen-

Macaulay (and thus Gorenstein). (One of the pro ofs we give of this fact involves

knowing the structure of the normal bundle of the rational normal curve explicitly.

This structure is folklore, but we know no reference; we provide a pro of, together

with the corresp onding results for all the osculating bundles and their quotients,

in an app endix at the end of the pap er.) The canonical emb edding gives a third

view of the the classi cation of ribb ons: giving a non-hyp erelliptic ribb on of genus

g is the same as giving a line bundle contained in the normal bundle of the rational

normal curve of degree g 1.

In section 6 we present a result obtained jointly with Jo e Harris which shows

that ribb ons always represent smo oth points on the Hilb ert scheme of canonically

emb edded curves.

Unlike the case of smo oth curves, it is p ossible to deal with canonical emb eddings

of ribb ons by induction on the genus. To do this, we prove in section 7 that the

g 2 g 1

image in P of the pro jection of a canonically emb edded ribb on in P from

a p oint on the ribb on is the canonical embedding of the ribb on obtained from C

by blowing up the p oint. This also leads to an easy pro of of one inequality of the

canonical ribb on conjecture: The resolution Cli ord index is always the Cli ord

index.

In this section we also show that the only nonhyp erelliptic ribb on that can be

emb edded in a smo oth surface is the double conic in P . This suggests one reason

why the theory of ribb ons has not b een pursued so much b efore: The double struc-

tures on P that one sees most often are all split ribb ons, and thus not of much

interest.

We have already mentioned that all ribb ons are hyp erplane sections of suitable

(nonreduced) K3 surfaces. In the last section of the pap er we give the part of the

theory of these K3 carp ets which is most relevant to the canonical ribb on conjec-

ture, explaining in particular how to construct a nonminimal free resolution for the

homogeneous co ordinate ring of a ribb on, and | conjecturally | how to make it

minimal. We include some numerical evidence, pro duced by the program Macaulay

of Bayer and Stillman [1990] for our conjectures, and thus for Green's Conjecture.

A quasi-mathematical remark: Each of the p eople who has worked on Green's

Conjecture probably has his/her own favorite nonmimal resolution of some degen-

erate curve of genus g, and a conjecture ab out minimalizing it that would imply

at least the generic case of Green's conjecture. So far as we know the metho d of

ribb ons is the only one that gives such p ossibilities for the generic curve of each

Cli ord index. There are several ways of making nonmiminal resolutions of rib-

b ons, and attempting to minimalize them. The metho d of K3 carp ets, presented in

section 8, seems to us the one with the fewest choices involved.

We are grateful as always to Jo e Harris, for numerous helpful comments and

suggestions, as well as for anumber of sp eci c results, which are attributed to him

6 DAVE BAYER AND DAVID EISENBUD

b elow.

Throughout this paper, we work over a xed eld k . By a scheme we shal l mean

a scheme of nite type over k .

RIBBONS AND THEIR CANONICAL EMBEDDINGS 7

1. General ribb ons and their morphisms

We b egin with some basic de nitions and remarks. Throughout this section, D

will denote a reduced connected scheme over the ground eld k .

A ribbon on D is a scheme C equipp ed with an isomorphism D ! C ; the

red

reduced scheme of C; such that the ideal sheaf L of D in C satis es

L =0:

Because of this condition, L may be regarded as a sheaf on D , and we further

require that

L is a line bundle on D:

Note that the subscheme D is determined by C as C : The line bundle L is the

red

conormal bundle of D in C .

A ribbon is simply a scheme C which is a ribb on on C .

red

We shall say that the the ribb on C is split if the inclusion D ,! C admits a

retraction C ! D .

The following result is an elementary but imp ortant sp ecial case of the Classi -

cation Theorem b elow:

Prop osition 1. Given a reduced connected scheme D and a line bund le L on D

there is a unique split ribbon on D with conormal bund le L:

Proof. If we embed D in the total space X of L, then the rst in nitesimal neigh-

borhood of D in X is a split ribb on on D with conormal bundle L.

Toprove uniqueness, let C be any split ribb on with conormal bundle L. Because

C is split, the natural exact sequence

0 ! L ! O ! O ! 0

C D

is a sequence of O -mo dules. It is split as a sequence of O -mo dules because the

D D

identity elementof O lifts the identity elementof O . Because L = 0, the algebra

C D

structure of O is determined by the mo dule structure of L.

There are also lots of ribb ons that are not split|we shall see in the next section

that the simplest example is given by the quartic plane curve whose equation is

the square of that of a nonsingular conic in P . To classify the nonsplit ribb ons,

we will say that two ribb ons C and C on D are isomorphic over D if there

is an isomorphism between them that extends the identity on D: More generally,

a morphism C ! C over D is by de nition a morphism extending the identity

morphism D ! D .

Given a ribb on C on D with conormal bundle L we de ne the restricted

cotangent sequence of C to be the natural short exact sequence

0 !L ! j ! ! 0:

C D D

8 DAVE BAYER AND DAVID EISENBUD

The restricted cotangent sequence de nes the extension class

e 2 Ext ( ; L)

C D

of C .

The following classi cation is closely related to the ideas of Lichtenbaum and

Schlessinger [1967]:

Theorem 1.2 (Classi cation Theorem). Given any line bund le L on a reduced

connected scheme D , and any class

e 2 Ext ( ; L)

there is a unique ribbon C on D with e = e :

2 Ext ( ; L) is another class, corresponding to a If D is proper over k and e

0 0 0

C i e = ae for some a 2 k . ribbon C , then C

Proof. Let d : O ! be the canonical derivation, and consider an extension

D D

e : 0 !L!E ! ! 0:

De ne O as a sheaf of ab elian groups to be the pullback

O ! O

C D

? ?

y y

E !

so that we have the commutative diagram

0 ! L ! O ! O ! 0

C D

? ?

y y

e : 0 ! L ! E ! ! 0:

We make O into a sheaf of k {algebras as follows: if a ;a are sections of O over

C 1 2 D

an op en set U of D, and x ;x are sections of E over U with

1 2

da = x

i i

so that each (a ;x ) is a section of O on U , we de ne

i i C

(a ;x )(a ;x )= (a a ;a x + a x );

1 1 2 2 1 2 1 2 2 1

RIBBONS AND THEIR CANONICAL EMBEDDINGS 9

the last b eing again a section of O b ecause d is a derivation. We maynow de ne

C =SpecO

and check that it is a ribb on on D . It is also easy to check that

d : O !E

is the universal k -linear derivation of O to an O -mo dule. Thus d is the restriction

C D

of the universal k -linear derivation, and

E j

C D

is the restriction of the mo dule of di erentials . This shows that C is a ribb on

with extension class e, proving the rst part of the Theorem.

To prove the second part, note rst that if C and C are ribb ons over D then a

morphism f : C ! C over D is a map f : O ! O of sheaves of k -algebras

C C

inducing the identityonO . Such a map induces a map of the restricted cotangent

sequences

0 ! L ! j ! ! 0

C D D

? ?

y y

0 ! L ! j ! ! 0

C D D

where is the map induced by df and is the map induced by . If f is an

isomorphism, then the is an isomorphism and it follows that is to o. Since D is

reduced, connected, and prop er over k , and L is a line bundle, the only automor-

phisms of L are the elements of k , so 2 k . The given map of exact sequences

corresp onds to the map induced by on Ext ( ; L), so that e and e di er byan

element of k as required.

Conversely,any element of k , regarded as a map

; : L! L

induces an isomorphism of exact sequences as ab ove. Using the constructions of C

and C from their restricted cotangent sequences, we may reverse the pro cess and

0 0

see that if e and e di er by an element of k then C C .

Corollary 1.3. If D is a smooth ane variety over k, then every ribbon on D is

split.

Proof. In this case is a pro jective O {mo dule, so

D D

Ext ( ; L)= 0:

D D

10 DAVE BAYER AND DAVID EISENBUD

Corollary 1.4. If D is reduced, connected, and proper over k , then the set of

nonsplit ribbons on D with conormal bund le L; up to isomorphism over D; is

in one{to{one correspondence with the points of the projective space of lines in

Ext ( ; L):

So far our classi cation has been up to \isomorphism over D ", that is, up to

isomorphisms inducing the identityon D . It is easy to turn this into a classi cation

up to abstract isomorphism:

Corollary 1.5. If D is reduced, connected, and proper over k , then the set of

isomorphism classes of nonsplit ribbons C such that C D; and such that the

red

conormal bund le of C is isomorphic to a given line bund le L on C ; is the

red red

projective space of lines in Ext ( ; L) modulo the group of automorphisms of D

preserving L.

Proof. Given a ribb on C as in the Corollary,twochoices of the structure of a ribb on

on D di er by a unique element of Aut D:

The morphisms from a ribb on C to a scheme X admit a simple description in

terms of the induced map f : D = C ! X: Given such a map f , we write

red

df : f ! for the induced map on the sheaves of di erentials, and we write

X D

1 1

df : Ext ( ; L) ! Ext (f ; L)

D X

D D

for the map induced by df on Ext. Note that if C is a ribb on, and f : D ! C is

the inclusion, then f = j . We shall often write j for f also in the

X C D X D X

general case.

Theorem 1.6. If C is a ribbon on D with restricted cotangent sequence

0 !L! j ! ! 0;

C D D

then the morphisms from C to a scheme X extending a given morphism f : D ! X

are in one to one correspondence with the splittings of the exact sequence df e ;

that is, with the maps of sheaves

g : j ! j

X D C D

making the diagram

! j

D X D

j !

C D D

commutative.

RIBBONS AND THEIR CANONICAL EMBEDDINGS 11

In particular a morphism extending f exists i

df e =0:

Proof. Since O is a sheaf of k -algebras on D , f O is asheaf of k -algebras on X ,

C C

and the morphisms f : C ! X extending f are in one-to-one corresp ondence with

# #

the maps of algebras f : O ! f O lifting the map f : O ! f O :

X C X D

Since f is left exact, f O is the pullback of the diagram

f O

f j ! f :

C D D

Thus the desired algebra maps are in one-to-one corresp ondence with the derivations

of sheaves O ! f j lifting the derivation

X C D

! f O ! f : O

D D X

Such derivations corresp ond uniquely to maps of sheaves of mo dules

g : ! f j

X C D

making the diagram

! f

D X

f j ! f

C D D

commute. By the adjointness of f and f ,such g are in one-to-one corresp ondence

with the maps g describ ed in the Theorem.

Corollary 1.7. If C is a ribbon on D with restricted cotangent sequence e , then

the maps f : C ! D such that the composite

D,! C ! D

is the identity are in one to one correspondence with the splittings of e .

The following gives a useful criterion for a map from a ribb on to be a closed

immersion. Given a closed immersion into pro jective space, it also tells us when the

homogeneous co ordinate ring of the image has depth 2, which is the analogue for

ribb ons of pro jective normalityfor smo oth varieties.

If f : C ! X is a morphism such that f j is a closed immersion, then we shall

write N for the pullback to D of the conormal sheaf of f (D ) in X ; that is,

D;f

def

0 2

N = f (I =I );

f (D )=X

D;f

f (D )=X

where I is the ideal sheaf of f (D ) in X . In this situation we de ne :

f (D )=X

!L to b e the pullback of the quotient map I !I . N

f (D )=X f (C )=X D;f

12 DAVE BAYER AND DAVID EISENBUD

Theorem 1.8. Let C be a ribbon on D with conormal bund le L.

(1) If

f : C ! X

is a morphism, then f is a closed immersion i the restriction of f to D is

a closed immersion and is an epimorphism.

(2) If

f : C ! P

is a closed immersion, and the homogeneous coordinate ring of f (D ) has

depth 2, then the homogeneous coordinate ring of f (C ) has depth 2 i

the map

0 0

(n):H (N (n)) ! H (L(n))

D;f

induced by is surjective for al l integers n.

Proof. (1) f is a closed immersion i the induced map f : O ! f O is an

X C

epimorphism, and similarly for the restriction of f to D . Since D is a subscheme of

C , it is clear that f can b e a closed immersion on C only if it is a closed immersion

on D . Now supp ose that f is a closed immersion on D and consider the diagram

0 ! I ! O ! f O ! 0

D X D

? ?

y y

0 ! f L ! f O ! f O ! 0:

C D

From the snake lemma we see that the middle vertical map is an epimorphism i

the left-hand vertical map is. Since f L is supp orted on f (D ) D , this map is an

epimorphism i is, proving part (1).

(2) The homogeneous co ordinate ring of f (C ) has depth 2 i the map

0 0

(n)) ! H (O (n)) f (n): H (O

C P

induced by f is surjective for all integers n. Note that, b ecause of our hyp othesis

on D , the maps

0 0

(n)) ! H (O (n)) f (n):H (O

D P

are all surjective. Thus setting X = P in the ab ove diagram, twisting by O (n),

and taking H , the desired result follows again from the snake lemma.

The maps between ribb ons on D which induce the identity map on D have a

particularly nice description: they are just the blow-ups of Cartier divisors on D

(these are Weil divisors on C ). First we analyze such blow-ups.

Let D be an e ective Cartier divisor in D , and let C be a ribb on on D with

conormal bundle L and extension class e . Let : L!L( ) be multiplication by C

RIBBONS AND THEIR CANONICAL EMBEDDINGS 13

a section of O ( ) corresp onding to , and let C be the ribb on corresp onding to

the extension class

e = (e );

C C

where we have written again for the induced map

1 1

Ext ( ; L) ! Ext ( ; L ):

D D

With this notation we have:

Theorem 1.9. If X ! C is the blow up of C along , and C is the ribbon corre-

C , and the blow up map corresponds to the sponding to (e ) as above, then X

map of exact sequences

0 ! L( ) ! j ! ! 0

C D D

x x

? ?

0 ! L ! j ! ! 0

C D D

induced by .

Proof. The matter is lo cal on C , so we may assume that C = Sp ec A, that L is

the trivial bundle, so that the nilp otent ideal of A is generated by one element y ,

and that the ideal of in D is principal. Let x 2 A be any element of A lifting

the generator of the ideal of in O (D ) = A=(y ). Since is Cartier, x is a

nonzero divisor on A=(y ) and thus on A. Of course y =0. With these choices it is

easy to check that

C =SpecA[y=x];

where A[y=x]is the subring generated by y=x in the lo calization A[x ].

Consider now the blowup. Let B = A (x; y ) (x; y ) :::, so that X =

S pec(B [x ] ). Since Proj(B ). Since y is nilp otent X is ane, and we have X

n n1 1

(x; y ) = x (x; y ), we get B [x ] = A[y=x] as required.

It follows that we can describ e all maps of ribb ons over D in these terms. For

simplicitywe stick to the case where D is irreducible:

Corollary 1.10. Suppose that D is irreducible, and let f : C ! C be a map of

ribbons over D . If the image of f is contained in D , then C is split and f is the

projection. Otherwise, f is the blowup of C along a subscheme that is a Cartier

divisor in D .

0 0

Proof. Let L and L be the conormal bundles of C and C . The map

j ! j

C D C D

corresp onding to f induces a map : L!L as in the preceding Theorem. Since

D is irreducible, is an inclusion corresp onding to some Cartier divisor . The rest

follows as in the Theorem.

14 DAVE BAYER AND DAVID EISENBUD

2. Rational ribb ons 1: The restricted cotangent sequence

In this section we shal l suppose that the eld k is algebraical ly closed, and we shal l

1 1

. Here the classi cation describ ed in consider only ribbons C on D = P = P

Section 1 becomes much more concrete.

As usual with curves, the fundamental invariant of C is its genus, here de ned as

the arithmetic genus

0 1

g (C )= 1 (O )= 1 H O +H O :

C C C

From the additivity of and the fact that (O (n)) = n +1 we see that C has

genus g i the conormal bundle L of P = C in C is O (g 1).

red

C is split i the the inclusion D ,! C admits a section. Such a section is a

map C ! P , which will have degree 2 in the sense that the scheme-theoretic

1 1

b er of a p oint in P has length 2. Conversely, any degree 2 map f : C ! P ,

. Comp osing f with the inverse of this induces a degree 1 map f : D ,! C ! P

red

isomorphism of P we obtain a section of D ,! C . Because of this we will call C

hyperelliptic if C is split.

We have used the word hyp erelliptic b ecause hyp erelliptic ribb ons have many

prop erties in common with smo oth hyp erelliptic curves. As a rst example, we

1 1

note that if g 2, then there are no nontrivial maps ! O (g 1), so by

P P

Corollary 1.7 there is at most one splitting of e , and the two to one map C ! P

is unique if it exists. On the other hand, in the cases g = 0 and g = 1 there are,

again by Corollary 1.7, one-parameter and two parameter families of such two to

one maps, resp ectively, just as in the case of smo oth curves.

From Corollaries 1.4 and 1.5 we deduce:

Theorem 2.1. The set of nonhyperel liptic ribbons of genus g on P ,uptoisomor-

phism on P , is the set

g 3

(g 3))); P = P(H (O

the space of 1-quotients of H (O (g 3)).

The set of abstract isomorphism classes of nonhyperel liptic ribbons of genus g on

1 1

P is thus this set modulo Aut P .

Proof. By Corollary 1.4, it is enough to identify the lines in

1 1

Ext ( ; O (g 1)) = H ( O (g 1))

P P

D D

(g +1)); =H (O

(g 3)), and this is simply Serre duality. with the 1-quotients of H (O

For ribb ons on P we maywrite the restricted cotangent sequence in the simple

form

1 1 1 1

0 !O (g 1) !O (a 2) O (b 2) !O (2) ! 0

P P P P

RIBBONS AND THEIR CANONICAL EMBEDDINGS 15

for some integers a and b with 0 a b g 1 and a + b = g 1. We de ne the

Clifford index of C to be the integer a. Note that the sequence and thus the

ribb on is split, that is, hyp erelliptic, i a = 0, as for a smo oth curve. And just as

in the case of smo oth curves, the Cli ord index takes values from 0 to (g 1)=2.

To give a sequence of the sort ab ove, it suces to sp ecify the right-hand map

1 1 1

O (a 2) O (b 2) !O (2);

P P P

whichmust b e an epimorphism of sheaves. If wecho ose co ordinates, and thus iden-

tify the homogeneous co ordinate ring of P with the p olynomial ring in 2 variables

S = k [s; t], then such a map is given by a pair of homogeneous p olynomials and

of degrees a and b resp ectively. The condition that the map be an epimorphism is

then simply the condition that ; is a regular sequence, and the restricted cotan-

gent sequence itself is the Koszul complex of ; twisted by 2. We shall write

I = ( ; ) for the ideal asso ciated in this way to the ribb on C . It is easy to see

that I is an invariant of the sequence e and thus of C .

C C

To make a connection with Theorem 2.1 we recall a result that seems to have b een

(d)), the vectorspace discovered by Macaulay. To express it, we write S for H (O

of homogeneous forms of degree d.

Theorem 2.2. There is a one-to-one correspondencebetween hyperplanes in S

g 3

and ideals generated by regular sequences ( ; ) S whose generators have degrees

a; b with a + b = g 1, given as fol lows: If I =( ; ) S is such an ideal, then the

subspace

H (I )= ( ; ) \ S

g 3

is a hyperplane; and if H S is a hyperplane, then the ideal I (H ) consisting of al l

homogeneous polynomials of degree c g 3 such that

g 3c

(s; t) H

together with al l forms of degrees > g 3 is generated by a regular sequence of

elements of suitable degrees.

In the situation of the Theorem, we will call any functional : S ! k with

g 3

kernel H (I )a dual socle generator for I. The following additional information

is also well-known:

Prop osition 2.3. The lines in

0 1 1

1 1 1

(g 3))) (g 1)) = (H (O (g 1)) = H ( O Ext ( ; O

P P P

D D

spanned by the class of the Koszul complex of ( ; ) and the class of a dual socle

generator for ( ; ) are the same.

Thus the element in

P(H (O (g 3)));

asso ciated by Theorem 2.1 to a ribb on C is the same as the element corresp onding

to a dual so cle generator f for I .

Using this, we can makethe classi cation theorem more geometric:

16 DAVE BAYER AND DAVID EISENBUD

1 1

Corollary 2.4. A ribbon C on P is determined by the set of divisors P such

that the blowup of C along is hyperel liptic.

In fact, if we write 2 S for a polynomial de ning the divisor , then the set of

divisors

f jthe blow up of C along is hyperel lipticg

is the same as the set of forms

f j 2 I g:

In particular, the Cli ord index of C is the minimal number of blowups of C at

reduced points necessary to reach a hyperel liptic ribbon.

Proof. If we write C for the blown up ribb on, then e is obtained by pushing

forward the sequence e along the map induced by ,

0 ! L( ) ! j ! ! 0

C D D

x x

? ?

) (

( ; )

1 1 1

! O 0 ! L (a 2) O (b 2) ! O (2) ! 0

P P P

(g 1), and it is obvious that the upp er sequence splits i 2 where L = O

I =( ; ).

Corollary 2.4 provides another signi cant justi cation for using the name \Cli ord

index" for the invariant of a ribb on that we have de ned:

Wede ne a generalized linear series of degree n and dimension r on a

ribb on C to b e an ordinary linear series (line bundle and space of sections) of degree

n d on the blowupofC at some divisor of length d in D . Note that blowing up

C corresp onds to removing base points; if C were smo oth, we could remove base

p oints without changing C , so that the given de nition is a natural extension of the

smo oth case.

From Corollary 2.4 we can say that a ribb on of Cli ord index a has a generalized

linear series of degree a +2 and dimension 1, corresp onding to a linear series of

Cli ord index a (in the usual sense!). For this we need only take the degree 2 map

to P of the hyp erelliptic a-fold blowup guaranteed by the Corollary.

We shall see that there is a go o d notion of torsion free sheaf on a ribb on corre-

sp onding to the notion of generalized linear series, and also that C do es not have

any generalized linear series of dimension 1 and degree n for n< a+2 | that is,

none of Cli ord index less than the Cli ord index of the ribb on. The same is true of

generalized linear series of higher dimension, as is proved in Eisenbud-Green [1994].

The same technique proves:

of C along is the corresponding to the blow up C Corollary 2.5. The ideal I

\quotient"

def

(I )= (I : I ) = f 2 S j I I g;

C C C

RIBBONS AND THEIR CANONICAL EMBEDDINGS 17

where I is the ideal in S of the subscheme .

In particular, when is a single point, we have

Cli C < Cli C

i either Cli = (g 1)=2 or is a zero of the (unique) lowest degree form in

I .

Using the dual so cle, wecanmakeexplicitthe strati cation of the set of ribb ons

P(H (O (g 3))) by Cli ord index. The following necessary results are well-

known in the theory of vector bundles. Let

X P(H (O (g 3)))

be the rational normal curve of one-quotients H (O (g 3)) ! k corresp onding

to evaluations at p oints of P .

Prop osition 2.6. Let f be an element of H (O (g 3)) , regarded as a point in

P(H (O (g 3))). If I is the ideal whose dual socle is f , then I contains a form

of degree a i f lies in an a-secant (a 1){plane to X .

Thus the set of ribb ons of Cli ord index a corresp onds to the union of the a 1-

planes a-secant to X. (Here we count limits of such planes as also b eing a-secant|

for example tangent lines are considered 2-secant lines.)

i g 3i ?

Writing f in terms of a basis (s t ) of H (O (g 3)) dual to the monomial

basis of H (O (g 3)) determined by the choice of co ordinates s; t, say as

g 3

i g 3i ?

f = f (s t ) ;

i=0

we can express the secant lo ci, and thus the sets of ribb ons of Cli ord index a

given number, as sp ecial determinantal varieties:

Prop osition 2.7. With notation as above, f lies in an a-secant (a 1){plane to

X i the rank of the \catalecticant" matrix satis es

1 0

f f ::: f

0 1 a1

f f ::: f

C B

1 2 a

C B

: a rank

A @

::: ::: :::

f f ::: f

b1 b a+b2

In fact the ideal of a a minors of the catalecticant matrix is the whole homoge-

neous ideal of the corresp onding secant lo cus, and this remains true for the a a

0 0

minors of any a b catalecticant matrix

0 1

f f ::: f

0 1 a 1

f f ::: f

B C

1 2 a

B C

;

@ A

::: ::: :::

0 0 0 0

f f ::: f

b 1 b a +b 2

0 0

as long as a a ;b ; see Gruson-Peskine [1982] for a pro of.

18 DAVE BAYER AND DAVID EISENBUD

3. Rational ribb ons 2: Gluing

Again in this section we shal l consider ribbons C on D = P . We shal l write g

for the genus of C .

Since ribb ons on the ane line are all split, it is useful to regard ribb ons on D

as b eing obtained by gluing together ribb ons on the ane line. In this section we

use this gluing to give another view of the classi cation of ribb ons.

In the next section we shall exploit gluing to analyze line bundles and their

sections, to lo cate the canonical bundle, and to form the canonical map of a ribb on

to pro jective space.

We b egin by xing notation: we shall regard D as glued together out of twoopen

sets

u =Spec k [s];

u =Spec k [t]

via the identi cation

s = t

on u \ u .

1 2

If C is a ribb on on D , then by Corollary 1.3, we may write

def

Sp ec k [s; ]= U = C j

1 u

def

Sp ec k [t; ]= ; U = C j

2 u

and C may b e sp eci ed by giving an appropriate gluing isomorphism b etween these

two schemes over the set u \ u .

1 2

Since the ideal sheaf L O (g 1)of D in C is generated on u by and on

u by , and since the gluing isomorphism must restrict to the one already sp eci ed

on D , we see at once that it can be written in the form:

g 1

= t

s = t + F (t)

on u \ u , with some

1 2

F (t) 2 k [t; t ]=O (u \ u ):

1 2

Conversely, any such gluing data de nes a ribb on of genus g on P .

0 0

If we change the co ordinates on U and U to s and t with

1 2

s = s + p(s)

t = t + q (t)

RIBBONS AND THEIR CANONICAL EMBEDDINGS 19

where p(s) and q (t ) are p olynomials, then we have

01 1 2

s = s s p(s)

2 g 1

= t + F (t) +(t + F (t) ) p(s)t

g +1 1

= t +(F (t)+ t p(t ))

0 0 0 g +1 0 1

= t + q (t) +(F (t + q (t) )+ (t + q (t) ) p((t + q (t) ) ))

0 g +1 1

= t +(F (t)+ t p(t )+ q (t));

where we have rep eatedly used the facts that

s = t

t = t

etc. Also, if we multiply s or t by a scalar then F will be multiplied by the same

scalar. From this we see that F is determined (at b est) as an element of the

pro jective space of lines in the quotient

1 g +1 1

k [t; t ]=(k [t]+ t k [t ]):

On the other hand, using the covering of P by u and u , this quotient may

1 2

be identi ed via Cech cohomology as H (O (g 1)). We have seen in the last

section that the lines in this vector space classify the ribb ons on P . The main

result of this section is that these two classi cations are the same:

Theorem 3.1. Let

F 2 k [t; t ]

be a Laurent polynomial. If C is the ribbon de ned by gluing U and U as above,

1 2

then F is proportional to the class e of the restricted cotangent sequence of C in

1 g +1 1

H (O (g + 1)) = k [t; t ]=(k [t]+ t k [t ]):

We must exhibit a construction of the restricted cotangent sequence by gluing.

On U we have

j =(O ds O d)=2d

C U C C

so that

( j ) j = O ds O d;

C D u D D

and the restricted cotangent sequence, restricted further to u , takes the form

0 !! !! 0;

20 DAVE BAYER AND DAVID EISENBUD

where we have written < x> for the free mo dule with basis element x. Of course

wehave a similar sequence, with t and on u . On the intersection u \ u wehave

2 1 2

dt = d =0 so

g 1

d = t d

2 2

ds = t dt t Fd:

Thus the gluing takes the form

! !

? ? ?

g 1

()

y y y

where we have written for the matrix

g 1 2

t t F

0 t

Toshow that F is prop ortional to the class of restricted cotangent sequence in

1 g +1 1

H (O (g + 1)) = k [t; t ]=(k [t]+ t k [t ]);

it suces to show that F is prop ortional to the image of 1 2 H O under the

connecting homomorphism

0 1

1 1

:H O ! H O (g +1)

P P

(2). induced by the restricted cotangent sequence twisted by O

The e ect of the twist is to replace the gluing diagram (*) by the diagram

! !

? ? ?

g +1

y y y

2 2 2 2

! !

where now is the matrix

g +1

t F

0 1

We may compute the connecting homomorphism from this gluing description:

the Cech 1-co cyle representing (1) = (ds) 2 H (O (g +1)) is obtained by

taking

t dt (ds) 1

RIBBONS AND THEIR CANONICAL EMBEDDINGS 21

as an element of

O (g +1) j :

u \u

1 2

Since this is F , we are done.

Using the ideas develop ed in the last section, wemay rephrase this result in terms

of the ideal I asso ciated to the ribb on. The Laurent p olynomial F determines a

linear form on the set of p olynomials in t by the rule

h 7! residue Fh:

t=0

1 0

1 1

The usual duality between H (O (g +1)) and H (O (g 3))is given by re-

P P

stricting this functional to the p olynomials of degree g 3, and identifying these

with the forms of degree g 3 in s and t. In this way we may regard F as a

functional on S ,and as such we have:

Corollary 3.2. F is a dual socle generator for the ideal I of the ribbon de ned

by F .

22 DAVE BAYER AND DAVID EISENBUD

4. Line bundles on ribb ons

Again in this section we shal l consider ribbons C on D = P , and we shal l write g

for the genus of C .

In this section we shall explain how to classify the line bundles on a rational

ribb on, and we shall compute the sections of a given line bundle. In the next

section we shall apply these ideas to nd and study the canonical bundle.

Unfortunately,we do not know a metho d of describing a line bundle on a ribb on

C in terms of some structure on the underlying P analogous to the metho d of the

restricted cotangent sequence for describing O itself. Thus we shall use a gluing

description analogous to the ideas develop ed in the last section. In particular, we

shall make use of the notation develop ed in the last section for the gluing of C by

means of the Laurent p olynomial F . However, we can take a few steps b efore using

co ordinates:

First we compute the Picard group of C:

Prop osition 4.1.

(g 1)) Z Pic C =H (O

k Z;

where the projection to the Z factor is given by associating to a line bund le L the

degree of the restriction L j .

Proof. From the exact sequence

0 !L!O !O ! 0

C D

we derive an \exp onential sequence"

0 !L!O !O ! 1

C D

. Taking cohomology,we get by sending a lo cal section 2L to 1+ 2O

0 ! H (L) ! Pic C ! Pic D ! 1;

where the last map represents restriction to D . Since

Pic D Z

via the degree and

1 1

H (L)= H (O (g 1))

k ;

this gives the desired conclusion.

As in the smo oth case, the fundamental invariant of a line bundle L is its degree,

here de ned as

def

deg L = (L) (O ):

From the restriction sequence

0 !LL ! L ! Lj ! 0

for L we see that the degree of L may be computed from the knowledge of LL

provided by the following result:

RIBBONS AND THEIR CANONICAL EMBEDDINGS 23

(n) then LL O Prop osition 4.2. If L is a line bund le on C and Lj

= =

O (n g 1), and deg L =2n.

Proof. Since L =0 we have

L L = L Lj = O (n g 1):

On the other hand, we see by restriction to an ane op en set that LL is a line

bundle, so the epimorphism L L ! LL must be an isomorphism. The degree

computation now follows trivially.

To replace line bundles of o dd degree, it seems that one must turn to torsion free

sheaves, which are line bundles on blowups of C , as we shall see later.

Next we turn to the question of sections of a line bundle. What we need is a

formula for the connecting homomorphism

0 1

:H (Lj ) ! H (LL)

L D

asso ciated to the restriction sequence. We shall get it in terms of gluing data for

the line bundle.

To construct line bundles by gluing, note rst that all line bundles on one of the

sets U are trivial (one can see this, for example, from the analogue of the exact

sequence for the Picard group ab ove, or by standard commutative algebra) so that

(n) it is enough to sp ecify to give a line bundle L on C whose restriction to D is O

that

L = k [s; ]e on U

1 1

L = k [t; ]e on U

2 2

e =(t + F) (1 + G )e on U \ U

1 2 1 2

1 1

for some G 2 k [t; t ]. Conversely, any G 2 k [t; t ] may be used in this way to

construct a line bundle L on C .

If we change co ordinates on U and U , say by

1 2

1 0

e =(1+m(s)) e

e =(1+n(t) )e ;

then we get

0 n G1 0

e =(t + F) (1 + G )(1 + m(s)t )(1 + n(t) )e

1 2

n 1 g 1 0

=(t + F) (1 + (G + m(t )t + n(t)) )e

so that to sp ecify L it is enough to give G as an element of

1 g 1 1

k [t; t ]=(k [t]+ t k [t ]) = H (O (g 1)):

We can now state the main result of this section:

24 DAVE BAYER AND DAVID EISENBUD

Theorem 4.3. Let L is a line bund le on C of degree 2n given by gluing data as

above. If p = p(t) is a polynomial of degree n, so that pe j de nes an element

2 D

0 0

2 H (Lj )=H (O (n));

then

( )= (p F + pG)

1 ng 1 1

2 k [t; t ]=(k [t]+ t k [t ])

(n g 1)) =H (O

where p = @p(t)=@ t:

Further, the space of sections of L restricted to U = Sp ec k [t; t ] is the direct

sum of the space of elements q (t) , for q a polynomial of degree n g 1, and

the space of expressions of the form

p(t)+ p (t)

where p(t) is a polynomial of degree n satisfying ( )= 0 and p (t) 2 k [t] is the

L 1

\polynomial part" of p F + pG, that is,

0 1 1

p (t) p (t)F (t)+ p(t)G(t) mo d t k [t ]:

0 1

Proof. The connecting homomorphism H Lj ! H (LL) is obtained by comparing

liftings of on U and on U . We have

1 2

1 n

p(t)e j = p(s )s e j

2 u \u 1 u \u

1 2 1 2

so that ( )is given as a Cech co cycle by the element

1 n

p(t)e p(s )s e

2 1

2LL(U \ U )

1 2

= O (n g 1)j :

u \u

1 2

Using the gluing formulas we get

1 n

p(t)e p(s )s e

2 1

n n

= p(t)e p(t + F)(t + F) (t + F) (1 + G )e

2 2

= p(t)e p(t + F)(1 + G )e

2 2

= p(t)e (p(t)+p (t)F)(1 + G )e

2 2

= [p(t)G + p (t)F ]e ; 2

RIBBONS AND THEIR CANONICAL EMBEDDINGS 25

which gives the desired formula for .

The elements q (t)e , with q a p olynomial of degree n g 1, clearly represent

the sections of LL. The rest of the sections of L are obtained by lifting sections

of Lj that go to 0 under . Given such a section, represented sayby p(t)e on u

D L 2 2

as ab ove, we must nd an expression

p(t)+ p (t)

1 1

that is equal in k [t; t ;]= k [s; s ;] to some element coming from k [s; ]e . But

we have just shown that

1 n 0

p(t)e p(s )s e = [p(t)G + p (t)F ]e :

2 1 2

Take p as in the Theorem. Setting

1 n

r (t)e =(p(t)+p (t) )e p(s )s e ;

2 1 2 1

we see that r (t) will have no p olynomial part. That is, r (t) 2 k [t ]: In addition, if

ng 1 1

=0 then r (t) 2 t k [t ]. Thus

1 g 1 n

r (t)e = r (s )(s )s e

2 1

ng 1 1

= s r (s )

2 k [s]e ;

so that (p(t)+p (t) )e represents a section as claimed.

1 2

26 DAVE BAYER AND DAVID EISENBUD

5. The canonical emb edding

Again in this section we shal l consider ribbons C on D = P , and we shal l write g

for the genus of C . We shall continue to use the notation intro duced in the previous

section.

As an application of the work done in the previous section, we can determine the

canonical line bundle and its sections. Note that a ribb on is lo cally Gorenstein,

so that the canonical sheaf really is a line bundle, and has degree 2g 2 by the

Riemann-Ro ch formula.

If G(t)= a t 2 k (t) is a rational function, then we write

def

G = a t

p olynomial i

for the \p olynomial part" of G.

Theorem 5.1. The restriction sequence of the canonical line bund le K has the

form

1 1

0 !O (2) ! K !O (g 1) ! 0:

P P

The associated connecting homomorphism is 0, so that the induced linear series

on D is the complete series of degree g 1. The gluing data of the canonical bund le

of C is given by

G = @ F =@ t;

so that the sections of K are represented by the expressions

p +(@ (pF )=@ t)

p olynomial

where p ranges over the polynomials of degree g 1 in t. Thus if we take F of

the form

g 2

F (t)= F t

i=1

then the elements of a basis of global sections of K restrict on U to the elements

C 2

i j

t + (j +1)F t (i =0;:::;g 1):

ij 2

j =0

Proof. The form of the restriction sequence comes simply from the degree of K ,

and thus from Riemann-Ro ch. From Riemann-Ro chwealsoknow that h (K )= g .

0 0

1 1

(2)) = 0, this implies that (g 1)) = g while h (O Since h (O

P P

0 1

1 1

:H (O (g 1)) ! H (O (2))

P P C

RIBBONS AND THEIR CANONICAL EMBEDDINGS 27

is 0. Further, since no other line bundle of degree 2g 2 has as many sections as

the canonical bundle, this vanishing actually characterizes K .

By the formula for given in Theorem 4.3, = 0 means that if G = G(t)

K K

C C

is the gluing data for K , then

residue p F + pG =0

t=0

for all p olynomials p = p(t) of degree g 1. Since the residue of the derivativeof

a rational function is automatically 0, this can be achieved by taking

G = @ F =@ t;

so that

p F + pG = @ (pF )=@ t:

By the remark ab ovecharacterizing K , this establishes the formula for G given in

the Theorem. The rest of the Theorem follows by direct computation from Theorem

4.3.

Wenextwishtoshow that the canonical linear series de nes an emb edding of C

g 1

as an arithmetically Cohen-Macaulay (even Gorenstein) subscheme of P . As a

rst step we have:

Corollary 5.2. The canonical series on C is base point free.

Proof. It induces the complete series on D .

The following is the main result of this section. It continues the strict analogy

with the theory of smo oth curves, and is the rst stage of what might be called

\Green's conjecture for ribb ons", which will b e discussed below.

Theorem 5.3 (No ether's Theorem for Ribb ons). Let the genus of C be g 2.

If C is hyperel liptic, then the canonical map is the degree 2 projection onto P

1 g 1

composed with the embedding of P into P as the rational normal curve.

If C is not hyperel liptic, then the canonical series embeds C as an arithmetical ly

g 1

Gorenstein subscheme of P .

Proof. If C is hyp erelliptic, then the form of the canonical sections given in Theorem

5.1 makes it clear that the canonical image of C is the rational normal curve of degree

g 1. In particular, the map of C onto this image is 2 to 1, and since g 2 there

is only one such map (see the remark at the b eginning of section 2). Conversely,

we see that if the canonical image of C is the rational normal curveofdegree d 1,

then C is hyp erelliptic.

Now supp ose that C is not hyp erelliptic. It is enough to show that the canonical

series de nes an emb edding of C as an arithmetically Cohen-Macaualay curve; the

fact that the emb edding line bundle is the canonical bundle then implies that the

image is arithmetically Gorenstein.

28 DAVE BAYER AND DAVID EISENBUD

Since the restriction of the canonical series to D C is the complete series, it

de nes an emb edding of D as an arithmetically Cohen-Macaulay curve. Writing

g 1

I for the ideal sheaf of D P , it is enough to show that the induced maps

0 0

H (I =I (n)) ! H (L(n))

are onto for all n.

The following well-known lemma identi es the sheaf involved:

Lemma 5.4. The conormal bund le to the rational normal curve D P of degree

g 1 is

1 1

I =I H (O (g 3)) O (g 1)

P P

equivariantly for the action of SL (2).

Proof of Lemma 5.4. This is just the dual of a sp ecial case of Prop osition 5A.2.

Returning to the pro of of Theorem 5.3, we see that the natural map I =I !L

has the form

g 2 2

1 1

((g 1)) O I =I L ((g 1)) !O

= =

P P

and is thus either identically 0 or a split epimorphism. The image of this map is in

either case I =I , where I is the ideal sheaf of C . In the latter case we are done

D C C

by Theorem 1.8. In the former case we see that I = I , so that the canonical

D C

image of C is the rational normal curve, and C is hyp erelliptic by the remarks

ab ove, contradicting our assumption.

As was p ointed out to us by Jo e Harris, one can also give a pro of of \No ether's

theorem for ribb ons" along the lines of No ether's original pro of: First one may use

Riemann-Ro ch to estimate the number of conditions imp osed by a subscheme of

length 2 and thus show that the canonical map is a closed immersion.

Next one checks the numb er of quadratic (and then cubic ::: ) equations satis ed

by the curve by checking the number of conditions imp osed on quadrics by the

general hyp erplane section. For a reduced nonhyp erelliptic curve of genus g the

general hyp erplane section of the canonical emb edding consists of 2g +2 points in

g 2

linearly general p osition in P , and it is easy to see that such a set of p oints

imp oses at least 2g +1 conditions on quadrics. In the case of a ribb on, the hyp erplane

section consists of g +1 double p oints. But these are again in general p osition in

a suitable sense, as one may show by a mono dromy argument; see for example

Eisenbud-Harris [1992] and, for a more general study, Chandler [1994].

We isolate for future use a piece of information from the pro of of Theorem 5.3:

g 1

Corollary 5.5. If C P is a nonhyperel liptic ribbon in its canonical embed-

ding, and I I are its ideal sheaf and that of the underlying rational normal

C D

curve D = P , then

O (g 1) I =I

D C P

RIBBONS AND THEIR CANONICAL EMBEDDINGS 29

is a direct summand of the conormal bund le

2 g 2

I =I O (g 1)

We remark that the pro of of the Theorem indicates a third view of the classi -

cation of ribb ons: To sp ecify a non-hyp erelliptic ribb on of genus g in its canonical

emb edding one must simply give a corank 1 direct summand of the conormal bun-

dle of the rational normal curve of degree g 1 | this subbundle, together with

the square of the ideal of the rational normal curve, generates the ideal of the rib-

bon. By Lemma 5.4, such a summand is sp eci ed by an element of the dual of

(g 3)), up to scalars. This is the same element that sp eci es the gluing H (O

data or the dual so cle generator, as the reader maycheck.

30 DAVE BAYER AND DAVID EISENBUD

6. Canonical ribb ons are smo oth p oints of Hilb

Again in this section we shal l consider ribbons C on D = P , and we shal l write g

for the genus of C .

In this section we shall show that ribb ons all lie in the smo oth lo cus of the Hilb ert

scheme of canonical curves. The result is from joint work with Jo e Harris.

g 1

Theorem 6.1. If C P is a nonhyperel liptic ribbon in its canonical embedding,

then C represents a smooth point of the Hilbert scheme of curves of genus g and

degree 2g 2, lying on a component of dimension (3g 3) + (g 1)

The given dimension is of course also the dimension of the comp onent containing

the smo oth curves, and in fact Fong [1993] shows that they are the same.

Proof. Let D = C P as usual, and write I and I for the ideal sheaves of

red C D

g 1

C and D in the canonical emb edding of C in P . Let

N = Hom (I =I ; O )

C C C C

be the normal sheaf.

Because C is lo cally a complete intersection, N isavector bundle of rank g 2

0 1

on C and it suces to show that h (N ) = g +3g 4 and h (N ) = 0 (see

C C

for example Sernesi [1986, Corollaries 8.5 and 8.6].) To do this, we shall use the

restriction sequence

() 0 ! N j O (g 1) ! N ! N j ! 0;

C D C C D

and we must determine the bundle N j . Since I =I is a vector bundle, the

C D C

op erations of taking its dual and restricting it to D commute, and we get

N j =Hom (I =I ; O )j

C D C C C D

=Hom (I j =I j ; O )

D C D C D

=Hom (I =I I ; O ):

D C C D D

g 3 2

O (g 1) , so Hom (I =I I ; O ) has a sub- By Corollary 5.6, I =I

D C C D D C

g 3

bundle isomorphic to O (g +1) . To compute the quotient, note that it is a

line bundle; thus to make the computation, it suces to know the rst chern class

of N j .

C D

Again b ecause C is lo cally a complete intersection, we may calculate the chern

class of N just as we would in the case where C is smo oth: that is, we have

O (1) = !

C C

g 2

g 1

= ^ N !

g 2

g 1

= ^ N O (g )

g 2

= ^ N O (g );

C C

RIBBONS AND THEIR CANONICAL EMBEDDINGS 31

g 2

c (N )=c (^ N )

1 C 1 C

= O (g +1):

Of course the restriction of the rst chern class is the rst chern class of the restric-

tion, so

c (N j )= O (g +1)

1 C D D

= O ((g +1)(g 1)):

Subtracting the rst chern class of the subbundle we already know, we see that

N j ts into an exact sequence

C D

g 3

1 1

() 0 !O (g +1) ! N j !O (2g +2) ! 0:

C D

P P

Putting the exact sequences (*) and (**) together, we see at once that

h (N )=g +3g 4

h (N )=0;

as required.

32 DAVE BAYER AND DAVID EISENBUD

7. Surfaces containing ribb ons

1 1

, and we shal l Again in this section we shal l consider ribbons C on D = P = P

write g for the genus of C . Moreover, we assume that k is algebraical ly closed.

In a certain sense the singularities of a ribb on are quite mild: they are \lo cally

planar", so that for example a ribb on is lo cally a complete interesection (in any

emb edding).

However, the fact that every p oint is singular leads to some signi cant di erences

from the theory of reduced curves with lo cally planar singularities. In this section

weprove a theorem that highlights such a di erence. While any reduced pro jective

curve with lo cally planar singularities is contained in many smo oth surfaces, this is

not the case for ribb ons:

Theorem 7.1. Up to isomorphism of ane neighborhoods, the only pair

C S

where C is a nonhyperel liptic ribbon and S is a surface that is smooth along C , is

the double conic in the projective plane.

Proof. If D = C S ,withS a smo oth surface, then the ideal sheaf L of D C is

red

the conormal bundle of D in S . If the arithmetic genus of C is g then deg L = g +1,

so D = g +1. If g 2 then C is automatically hyp erelliptic, so it suces to treat

the case g 3. Hartshorne's Theorem on curves of high self-intersection [1969,

Thm. 4.1] says that if D 4 genus(D ) + 5 = 5, then either D S is a nonsingular

cubic in the pro jective plane, or else S is ruled and D is a section. In the former

case, D would not be rational. In the latter case there is a pro jection from S back

to D that induces a morphism C ! D , showing that C is hyp erelliptic. This proves

Theorem 7.1 in the case D = g +1 5.

It remains to treat the case D = 4; g = 3. We claim that a smo oth rational

curve D of self-intersection 4 on a smo oth surface S is (up to equivalence of ane

neighb orho o ds) either a smo oth conic in the plane or a section on one of the rational

ruled surfaces F ;F ;F . This suces to prove Theorem 7.1, since in the latter case

0 2 4

the pro jection from S to D de nes a a two-to-one map from C to D , so that C is

hyp erelliptic.

The claimed result is certainly known to exp erts on surfaces, but for want of a

reference we sketch a pro of. Let K be the canonical divisor class of S . Since D is

rational, we see from the adjunction formula that K D = 6. Thus no multiple

of K can b e e ective. By the Enriques classi cation, S is rational or ruled. If S is

ruled with base B , then since D has p ositive self-intersection it cannot b e contained

in a b er; thus B is rational, so S is rational in any case. In particular, (O )= 1.

Now H (O (K )) = 0, and thus

S S

2 0

H (O (D )) = H (O (K D )) = 0:

S S

By the Riemann-Ro ch formula on S ,

H (O (D )) D (D K )=2+ (O )= 6:

S S

RIBBONS AND THEIR CANONICAL EMBEDDINGS 33

The restriction of O (D ) to D has degree 4, and thus the asso ciated line bundle

O (D ) has 5 indep endent global sections. The exact sequence

0 0 0

! H (O (D )) ! H (O (D )) 0 ! H

S D

0 0 0

shows that h (O (D )) = 6, and the restriction map H (O (D )) ! H (O (D )) is

S S D

surjective.

It follows that the complete linear series jD j asso ciated to D on S has no base

p oints on D , and thus no base points anywhere; it de nes a morphism from

S to P of degree 4. Since (S ) is nondegenerate it cannot have degree < 4, so

we see that is birational, that its image is a surface of degree 4 in P . Since

the self-intersection of D is the same as the self-intersection of the the hyp erplane

section of (S ) that is the image of D ,we see that do es not blowdown anycurves

meeting D . That is, is biregular in a neighborhood of D ,sowemay assume that

S = (S ) P from the outset.

According to the Del Pezzo-Bertini classi cation of surfaces of minimal degree,

S is either a ruled surface F or F , or a cone over the rational normal quartic,

0 2

2 5

which away from the vertex is F or S is the Veronese embedding of P in P ; see

for example Eisenbud-Harris [1987]. In the former cases D is a section of S (not

containing the vertex, in the case where S is a cone) so C is hyp erelliptic by the

argument ab ove. In the latter case C S is isomorphic to the conic in the plane,

as required.

Of course any ribb on can, by Bertini's theorem, be emb edded in a surface with

only isolated singularities, and these can b e kept away from any nite set of p oints

on the ribb on. It would be interesting to know more ab out the number and typ e

of singularities that a ribb on imp oses on a surface containing it (Our attention was

drawn to this question by M. Boratynsky). The easiest way to pro duce a surface

containing a given ribb on in pro jective space is to pro ject the ribb on from a p oint,

and take the cone over the image. This image is the canonical image of another

ribb on:

g 1

Theorem 7.2. If C P is a ribbon in its canonical embedding, and p 2 C is

a reduced point on C , then the image of C under projection from p is the image of

the canonical map from the ribbon C obtained by blowing up C at p.

g 1

Proof. Pro jection de nes a morphism from the blowup of P , and thus also from

0 0 1

the blowup C of C . The restriction of this morphism to C = C = P is given

red

by the complete series of degree g 2. It follows that the corresp onding series on

0 0

C has degree 2(g 1) 2 and (linear) dimension g 1; since the genus of C is

g 1, this must by Riemann-Ro ch be the canonical series.

As a consequence we see inductively that the Cli ord index is related to the

length of the 2-linear part of the free resolution of a canonical ribb on, at least by

an inequality. The result is the \easy half of the canonical ribb on conjecture":

34 DAVE BAYER AND DAVID EISENBUD

Corollary 7.3. The 2-linear part of the free resolution of a canonical ribbon of

genus g and Cli ord index c has length at least g 2 c.

Proof. The 2-linear part of the resolution of a subscheme of pro jective space is

always at least as long as the 2-linear part of the resolution of any subscheme

g 1 0 g 2

containing it. Since the resolution of the cone in P over a subscheme C of P

0 g 2

has the same graded betti numbers as the resolution of C in P , and since the

blowup of a non-hyp erelliptic ribb on C at a suitable point will have Cli ord index

one less than that of C by Corollary 2.5, we are inductively reduced to the case of

the resolution of the canonical image of a hyp erelliptic ribb on, which is of course

the rational normal curve. But in this case | the case c = 0 | the resolution is

well known (it is given by the Eagon-Northcott complex) and the result is true.

More concretely, the metho d shows that a ribb on of genus g and Cli ord index a

is contained in the cone over a rational normal curve of degree g 1 a,and thus

that the free resolution of the ribb on contains that of the rational normal curve.

A simple example will illustrate the results of this section:

Example: The canonical ribb on of genus 4

Let C P be a canonically emb edded ribb on of genus 4. The Cli ord index of C

is necessarily 1, and C is the complete intersection of a cubic and quadric. The ideal

I asso ciated to C as in section 2 is generated by a linear form and quadratic form

1 2

on P . By cho osing co ordinates s; t appropriately wemay assume that I =(s; t ).

It follows that the blowup of C at the reduced point p given by s =0 in C is a

red

hyp erelliptic ribb on. By Theorem 7.2, the image of C under pro jection from p is

the reduced conic in P . The ribb on thus lies on the cone over this conic, and this

is the unique quadric containing C .

Cho osing co ordinates on P appropriately wemay assume that C is the ratio-

red

nal normal curve D de ned by the 2 2 minors of the matrix

x x x

0 1 2

x x x

1 2 3

and the quadric cone containing C is given bythe equation x x x =0,while C

0 2

itself is closure in the cone of the \double" of the divisor that is 2D away from the

vertex of the cone (D is not Cartier at the vertex.) The cubic form necessary to

generate the homogeneous ideal of C (which is unique mo dulo the ideal of D )may

be written as the determinant of the matrix

0 1

x x x

0 1 2

@ A

x x x :

1 2 3

x x 0

2 3

The corresp onding cubic surface has a double line, and a total of 4 singular p oints

on the rational normal curve, none of them at the vertex of the quadric cone; the

general cubic in the ideal of C has 4 singular points, all on the rational normal

curve, b earing out Theorem 7.1.

RIBBONS AND THEIR CANONICAL EMBEDDINGS 35

8. Free resolutions

We assume for simplicity that the ground eld k has characteristic 0, .

The material in this section was partially develop ed in conversation with Frank

Schreyer and Jo e Harris.

It turns out that all canonically emb edded rational ribb ons of given genus and

Cli ord index are hyp erplane sections of a single surface, which is itself a ribb on

on a rational normal scroll. This surface has the same numerical invariants as a

smo oth K3 surface; Following terminology suggested by Frank Schreyer, we shall

call it a K3 carp et. In this section we explain the construction of K3 carp ets, and

we prove that the natural emb edding of a K3 carp et of sectional genus g into P

is arithmetically Cohen-Macaulay. It follows that all ribb ons of given genus and

Cli ord index have minimal free resolutions with the same graded b etti numb ers;

in particular, the canonical ribb on conjecture is true for all of them if it is true for

one. Tocheck the conjecture, it would suce to compute a minimal free resolution

of each K3 carp et.

In this secction we shall compute a nonminimal resolution of a K3 carp et, and we

shall explain how to measure its nonminimality in terms of certain maps of vector

spaces de ned by elementary multilinear algebra. In particular, we explain some

conjectures that would imply the canonical ribb on conjecture.

The description of the nonminimal resolution is facilitated by the observation that

any K3 carp ets is an anti-canonical divisor on a certain (reduced) 3-fold (dep ending

only on the genus and degree of the plane section of the K3 carp et) that thus

g +1

app ears as a degenerate Fano 3-fold J in P . The Fano 3-fold J is extremely

easy to describ e: is simply the join variety of a pair of rational normal curves. It is

equally easy to describ e its minimal free resolutions F ,which is the tensor pro duct

of two Eagon-Northcott complexes. The minimal free resolution of the canonical

line bundle ! on this Fano 3-fold is F , the dual of F , up to a shift in degrees. It

follows easily that a (nonminimal) free resolution of a K3 carp et may b e constructed

as a mapping cone of a map of complexes F ! F . This map is only unique up

to homotopy, but it has a canonical representative that is equivariant with resp ect

to the SL(2) actions on the two rational normal curves. Most of the work in this

section is devoted to an explicit construction of this canonical representative.

Let S (a; g 1 a) be a rational normal scroll, the union of lines joining corre-

sp onding p oints on a rational normal curve of degree a and a rational normal curve

of degree g 1 a in P According to [Hulek-Van de Ven, 1985] there is a unique

double structure X on S (a; g 1 a), a ribb on on S (a; g 1 a) in the sense

of section 1 of this pap er, with trivial canonical bundle. We shall call this double

structure a K3 carp et. We give a construction of this carp et below.

Fix integers 1 a b and let g = a + b +1. In P we consider apair of disjoint

a b a b

linear spaces P and P , and the rational normal curves D P and D P .

a b

Let J be the join of these two rational normal curves; that is, J is the union of

the lines joining p oints of D to points of D . The K3 carp et X in which we are

a b

interested lies as an anticanonical divisor on J .

36 DAVE BAYER AND DAVID EISENBUD

If we write R and R for the homogeneous co ordinate rings of D and D in

a b a b

a b g

P and P resp ectively, then the homogeneous co ordinate ring of J in P is R =

R R , graded by total degree. We regard R and R as homomorphic images

a k b a b

of p olynomial rings T and T in a +1 and b +1 variables, resp ectively.

a b

Writing ! and ! for the canonical mo dules of R and R , the canonical mo dule

a b a b

of R is given by ! = ! ! . Further, we have

J J a k b

Hom (! ;R )= Hom (! ;R ) Hom (! ;R ):

R J J R a a k R b b

J a

Identifying D with P = P(V ), where V is a 2-dimensional vector space, we have

! = O (2 ), and a moment's argument gives

Hom (! ;R ) =H (O (2)) = S (V );

R a a 0 2

the second symmetric p ower of V . Making the corresp onding identi cations for D

with P = P(W ) for another 2-dimensional vectorspace W , we get

Hom (! ;R ) = S (V ) S (W ):

R J J 0 2 k 2

Since ! is a torsion free R -mo dule of rank 1, every nonzero map is a monomor-

J J

phism. We set

X =ProjR = (! ):

J J

Since R = (! ) is a 3-dimensional Gorenstein ring with trivial canonical divisor,

J J

X may b e regarded as a K3 surface, and it's hyp erplane sections will b e canonically

emb edded curves.

As wehave already remarked, the reduced structure on a K3 carp et is a rational

normal scroll. In the terms ab ove, such a scroll is determined by making an identi-

cation of D with D , that is, by identifying V with W . We may regard such an

a b

identi cation as an element of Hom (V; W ), or, using the identi cation of V and

V that we may make b ecause V is 2-dimensional, with an element of V W .

Squaring this, we get an element of S (V ) S (W ) corresp onding to the carp et

2 2

X . (Note that if we were not working in characteristic 0, some further care would

be necessary!) Thus we see that the K3 carp ets are quite sp ecial elements of the

family X . It is not to o hard to show that the general element may be describ ed

1 1

as follows: wemay asso ciate to a divisor E of typ e (2; 2) on P P = C C ,

a b

or equivalently as a corresp ondence of typ e (2; 2) from C to C ; generally, E will

a b

be an elliptic curve, but it may degenerate to twice a conic, or b ecome reducible.

The union of the lines in P joining corresp onding points is X . A hyp erplane H

in P cuts C in a points and C in b points. Thus it determines on E a set of a

a b

pairs of p oints and aset of b pairs of p oints. The hyp erplane section H \ X is an

emb edding of E with these a + b pairs of points identi ed.

It is easy to write down a minimal free resolution of R over the p olynomial ring

T = T T , the homogeneous co ordinate ring of P : it is obtained by tensor-

a b

ing together (over k ) a minimal free resolution of R over T and a minimal free

a a

RIBBONS AND THEIR CANONICAL EMBEDDINGS 37

resolution of R over T . These resolutions may be written, equivariantly for the

b b

action of GL(V )and GL(W ), as Eagon-Northcott complexes. Further, the minimal

resolution of ! is the dual of the minimal resolution of R . A chosen element

J J

of S (V ) S (W ), regarded as a map ! ! R , lifts (uniquely up to homotopy)

2 2 J J

to a map of these resolutions. The mapping cylinder of is a resolution of the

homogeneous co ordinate ring R of the K3 surface.

To obtain from this construction the graded b etti numb ers in a minimal resolution

of R , it suces to nd the rank of the degree 0 (in the sense of the grading from

T ) part of the map of complexes. Because the Eagon-Northcott complexes and

their tensor pro duct are minimal, the degree 0 part of is actually unique, and will

thus be GL(V ) GL(W )-equivariantly de ned from . Wenowmake this explicit.

Anticanonical divisors on the rational normal curve

We rst recall the resolution of R over T . To simplify the notation, write S

a a i

for the free T -mo dule S (V ) T . The resolution F has the form:

a i k a a

m+1

1 m

2 m+1

T ^ S (2) ::: ::: ^ S S (m 1)

a a1 a1 m1

^ S S (a) 0:

a1 a2

The resolution of ! is the dual complex F (a 1), whichmay conveniently b e

m am

rewritten, using the isomorphisms (^ S ) ^ S given by a choice of a

a1 a1

free generator for (^ S ) as

m+1

1 m

S (1) ^ S S (2) ::: ::: ^ S S (m 1)

a2 a1 a3 a1 a2m

a2 a1

^ S S (a +1) T (a 1) 0:

a1 1 a

The map ! ! R corresp onding to an element x 2 S (V )isgiven on generators

a a 2

(x)

S (V ) ! (T ) = S (V ) e 7! xe

a2 a 1 a

and lifts to a map (x):F (a 1) !F given by the formulas

(x)

m m+1

^ S S (m 1) !^ S S (m 1)

a1 a2m a1 m1

e f 7! e ^ (xu f ) u

for 1 m a 2, where for some basis v ;v 2 V the elements u and u are

1 2 i

de ned by the condition that

u u 2 S (V ) S (V )

i m1 m1

i i

38 DAVE BAYER AND DAVID EISENBUD

is the \trace" element

(v v v v ) :

1 2 2 1

The key p oint of the pro of that the (x) commute with the and is the

i i i

formula

e (e )= 0;

j i i

i;j

0 0

where 2 V V is the usual trace element, and the action (e )of V

j i

j j

on S (V ) is by derivations. We omit the details.

We can get a b etter idea of the nature of the maps (x)asfollows. The mapping

cylinder of (x) is a (nonminimal) resolution of an anticanonical divisor A = A(x)

on the rational normal curve C . Nowananticanonical divisor A on C is simply a

a a

divisor of degree 3 | that is, a scheme of three p oints. Of course A spans a 2-plane.

The minimal free resolution of O in P is the tensor pro duct of the resolution of

A in the plane and the resolution of the plane itself, which is a Koszul complex.

Note that for 1 m a 3 the maps (x) are matrices of scalars. Their kernels

and cokernels must add up to the minimal free resolution of O . Comparing these

sequences we deduce a self-dual family of natural exact sequences

(x)

m+1 m m+1 m1

0 !^ S !^ S S !^ S S !^ S ! 0:

a2 a1 a2m a1 m1 a2

asso ciated with x 2 S V . (To prove this, rst compute the kernel of (x) by

2 m

comparing the minimal resolution and the mapping cylinder, as ab ove; then use the

fact that (x) is isomorphic to the dual of the map (x).)

m a1m

As suggested by the notation, we may regard (x) as a family of maps of free

mo dules de ned over the p olynomial ring in 3 variables Q := k [S (V )], and in these

terms the exact sequences ab ove b ecome complexes

m+1 m

0 !^ S (m) !^ S S !

a2 a1 a2m

m+1 m1

^ S S (1) !^ S (a m):

a1 m1 a2

over Q. For example, if we take m =1, we get the resolution of the (a 1 m)

power of the maximal ideal of Q, written in a p eculiar way. It wouldbeinteresting

to understand these complexes in general.

Anticanonical divisors on J

The minimal free resolution of R over T is

F := F F ;

a k b

and the minimal free resolution of ! is

G := F F (g 1):

a b

RIBBONS AND THEIR CANONICAL EMBEDDINGS 39

Given an element := x y 2 S (V ) S (W ), we get a (nonminimal)

i i 2 2

resolution of the homogeneous co ordinate ring of the corresp onding anticanonical

divisor X by taking the mapping cylinder of the map of complexes

= (x ) (y ):G !F:

i j

i;j

The canonical ribb on conjecture, whichsays that the minimal free resolution of X

has no (a 1) syzygies of degree a +1 when is a square of an element of rank

2 is thus equivalent to the statement that for generic the map

:(G ) ! (F )

;a1 a1 a+1 a1 a+1

is surjective. Of course proving the surjectivity for general would prove the analogue

of Green's conjecture for some other degenerate K3 surface of Cli ord index a,which

would be just as interesting.

Writing out the terms in question, one must show that for all m; n with m

1;n 1;m + n = a 1

m n

 (x ) (y ): ^ S S ^ S S !

m i n j a1 a2m b1 b2n

i;j

m+1 n+1

^ S S ^ S S

a1 m1 b1 n1

is surjective for suitable x and y .

i j

The star construction

We can abstract the construction ab ove as follows: Since we are interested in

these things for generic values of the x and y we might as well take these as

i j

variables. The maps and used ab ove are then de ned over the p olynomial

m n

rings Q = k [x ;x ;x ] and Q = k [y ;y ;y ] resp ectively.

1 2 3 1 2 3

Supp ose, in general, that we are given a map d : F ! G of free mo dules over Q =

0 0 0 0

k [x ;:::;x ] and another map d : F ! G of free mo dules over Q = k [y ;:::;y ].

1 s 1 t

Supp ose further that, as in the case ab ove, each of these maps is represented by a

matrix of linear forms. We de ne a map

0 0 0

d d : F F ! G G ;

k k

the \star pro duct" of d and d , over the ring Q := k [z ] , by taking

i;j 1is;1j t

0 0

d d over the ring Q Q , and then replacing the pro duct x y by the variable

k k i j

z . This is legitimate b ecause the natural map Q ! Q Q maps the linear forms

i;j k

of Q isomorphically to the bilinear forms of Q Q . To show that the maps ab ove

are surjective for generic choice of x and y is to show that are the *-pro ducts of

i j

the \easy" maps and have maximal rank over Q.

m n

40 DAVE BAYER AND DAVID EISENBUD

It may clarify matters to give a simple example. If we are to have a nontrivial

computation then wemust have m 1;n 1;m+ n = a 1 and a b; g = a + b +1.

The rst case in which this is p ossible is a = b = 3;g = 7. Here we must take

m = n = 1. In this case d is just the middle map of the koszul complex in 3

variables

1 0

0 v v

3 2

2 2

A @

: v 0 v (x v + x v v + x v )=

3 1 2 1 2 1 2 3

1 2

v v 0

2 1

Of course these maps have rank only 2; they are not themselves of maximal rank

since they are skew symmetric and of odd size. However, the star pro duct

2 2 2 2

 (x v + x v v + x v ) (y v + y v v + y v )=

2 1 2 1 2 3 2 1 2 1 2 3

1 2 1 2

0 1

0 0 0 0 z z 0 z z

3;3 3;2 2;3 2;2

0 0 0 z 0 z z 0 z

B C

3;3 3;1 2;3 2;1

B C

0 0 0 z z 0 z z 0

B C

3;2 3;1 2;2 1;1

B C

0 z z 0 0 0 0 z z

B C

3;3 3;2 1;3 1;2

B C

z 0 z 0 0 0 z 0 z

B C

3;3 3;1 1;3 1;1

B C

z z 0 0 0 0 z z 0

B C

3;2 3;1 1;2 1;1

B C

0 z z 0 z z 0 0 0

B C

2;3 2;2 1;3 1;2

@ A

z 0 z z 0 z 0 0 0

2;3 2;1 1;3 1;1

z z 0 z z 0 0 0 0

2;2 2;1 1;2 1;1

is of maximal rank as long as the characteristic of k is not 2; it is a symmetric 9 9

matrix with zeros on the diagonal, so in characteristic 2 it can have rank only 8. The

exception for characteristic 2 corresp onds precisely to Schreyer's observation that

the general canonical curve of genus 7 do es NOT in fact satisfy Green's conjecture.

RIBBONS AND THEIR CANONICAL EMBEDDINGS 41

Numerical evidence

We nish by exhibiting the b etti numbers for canonical ribb ons of Cli ord index

 2 and genus 12, as computed by the program Macaulay [1990], in characteristic

31,991. We give only the \2-linear" part. The rest may b e repro duced by using the

symmetry of the resolution. Thus a listing

g enus index

7 2 10 16 9

is to b e read as the assertion that a rational ribb on of genus 7 and Cli ord index 2

has a minimal free resolution of the form

10 16 9 9 16

O O(2) O(3) O(4) O(4) O(5)

O(6) O(8) 0;

g 1

(a). In the notation used by the \b etti" op eration (a)= O where O denotes O

of the program Macaulay this would be written

1 - - - - -

- 10 16 9 - -

- - 9 16 10 -

- - - - - 1

The following b etti numbers were computed by Macaulay:

genus index

5 2 3

6 2 6 5

7 2 10 16 9

8 2 15 35 35 14

9 2 21 64 90 64 20

10 2 28 105 189 189 105 27

11 2 36 160 350 448 350 160 35

12 2 45 231 594 924 924 594 231 44

genus index

7 3 10 16

8 3 15 35 21

9 3 21 64 70 24

10 3 28 105 162 119 35

11 3 36 160 315 336 210 48

12 3 45 231 550 756 672 342 63

genus index

9 4 21 64 70

10 4 28 105 162 84

11 4 36 160 315 288 100

12 4 45 231 550 693 455 125

42 DAVE BAYER AND DAVID EISENBUD

genus index

11 5 36 160 315 288

12 5 45 231 550 693 330

These numb ers supp ort the Canonical Ribb on Conjecture stated at the b eginning

of this pap er.

It is interesting to compare these b etti numb ers with those computed bySchreyer

[1986] for smo oth curves of Cli ord index 2, genus 8, over a eld of charac-

teristic 0. For curves of Cli ord index 2 and genus 7 or 8, the betti numbers of

the smo oth curves agree with the corresp onding b etti numb ers of the ribb ons if the

2 2

curve has a g . In the case of a smo oth curve of Cli ord index 2 with no g , the

6 6

betti numb ers computed by Schreyer are instead:

genus index

7 2 10 16 3

8 2 15 35 25 4

RIBBONS AND THEIR CANONICAL EMBEDDINGS 43

App endix: Osculating bundles of the rational normal curve

We assume for simplicity that the ground eld k has characteristic 0, although our

results could be reformulated for the case of arbitrary k .

The following results identify the quotients of any two osculating bundles of the

rational normal curve in terms of the representation theory of SL(2). Though

sp ecial cases, at least, are well known, we do not know a convenient reference. For

simplicity,wework over a eld k of characteristic 0, although the second of the two

pro ofs we give for the main result may be adapted to work in any characteristic.

1 r

For each r , let P D P be the rational normal curve, and let T = T j

r P

be the restricted tangent bundle. There is an obvious emb edding T = T T ,

1 r

and in fact the T form a ag of bundles

0 T T T ::::

1 2 r

In terms of the emb edding in a given P , this ag is realized geometrically as the

ag of osculating bundles of D :

Prop osition 5A.1. The subbund le of T whose ber at a point p 2 D is the set

of tangent vectors lying in the osculating m-plane to D at p is isomorphic to T .

Further, the inclusions T T obtained in this way are independent of r .

m n

Proof. For the pro of we shall need a more formal description of the subbundle of

T whose b ers are the osculating m-planes: we shall temp orarily call it Osc .

r m

(r ) on D , and let P (L) be the bundle of If we write L for the line bundle O

principle parts of L of order m, then there is a natural map H (L) O !P (L)

that, lo cally at each p oint, takes a section to its Taylor series. Let E b e the image

of the dual map, twisted by L. The bundle that we have called Osc is the image

of E under the natural map H (L) L ! T j ; that is, there is a commutative

m P D

diagram

0 ! O ! E ! Osc ! 0

D m m

? ? ?

y y y

j ! 0; 0 ! O ! H (L) L ! T

D D P

see Piene [1977] for details.

We shall show that the osculating bundles and inclusions are the same in P and

r 1

P by pro jecting from a p oint p of the rational normal curve D . In particular,

T = Osc . By induction this proves the Theorem.

r 1 r 1

(r ) for the emb edding line bundle. The natural inclusion L(p) Write L = O

L induces diagrams

0 ! Osc L ! P (L) ! L ! 0

x x x

? ? ?

0 ! Osc L(p) ! P (L(p)) ! L(p) ! 0 n

44 DAVE BAYER AND DAVID EISENBUD

where we have written Osc for the osculating bundle of the rational normal curve

r 1

in P . To show that Osc Osc it suces to show that the map lab elled

n n

in the diagram is isomorphic to the inclusion of Osc L(p) in Osc L, or

n n

equivalently that the cokernel of is the the sheaf k (p) . Since the inclusions of

one osculating bundle in another are compatible with these commutative diagrams,

this will suce to show that the inclusions are also indep endent of r .

Since and are isomorphisms away from p,theyare inclusions of sheaves, and

it follows that the same is true for . Since the cokernel of is obviously k (p), it

n+1 n

suces to show that the cokernel of is k (p) . Since lo cally P (L) lo oks like

L dt this is clear.

To describ e the quotients T =T equivariantly we need a notation for the rep-

m n

resentations of SL(2). If V is a 2-dimensional vectorspace, S := Sym(V ) is the

symmetric algebra, and we identify the rational normal curve as D =ProjS , then

SL (2) = SL(V ) acts naturally on D and on

S := Sym (V )

(n)); =H (O

and these are the irreducible representations of SL (V ).

Prop osition 5A.2. With notation as above,

(m + n +1) S O T =T

mn1 m n

equivariantly for the action of SL (2).

We givetwo pro ofs { the rst, whichwas suggested by Jo e Harris, is app ealingly

geometric. The second, by free resolutions, gives slightly more information, and is

essentially characteristic free.

Geometric Proof. First we compute chern classes: from the exact sequences

n n1 n

L !P (L) !P (L) ! 0 0 !

and the \initial case"

P (L)= L

we easily derive

c (P (L)) = (n + 1)(r n);

from which we get

c (T =T )=(m n)(m + n +1):

1 m n

Next we shall show that T =T is a direct sum of equal line bundles so that, for

m n

some SL(2) representation U, we have

U O (m + n +1) T =T

m n P

RIBBONS AND THEIR CANONICAL EMBEDDINGS 45

From the construction, it will app ear that these line bundles can be chosen from

a single family parametrized by a 1-dimensional pro jective SL (2) orbit. Since the

only representations of SL(2) whose pro jectivization contains such a curve are the

irreducible representations S , this will conclude the pro of. (Here we are using the

characteristic 0 hyp othesis.)

1 m

It remains to pro duce the family of subbundles. The curve P = D P will

itself form the parameter space: for each point p 2 D we de ne a line bundle

M [p] T whose b er at a p oint x 2 D other than p is the line spanned by x

and p, regarded as a line in the tangent space to P at x mo dulo the osculating

n-plane to D at x. This de nes M [p] as a bundle on D p. There is of course a

unique extension of M [p] to a bundle on all of D : its b er at p is the osculating

(n + 1)-plane at p mo dulo the osculating m-plane.

We claim that for any set of m n distinct points p ;:::;p on D , we have

1 nm

M [p ]: T =T = 

i m n

This follows from the fact that any collection of osculating spaces to the rational

normal curve is \as linearly indep endent as p ossible": in our case, if the line bundles

in question failed to span at some p oint x, then the osculating n-space at x together

with the p oints p (or the osculating (n + 1)-space at x together with all the p if x

i i

is one of the p ) would be contained in a hyp erplane, which would then meet D at

least m +1 times, contradicting the fact that the degree of D is m.

Algebraic Proof. We shall actually make explicit the maps in the sequences de ning

the bundles T and in the exact sequences

0 ! T ! T ! S O (m + n +1):

n m mn1

All the maps of bundles with which we are concerned are of twotyp es, which we

shall rst desrib e abstractly. Let

 : S ! S S

a;b a a+b b

be the map of SL (2)-representations obtained by multiplying with the canonical

\trace" element in S S . As an element of S S this is (s t t s) where

b b b b

s; t is a basis of V = S . If wewrites^; t 2 V for the basis dual to s; t 2 V then the

usual trace element is s s^ + t t 2 V V . Under the equivariant identi cation

V = V , which sends V

s^ 7! t t 7! s

this trace element go es to the element s t t s,whichiswhywe call it the trace

as well.

This map induces an equivariant map of sheaves, for whichwe shall use the same

name:

 : S O ! S O (b):

a;b a P 1 a+b P

46 DAVE BAYER AND DAVID EISENBUD

p q

Explicitly, if i + j = a and we write [s t ] for the corresp onding basis element of

S O then

p+q

 

i j nj nj mi m n

 :[s t ] 7! (1) s t [s t ]:

a;b

m i

m+n=a+b

The second typ e of map that we shall need is closely related. Let

: D (V ) ! D (V ) S ;

b a+b a

a;b

where D (V ) denotes the b graded comp onent of the divided power algebra on

V , b e the map of SL(2)-representations obtained bymultiplying with the canonical

\trace" elementinD (V ) S . In terms of the basis and dual basis for V and V

a a

intro duced ab ove, this element is the divided power

(2) (a) (a) (a1) (a2)

^ ^

t + :::: (^s s + t t) = s^ +^s t +^s

This map induces an equivariant map of sheaves, for which we use the same name

1 1

(a): ! D (V ) O : D (V ) O

a+b b

P P

a;b

Dualizing, twisting with O (a), and using the canonical isomorphism D (V )

m n m n

S that makes fs t g the dual basis to fs^ t g we get the map we want,

1 1

(a): ! S O : S O

b a;b a+b

P P

Explicitly, with notation as ab ove: if m + n = a + b, then

 

m n

mi nj i j m n

s t [s t ]: :[s t ] 7!

a;b

i j

i+j =b

We claim that (always in characteristic 0) these maps form, for every a; b, an

exact sequence of sheaves:

a1;b

1 1

!S O (b) E :0! S O

a+b1 a;b a1

P P

(b)

a+b1;a

! S O (a + b) ! 0:

(a +1) is isomorphic to the sequence In fact, we claim that E O

a;b

() 0 ! T ! T ! T ! 0:

a a+b b

Further, the \Euler sequence" de ning T ,

1 1

0 !O ! S O (b) ! T ! 0;

P P b

RIBBONS AND THEIR CANONICAL EMBEDDINGS 47

is isomorphic to the sequence E . Of course this will prove the Prop osition.

1;b

First, to prove that the sequences E are exact, we app eal to the criterion of

a;b

exactness of Buchsbaum-Eisenbud [1973]. It is easy to check directly the the E

a;b

are complexes, and from insp ection the ideals of maximal minors of the two maps

contain powers of both s and t, so the conditions of the criterion are immediate.

Next, that E is the Euler sequence is also clear, since the map

1;b

1 1

O ! S O (b)

P P

in the Euler sequence is multiplication by the b power of the trace element.

Finally, to identify the sequence (*) with E O (a +1) it suces to show

a;b

that the diagram

0 0

? ?

y y

1 1

O O

P P

? ?

y y

E (a): 0 ! S (a) ! S (a + b) ! S (2a + b +1) ! 0

a+1;b1 a a+b b1

? ?

y y

E (a +1): 0 ! S (a +1) ! S (a + b +1) ! S (2a + b +1) ! 0

a;b a1 a+b1 b1

? ?

y y

0 0

commutes, where the two long vertical columns are the Euler sequences, and we

have written S (a) for S O (a), etc. Note that there is, up to scalar, only one

a a

inclusion of representations

S S S

a a+b b

so the inclusion T T induced by the diagram must b e the geometrically de ned

n m

one.

Since all the maps have b een given explicitly, this is presumably only an exercise.

But it is p ossible to say that the maps must commute. To see this, note that we

need only check the commutativity of the part involving the two Euler sequences,

since the commutativityofthelower right hand b ox (at least up to a scalar) is then

forced by the irreducibilityof S . Similarly,tocheck the commutativity of the part

involving the Euler sequences (up to scalar), we need only check the commutativity

of the upp er left hand box

1 1

O O

P P

? ?

 

y y

S (n) ! S (m)

n m

48 DAVE BAYER AND DAVID EISENBUD

b ecause S is irreducible. This is easy, since the maps lab elled are all given by

multiplication by powers of the trace element.

RIBBONS AND THEIR CANONICAL EMBEDDINGS 49

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Department of Mathematics, Barnard College, Columbia University, New York,

NY 10027

[email protected]

Department of Mathematics, Brandeis University, Waltham MA 02254

[email protected] typeset 6/6/1997