A Minicourse on Moduli of Curves

A minicourse on mo duli of curves

Eduard Lo oijenga

Faculteit Wiskunde en Informatica, University of Utrecht,

P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands

Lecture given at the

School on Algebraic Geometry

Trieste, 26 July { 13 August 1999

LNS001006

lo [email protected]

Abstract

These are notes that accompany a short course given at the Scho ol on

Algebraic Geometry 1999 at the ICTP, Trieste. A ma jor goal is to outline

various approaches to mo duli spaces of curves. In the last part I discuss the

algebraic classes that naturally live on these spaces; these can b e thoughtof

as the characteristic classes for bundles of curves.

Contents

1. Structures on a surface 271

2. Riemann's mo duli count 273

3. Orbifolds and the Teichmuller approach 274

4. Grothendieck's view p oint 275

4.1. Ko daira-Sp encer maps 275

4.2. The deformation category 276

4.3. Orbifold structure on M . 278

4.4. Stable curves 279

5. The approach through geometric invariant theory 280

6. Pointed stable curves 282

6.1. The universal stable curve 284

6.2. Strati cation of M 285

g;n

7. Tautological classes 286

7.1. The Witten classes 286

7.2. The Mumford classes 286

7.3. The tautological algebra 287

7.4. Fab er's conjectures 289

References 291

A minicourse of moduli of curves 271

1. Structures on a surface

We start with two notions from linear algebra. Let T be a real vector

space. A conformal structure on T is a p ositive de nite inner pro duct ( )

on T given up to multiplication by a p ositive scalar. The notion of length

is lost, but we retain the notion of angle, for if v ;v 2 T are indep endent,

1 2

then

(v v )

1 2

\(v ;v ):=

1 2

kv kkv k

1 2

do es not change if wemultiply ( ) with a p ositive scalar. A complex structure

on T is a linear automorphism J of T such that J = 1 ; this makes T

a complex vector space by stipulating that multiplication by 1 is given

by J . If dim T =2, then these notions almost coincide: if we are given an

' is a complex orientation plus a conformal structure, then `rotation over

structure on T . Conversely, if we are given a complex structure J , then

we have an orientation prescrib ed by the condition that (v; J v) is oriented

whenever v 6= 0 and a conformal structure by taking any nonzero inner

pro duct preserved by J .

Let S be an oriented C surface. A conformal structure on S is given

by a smo oth Riemann metric on S given up to multiplication by a p ositive

C function on S . By the preceding remark this is equivalent to giving

an almost-complex structure on S , i.e., an automorphism J of the tangent

bundle TS with J = 1 , that is compatible with the given orientation.

Given such a structure, then wehave a notion of holomorphic function: a C

function f : U ! C on an op en subset U of S op en is said to b e holomorphic

1df : T S ! C . This generalizes the familiar if for all p 2 U , df J =

p p p p

notion for if S happ ens to b e C with its standard almost-complex structure,

then we are just saying that f satis es the Cauchy-Riemann equations. It

is a sp ecial prop erty of dimension two that S admits an atlas consisting of

holomorphic charts. (This amounts to the prop erty that for every Riemann

metric on a neighborhood of p 2 S we can nd lo cal co ordinates x; y at p

2 2

such that the metric takes the form c(x; y )(dx + dy ).) Co ordinate changes

will b e holomorphic also (but now in the conventional sense), and we thus

nd that S has actually a complex-analytic structure.

A surface equipp ed with such a structure is called a Riemann surface.

We shall usually denote such a surface by C . I will assume you are familiar

with some of the basic facts regarding compact Riemann surfaces such as the

272

Riemann-Ro ch theorem and Serre-duality and the notions that enter here.

These give us for instance

Theorem 1.1. Acompact connected Riemann surface C of genus g can be

g +1

complex-analytical ly embedded in P such that the image is a nonsingular

complex projective curve of degree 2g +1. The algebraic structure that C

thus receives is canonical.

The pro of maybesketched as follows. Cho ose p 2 C . Then the complete

linear system generated by (2g + 1)(p) is of dimension g +2 and de nes a

g +1

complex-analytic emb edding of C in P of degree 2g +1. A theorem of

Chow asserts that a closed analytic subvariety of a complex pro jective space

is algebraic. So the image of this emb edding is algebraic. It also shows that

the algebraic structure is unique: if C is complex analytically emb edded

k l

into two pro jective spaces as C P and C P , then consider the

1 2

k l

diagonal emb edding of C in P P , comp osed with the Segre emb edding of

k l kl+k+l

P P in P ;by Chow's theorem the image is a complex pro jective curve

C . The curves C and C are now obtained as images of C under linear

3 1 2 3

pro jections. These are therefore (algebraic) morphisms that are complex-

analytic isomorphisms. Such morphisms are always algebraic isomorphisms.

The ab ove theorem shows in particular that a compact Riemann surface

of genus zero resp. one is isomorphic to P resp. to a nonsingular curve in

P of degree 3.

What can wesay ab out the automorphism group of a compact Riemann

1 1

surface C of genus g ? If g = 0, then we can assume C = P and Aut(P )is

then just the group of fractional linear transformations z 7! (az + b)(cz + d)

with ad bc 6= 0. If g = 1, then the classical theory tells us that C is

isomorphic to a complex torus, and so Aut(C ) contains that torus as a

`translation' group. This subgroup is normal and the factor group is nite.

In all other cases (g 2), Aut(C ) is nite. There are several ways to see

this, one could b e based on the uniformization theorem, another on the fact

that Aut(C ) acts faithfully on H (C; Z) (any automorphism acting trivially

has Lefschetz number 2 2g < 0, so cannot have a nite xed p oint set,

hence must b e the identity) and preserves a p ositive de nite Hermitian form

on H (C; C ).

We denote by M the set of isomorphism classes of nonsingular genus g

curves. For the moment it is just that: a set and nothing more, but our aim

is to put more structure on M when g 2. We will discuss four approaches

g to this:

A minicourse of moduli of curves 273

Riemann's original (heuristic) approach, that we will discuss very brie y.

The approach through Teichmuller theory. Actually there are several

of this typ e, but we mention just one. More is said ab out this in Hain's

lectures.

The intro duction of an orbifold structure on M in the spirit of Grothen-

dieck's formalization and generalization of the Ko daira-Sp encer theory.

The intro duction of a quasi-pro jective structure on M by means of

geometric invariant theory.

The last two approaches lead us to consider a compacti cation of M as

well.

2. Riemann's moduli count

Fix integers g 2 and d 2g 1. Let C be a smo oth genus g curve.

Cho ose a p oint p 2 C . By Riemann-Ro ch the linear system jd(p)j has

dimension g d. Cho ose a generic line L in this linear system that passes

through d(p), in other words, L is a p encil through d(p). The genericity

assumption ensures that this p encil has no xed p oints. Cho ose an ane

co ordinate w on L such that w = 1 de nes d(p). We now have a nite

morphism C ! P of degree d that restricts to a nite morphism f : C

fpg ! C . We invoke the Riemann-Hurwitz formula (which is basically an

euler characteristic computation):

(f )=2g1+d;

x2C fpg

where (f ) is the rami cation index of f at x (= the order of vanishing

(f )(f (x)) (so the of df at x). The discriminant divisor D is

x2C fpg

co ecient of w 2 C is the sum of the rami cation indices of the p oints of

f (w )). Its degree is clearly 2g 1+d. The passage to the discriminant

divisor loses only a nite amount of information: from that divisor we can

reconstruct C and the covering C ! P (up to isomorphism) with nite

ambiguity. Furthermore, it is easy to convince yourself that in the (2g 1+d)-

dimensional pro jective space of e ective degree 2g 1+d divisors on P the

discriminant divisors make up a Zariski op en subset. Wenow count mo duli

as follows: in order to arrive at f we needed for a given C , the choice of

p 2 C (one parameter), the choice of a line L in jd(p)j through d(p)(dg1

parameters) and the ane co ordinate w (2 parameters). Hence the number

of parameters remaining for C is

(2g 1+d) 1+(dg 1) + 2 =3g3:

274

This suggests that M is likea variety of dimension 3g 3.

3. Orbifolds and the Teichmuller approach

We b egin with a mo dest discussion of orbifolds. Let G be a Lie group

acting smo othly and prop erly on a manifold M . Proper means that the map

(g; p) 2 G M ! (g(p);p) 2 M M is prop er; this guarantees that the

orbit space GnM is Hausdor .

If G acts freely on M , then the orbit space GnM is in a natural way a

smo oth manifold: for every p 2 M ,cho ose a submanifold S of M through p

such that T S supplements the tangent space of the G-orbit of p at p. After

shrinking S if necessary, S will meet every orbit transversally and at most

once. Hence the map S ! GnM is injective. It is not hard to see that

the collection of these maps de nes a smo oth atlas for GnM , making it a

manifold.

If G acts only with nite stabilizers, then we can cho ose S in such a

way that it is invariant under the nite group G . After shrinking S in a

suitable way we can ensure that every G-orbit that meets S , meets it in

a G -orbit and that the intersection is transversal. So we then have an

injection G nS ! GnM . This is in fact an op en emb edding and hence GnM

is lo cally like a manifold mo dulo a nite group. It is often very useful to

rememb er the lo cal genesis of such a space, b ecause this information cannot

be recovered from the space itself (example: the obvious action of the nth

ro ots of unity on a one dimensional complex vector space has orbit space

isomorphic to R , so that we cannot read o n from just the orbit space).

This leads to Thurston's notion of orbifold: this is a Hausdor space X for

which we are given an `atlas of charts' of the form (U ;G ;h ) , where

U is a smo oth manifold on which a nite group G acts, and h is an

op en emb edding of the orbits space U nG in X . The images of these op en

emb eddings must cover X and there should be compatibility relations on

overlaps. It is understo o d that two such atlasses whose union is also atlas

de ne the same orbifold structure. So if F is a discrete space on which a

nite group H acts simply transitively, then wemay add the chart given by

U F with its obvious action of G H (its orbit space is U nG ) and h .

This allows us to express the compatibility relation simply by saying that

the atlas is closed under the formation of b ered pro ducts (`intersections'):

U U with its G G -action and the identi cation of the orbit space

with a subset of X should also be in it. It also implies that we have an

atlas of charts for which the group actions are e ective. I leaveittoyou to

A minicourse of moduli of curves 275

verify that GnM has that structure. There exist parallel notions in various

settings, e.g., complex-analytic and algebraic. In the last two cases, it is

often useful to work with a more re ned notion of orbifold (a `stack'), but

we will not go into this now.

The case that concerns us is in in nite dimensional analogue of the ab ove

situation: we x a closed oriented surface S of genus g and we let the space

of conformal structures on S take the role of M and the group of orientation

preserving di eomorphisms take the role of G. It is understo o d here that

these carry certain structures that allow us to think of an action of an in nite

dimensional Lie group on an in nite dimensional space. This action turns

out to be prop er with nite stabilizers. It turns out that all orbits have

co dimension 6g 6 and so it is at least plausible that M has the structure

of an orbifold of real dimension 6g 6. This heuristic reasoning has b een

justi ed by Earle and Eells.

4. Grothendieck's view point

It is worthwhile to discuss things in a more general setting than is strictly

necessary for the present purp ose, for the metho ds and notions that we need

come up in virtually all deformation problems.

4.1. Ko daira-Sp encer maps. Supp ose : C ! B is a prop er (holomor-

phic) submersion between complex manifolds. According to Ehresmann's

bration theorem, is then lo cally trivial in the C -category, that is,

for every b 2 B we can nd an op en U 3 b and a smo oth retraction

1 1

h : C = U ! C = (b) such that h = (h; ) : C ! C U is

U U

b b

a di eomorphism. In particular, when B is connected, then all b ers of

are mutually di eomorphic. If b oth B and the b ers are connected, we will

call a family of compex manifolds with smooth base. We assume that this is

the case and we wish to address the question whether the b ers C are mu-

tually isomorphic as complex manifolds. Supp ose that the family is trivial

over U , in other words, that the retraction h can b e chosen holomorphically

so that h is an analytic isomorphism. Then each holomorphic vector eld

on U lifts to C U in an obvious way and hence also lifts to C (via h).

Supp ose now the converse, namely, that every holomorphic vector eld at

b lifts holomorphically. Then is lo cally trivial at b. To see this, assume

for simplicity that dim B = 1. Cho ose a nowhere zero vector eld on an

op en U 3 b (always obtainable by means of a co ordiante chart) that lifts

holomorphically to a vector eld on C . The prop erness of ensures that U

276

this lift is integrable to a holomorphic ow on C . The ow surfaces (=

complex ow lines) pro duces the desired retraction h : C ! C . The case

of a higher dimensional base go es by induction and the induction step is a

parametrized version of the one dimensional case just discussed.

Liftability issues lead inevitably to cohomology. Let us b egin with the

noting that the fact that is a submersion implies that for every x 2C we

have an exact sequence

0 ! T C ! T C!T B!0; b:= (x):

x x

b b

This shea es as an exact sequence of O -mo dules

0 ! ! ! ! 0

C B

C =B

( stands for the sheaf of holomorphic vector elds on C , for the

C =B

subsheaf of of vector elds that are tangent to the b ers of ). Now take

the direct image under ; since the b ers are connected, we get:

0 ! ! ! ! 0:

C B

C =B

This sequence diplays our lifting problem: an elementof is holomorphi-

cally liftable i it is in the image of . But the sequence may fail to b e

C =B

exact at since is only left exact. We need the right derived functors

of in order to continue the sequence in an exact manner:

0 ! ! ! !R ! :

C B

C =B C =B

The sheaf R is a coherent O -mo dule, whose value at b is equal to

C =B

the cohomology group H (C ; ). So an elementof is holomorphically

C B

liftable i its image under vanishes. Hence gives us a go o d idea of how

nontrivial the family at a p oint b is: the family is lo cally trivial at b i is

zero in b. Both and its value at a p oint b, (b):T B !H (C ; ), are

b b

called the Kodaira-Spencer map.

4.2. The deformation category. Let us x a connected compact complex

manifold C . A deformation of C with smooth base (B; b ) is given by a prop er

holomorphic submersion : C ! B of complex manifolds, a distinguished

p oint b 2 B and an isomorphism : C C , with the understanding

that replacing by its restriction to a neighb orho o d of b in B de nes the

same deformation (in particular, B may be assumed to connected). From

the preceding discussion it is clear that wemay think of this as a variation

of complex structure on C parametrized by the manifold germ (B; b ). 0

A minicourse of moduli of curves 277

0 0

The deformations of C are ob jects of a category: a morphism from (; )

to (; ) is given by a pair of holomorphic map germs (; ) in the diagram

C ! C

? ?

y y

0 0

(B ;b ) ! (B; b )

such that the square is cartesian (this basically says that sends the b er C

isomorphically to the bre C 0 ) and = . So (; ) is more interesting

(b )

0 0

than (; )asevery b er of the latter is present in the former. In this sense,

the most interesting ob ject would b e a nal ob ject, if it exists (which is often

not the case). A deformation (; ) is said to b e universal if it is a nal ob ject

0 0

for this category. So this means that for every deformation ( ; )ofC there

0 0

is a unique morphism ( ; ) ! (; ). A universal deformation is unique

up to unique isomorphism (a general prop erty of nal ob jects). We also

observe that the automorphism group Aut(C ) acts on (; ): if g 2 Aut (C ),

then (; g ) is another deformation of C and so there is a unique morphism

~ ~ ~ ~

( ; )) :(; g ) ! (; ). The uniqueness implies that = and

g g g

gh h

that is the identity. So the action of Aut(C ) extends to (C ;C ). Similarly,

Aut(C ) acts on (B; b ) such that is equivariant.

Remark 4.1. The restriction to deformations over a smo oth base turns out to

b e inconvenient. The custom is to allow B to b e singular. The submersivity

requirement for is then replaced by the condition that b e lo cally trivial

on C : for every x 2 C , there is a lo cal holomorphic retraction h :(C;x) !

(C ;x) such that h = (h; ) is an isomorphism of analytic germs. This

enlarges the deformation category and consequently the notion of universal

deformation changes. Our restriction to deformations with smo oth base was

only for didactical purp oses: a universal deformation is always understo o d

to b e the nal ob ject of this bigger category (and therefore need not havea

smo oth base).

We can go a step further and allow C to b e singular as well (in fact, we

shall have to deal with that case). Then the right condition to imp ose on

is that it be at, whichis an algebraic way of saying that the map must

b e op en. In contrast to the situation considered ab ove, the top ological typ e

of C can nowchange (simple example: take the family of conics in P with

2 2

ane equation y = x + t). The Ko daira-Sp encer theory has to b e mo di ed

as well. For example, H (C; )must b e replaced by Ext ( ; O ).

C C C

278

Remark 4.2. As said earlier, a universal deformation need not exist. But

what always exists is a deformation (; ) with the prop erty that for any

0 0 0

deformation (; ) there exists a morphism (; ) : ( ; ) ! (; ) with

unique up to rst order only. This is called a semi-universal deformation

of C (others call it a Kuranishi family for C ). Such a deformation is still

unique, but may have automorphisms (inducing the identity on C and the

Zariski tangent space of the base).

4.3. Orbifold structure on M . We can now state the basic

Theorem 4.3. For a smooth curve C of genus g 2 we have

(i) C has a universal deformation with smooth base.

(ii) A deformation (; ) of C is universal i its Kodaira-Spencer map

1 1

T B ! H (C ; ) H (C; ) is an isomorphism.

C C

b b

0 0

(iii) A universal deformation of C can be represented by a family C ! B

such that Aut(C ) acts on this family (and not just on the germ) so that

every isomorphism between bers C ! C is the restriction of the

b b

1 2

action of an automorphism of C .

So the universal deformation of C is smo oth of dimension h ( ). A

simple application of Riemann-Ro ch shows that this numberis3g3.

A universal deformation as in (iii) de nes a map B !M that factorizes

over an injection Aut (C )nB ! M . We give M the nest top ology that

g g

makes all those maps continuous. It is not dicult to derive from the ab ove

theorem that with this top ology the maps Aut(C )nB !M b ecome op en

emb eddings. It is harder to prove that the top ology is Hausdor . Thus M

acquires the structure of a (complex-analytic) orbifold of complex dimension

3g 3.

Remark 4.4. The orbifold structure on M has the prop erty that every orb-

ifold chart (U; G; h : GnU ! M is induced by a family of genus g curves

over U , provided that g 3. For g =2 we run into trouble since every genus

2 curve has a nontrivial involution (it is hyp erelliptic). For this reason, the

orbifold structure as de ned here is not quite adequate and wehave to resort

to a more sophisticated version: ultimately wewant only charts that supp ort

honest families of curves, with a change of charts covered by an isomorphism

of families.

The space M is not compact. The reason is that one easily de nes

families of smo oth genus g curves over the punctured unit disk C!f0g

with mono dromy of in nite order. A classic example is that of degerating

A minicourse of moduli of curves 279

2 3 2

family of genus one curves given by the ane equation y = x + x + t, with

t the parameter of the unit disk. I admit that we dismissed genus one, but

it is not dicult to generalize to higher genus: for example, replace x by

2g +1

x . Such a family de nes an analytic map f0g!M such that the

image of its intersection with a closed disk of radius < 1 is closed. To see this,

supp ose the opp osite. One then shows that the map f0g!M must

extend holomorphically over . If the image of the origin is represented by

the curve C , then a nite rami ed cover of (; 0) will map to the universal

deformation of C so that the family on this nite cover will have trivial

mono dromy. This contradicts our assumption. The remedy is simple in

principle: compactify M by allowing the curves to degenerate (as mildly

as p ossible). This leads us to the next topic.

4.4. Stable curves. A stable curve is by de nition a nodal curve C (that

is, a connected complex pro jective whose singularities are normal crossings

(no des), analytically lo cally isomorphic to the union of the two co ordinate

axes in C at the origin) such that

the euler characteristic of every connected comp onent of the smo oth

part of C is negative.

The genus of such a curve can be de ned algebro-geometrically as h (O )

or top ologically by the formula 2 2g = e(C ). So it has to be 2. The

reg

itemized condition is equivalent to each of the following ones:

g 2 and C has no connected comp onent isomorphic to P f1g

reg

or P f0;1g,

Aut(C ) is nite,

C has no in nitesimal automorphisms.

Top ologically a stable genus g curve is obtained as follows. Let S be a

closed oriented genus g surface. Cho ose on S a nite collection of emb edded

circles in distinct isotopy classes and such that none of these is trivial in the

sense that it b ounds a disk. Then the space obtained by contracting eachof

the circles underlies a stable curve and all these top ological typ es are thus

obtained. (Note that removal of an emb edded circle from a surface do es not

alter its euler characteristic.) There is a deformation theory for stable curves

which is almost as go o d as if the curvewere smo oth:

Theorem 4.5. A stable curve of genus g 2 has a universal deformation

with smooth base of dimension 3g 3. This deformation can be represented

by a family C!B such that Aut(C ) acts on this family in such a way that

280

(i) every isomorphism between bers C ! C is the restriction of the

b b

1 2

action of an automorphism of C ,

(ii) al l bers are stable genus g curves,

(iii) for any singular point x of C , the locus B that parametrizes the

curves for which x persists as a singular point is a smooth hypersurface

and the ( ) cross normal ly.

x x2C

sing

So the complement of the normal crossing hyp ersurface := [

x2C x

sing

in B parametrizes smo oth genus g curves. If there is just one singular p oint,

then you may picture the degeneration from a smo oth curve C to the sin-

gular curve C C in metric terms as by letting the circumference of the

emb edded circle on S that de nes the top ological typ e of C go to zero. The

mono dromy around B is in this picture the Dehn twist along that circle

(see the lectures by Hain).

Let M be the set of isomorphism classes of stable genus g curves. The

ab ove theorem leads to a compact orbifold structure on this set:

Theorem 4.6 (Deligne-Mumford). The universal deformations of stable ge-

nus g curves put a complex-analytic orbifold structure on M of dimension

3g 3. The space M is compact and the locus @ M = M M parametriz-

g g g g

ing singular curves is a normal crossing divisor in the orbifold sense.

This is why M is called the Deligne-Mumford compacti cation of M . A

g g

generic p oint of the b oundary divisor corresp onds to a stable curve C with

just one singular p oint. The underlying top ological typ e of such a curve is

determined by a nontrivial isotopy class of an emb edded circle on S . The

following cases o ccur:

: C is irreducible (S is connected) or

0 g

0 00

0 00 : C is the one p oint union of two smo oth curves of p ositive genera g ;g

fg ;g g

0 00

with sum g + g = g (S disconnected with comp onents punctured

0 00

surfaces of genera g and g ).

These cases corresp ond to irreducible comp onents of @ M . We denote them

by and 0 00 .

fg ;g g

5. The approach through geometric invariant theory

Perhaps the most app ealing way to arriveatM and its Deligne-Mumford

compacti cation is by means of geometric invariant theory. Conceptually

this approach is more direct than the one discussed in the previous section.

Best of all, we stay in the pro jective category. The disadvantage is that we

do not know a priori what ob jects we are parametrizing.

A minicourse of moduli of curves 281

Let us b egin with a minimalist discussion of the general theory. Let a

semisimple algebraic subgroup G of SL(r +1;C) b e given. That group acts

r r r

on P . Let X P be a closed G-invariant subvariety. A G-orbit in P is

r +1

called semistable if it is the pro jection of a G-orbit in C f0g that do es

not have the origin in its closure. The union X of the semistable orbits

contained in X is a subvarietyof X. This subvariety need not b e closed and

may be empty (in which case there is little reason to pro ceed). The basic

results of Hilb ert and Mumford are as follows:

1. Every semistable orbit in X has in its closure a unique semistable

orbit that is closed in X .

2. There exists a p ositiveinteger N such that the semistable orbits that

are closed in X can be separated by the G-invariant homogeneous

p olynomials of degree N .

3. If R stands for the homogeneous co ordinate ring of X , then the sub-

ring of its G-invariants R is no etherian.

Let GnnX denote the set of semistable G-orbits in X that are closed in X .

Prop erty 1 implies that there is a natural quotient map X ! GnnX . If

ss ss ss

G happ ens to act prop erly on X , then every orbit in X is closed in X

and so GnnX will b e just the orbit set GnX . From prop erty 2 it follows

that if f ;::: ;f is a basis of the degree N part of R , then the map

0 m

ss m

[f : : f ] : X ! P is well de ned and factorizes over an injection

0 m

m m

GnnX ! P . The image of this injection is a closed subvariety of P and

thus GnnX acquires the structure of a pro jectivevariety. (A more intrinsic

waytogive it that structure is to identify it with Pro j(R ).)

So much for the general theory. For the case that interests us, you need

to know what the dualizing sheaf ! of a no dal curve C is: it is the coherent

subsheaf of the sheaf of meromorphic di erentials characterized by the prop-

erty that on C it is the sheaf of regular di erentials, whereas at a no de p

reg

we allow a lo cal section to have on each of the two branches a p ole of order

one, provided that the residues sum up to zero. It is easy to see that ! is

always a line bundle (as opp osed to ) and that its degree is 2g (C ) 2. It

is very ample for is ample precisely when C is stable and in that case !

k 3. (The name dualizing sheaf has to do with the fact that it governs

Serre duality. But that prop erty is of no concern to us; what matters here

is that every stable curve comes naturally with an ample line bundle.)

If C is stable of genus g , then a small computation shows that for k 2,

h (! )=(2k1)(g 1). Let us x for each k 2 a complex vector space C

282

V of dimension d := (2k 1)(g 1) (k 2). Cho ose k 3. Since ! is

k k

very ample, we have an emb edding of C in P(H (C; ! ) ). The choice of

an isomorphism : H (C; ! ) V allows us to identify C with a curve

C P(V ). It is a standard result of pro jective geometry that for m large

enough,

1. the degree m hyp ersurfaces in P(V ) induce on C a complete linear

system,

2. C is an intersection of degree m hyp ersurfaces.

In other words, the natural map

k mk

m m 0 0

Sym (V ) Sym H (C; ! ) ! H (C; ! )

C C

is surjective (prop erty 1) and its kernel de nes C (prop erty 2). So the

image W Sym V of the dual of this map is of dimension d and

k mk

determines C . Nothing is lost if we take the d th exterior power of W

and regard it as a p oint of P(^ Sym V ). Now let X be the set of

k k;m

p oints in P(^ Sym V ) that we obtain by letting C run over all the

stable genus g curves and over all choices of isomorphism. This is a (not

necessarily closed) subvariety that is invariant under the obvious SL(V )-

action on P(^ Sym V ). One can show that SL(V ) acts prop erly on

k k

X . It is clear from the construction that as a set, SL(V )nX may be

k;m k k;m

identi ed with M . The fundamental result is

Theorem 5.1 (Gieseker). For k and m suciently large, the semistable lo-

cus of the closure of X in P(^ Sym V ) is X itself.

k;m k k;m

Corollary 5.2. The set M is in a natural way a projective variety con-

taining M as an open dense subvariety

In particular, M acquires a quasi-pro jective structure. As one may ex-

p ect, the structure of pro jectivevariety M is compatible with the analytic

structure de ned b efore. Incidentally, geometric invariant theory also allows

us to put the orbifold structure on M , but we shall not discuss that here.

6. Pointed stable curves

It is quite natural (and very worthwhile) to extend the preceding to the

case of p ointed curves. If n is a nonnegativeinteger, then an n-pointed curve

is a curve C together with n numb ered p oints x ;::: ;x on its smo oth part

1 n

C . If (C ; x ;::: ;x )isann-p ointed smo oth pro jective genus g curve, then

reg 1 n

its automorphism group is nite unless 2g 2+n 0 (so the exceptions

A minicourse of moduli of curves 283

are (g; n) = (0; 0); (0; 1); (0; 2); (1; 0)). Therefore we always assume that

2g 2+n>0. In much the same wayasforM one shows that the set of

isomorphism classes M of n-p ointed smo oth pro jective genus g curves has

g;n

the structure of a smo oth orbifold of dimension 3g 3+n. Just as we did

for M ,we compactify M by allowing mild degenerations. The relevant

g g;n

de nition is as follows:

An n-p ointed curve(C;x ;::: ;x ) is said to b e stable if C is a no dal curve

1 n

C such that

the euler characteristic of every connected comp onentof

C fx ;::: ;x g is negative,

reg 1 n

a condition that is equivalent to each of the following ones:

2g 2+n> 0 (where g is the genus de ned as b efore) and C has

reg

1 1

no connected comp onent isomorphic to P f1g or P f0;1g,

Aut(C ; x ;::: ;x ) is nite,

1 n

(C ; x ;::: ;x ) has no in nitesimal automorphisms.

1 n

We will also refer to a stable n-p ointed genus g curve as a stable curve of

type (g; n). The underlying top ology is obtained as follows: x p ;::: ;p

1 n

distinct p oints of our surface S and cho ose on S fp ;::: ;p g a nite

g g 1 n

collection of emb edded circles ( ) in distinct isotopy classes relative to

e e2E

S fp ;::: ;p g such that none of these b ounds a disk on S containing at

g 1 n g

most one p , and contract each of these circles. There is a more combinatorial

way of describing the top ological typ e that we will use later. It is given by

a nite connected graph that may have multiple b onds and for whichwe

allow loose ends (that is, edges of which only one end is attached to a vertex,

some call them legs). We need the additional data consisting of

for eachvertex a nonnegativeinteger g and

anumb ering of the lo ose ends by f1; 2;::: ;ng.

Wesay that these data de ne a stable graph if

for every vertex v wehave2g 2 + deg (v ) > 0.

g + b (), and we call the pair (g ();n) We de ne the genus by g () :=

v 1

the type of the stable graph. The recip e for assigning a stable graph of

typ e (g; n) to (S ; p ;::: ;p ;( ) ) is as follows: the vertex set is the set

g 1 n e e2E

of connected comp onents of S fp ;::: ;p g[ , the set of b onds is

g 1 n e e

indexed by E : the two sides of de ne one or two connected comp onents

and we insert a b ond b etween the corresp onding vertices (so this might be

a lo op), and we attach the ith lo ose end to the vertex v if the corresp onding

connected comp onent contains p . You should check that the top ological i

284

typ e is faithfully represented in this way. Note that the stable graph of a

smo oth curve of typ e (g; n) is likea star: n lo ose ends attached to a single

vertex of weight g .

Let M denote the set of isomorphism classes of stable curves of typ e

g;n

(g; n). Wehave the exp ected theorem:

Theorem 6.1 (Knudsen-Mumford). The universal deformations of stable

curves of type (g; n) put a complex-analytic orbifold structure on M of

g;n

dimension 3g 3+n. The space M is compact and the locus @ M

g;n g;n

parametrizing singular curves is a normal crossing divisor in the orbifold

sense.

The construction of M can also be obtained by means of geometric

g;n

invariant theory, which implies that it is a pro jective orbifold. (The role of

the dualizing sheaf is taken by ! (x + + x ); details can be found in

C 1 n

a forthcoming sequel to [1].) The irreducible comp onents of the b oundary

@ M are in bijective corresp ondence with the emb edded circles in S

g;n

fp ;::: ;p g given up to orientation preserving di eomorphism. These are:

1 n

: C is irreducible or

0 0 00 00 : C is a one p oint union of smo oth curves of genera g and

f(g ;I );(g ;I )g

00 0

g with the former containing the p oints x with i 2 I and the latter

00 0 00

the p oints indexed by I (so fI ;I g is a partition of f1;::: ;ng). We

0 0 00

allow g to b e zero, provided that jI j2 and similarly for g .

6.1. The universal stable curve. Let (C ; x ;::: ;x ) be a stable curve

1 n

of typ e (g; n). Let us show that any x 2 C determines a stable curve

(C ;~x ;::: ;x~ )oftyp e (g; n + 1).

1 n+1

If x 2 C fx ;::: ;x g, then take(C;~x ;::: ;x~ )=(C;x ;::: ;x ;x).

reg 1 n 1 n+1 1 n

If x = x for some i,we let C b e the disjoint union of C and P with the

1 1

p oints x and 1 indenti ed. We let x~ =1 2P and x~ =0 2P ;

i i n+1

whereas for j 6= i; n +1, x~ = x , viewed as a p oint of C . We denote

j j

this (n + 1)-p ointed curveby(C;x ;::: ;x ).

i 1 n

If x 2 C , then C is obtained by separating the branches of C in x

sing

(i.e., we normalize C in this p oint only) and by putting back a copyof

1 1

P with f0; 1g identi ed with the preimage of x. Then x~ =1 2P

n+1

and for i n, x~ = x , viewed as a p ointof C.

i i

M that maps x to ((C ; x ;::: ;x ). Wethus have de ned a map C !

g;n+1 i i 1 n

There is also a converse construction: given stable curve(C;~x ;::: ;x~ )

1 n+1

of typ e (g; n + 1), then we can asso ciate to it a stable curve(C;x ;::: ;x )

1 n

A minicourse of moduli of curves 285

of typ e (g; n) basically by forgetting x~ ; this yields a stable p ointed curve

n+1

unless x~ lies on a smo oth rational comp onent which has only two other

n+1

sp ecial p oints. Let C b e obtained by contracting this comp onent and let x

b e the image of x~ (i n). This de nes a map : M ! M . Notice

i g;n+1 g;n

that the map C ! M de ned ab ove parametrizes the b er of over

g;n+1

the p oint de ned by(C;x ;::: ;x ).

1 n

Prop osition 6.2. The map : M ! M is morphism and so are its

g;n+1 g;n

sections ;::: ; . The ber of over the point de ned by (C ; x ;::: ;x )

1 n 1 n

can be identi ed with the quotient of C by Aut(C ; x ;::: ;x ).

1 n

This Prop osition says that in a sense the pro jection with its n sections

de nes the universal stable curve of typ e (g; n). For this reason we often

refer to M as the universal smooth genus g curve (g 2) and denote it

g;1

by C . Likewise C := M is the universal stable genus g curve.

g g g;1

6.2. Strati cation of M . The normal crossing b oundary @ M deter-

g;n g;n

mines a strati cation of M in an obvious way: a stratum is by de nition

g;n

M where the number of a connected comp onent of the lo cus of p oints of

g;n

lo cal branches of @ M at that p oint is equal to xed numb er. That number

g;n

may be zero, so that M is a stratum. It is clear that the strata decom-

g;n

M into subvarieties. A stratum of co dimension k parametrizes stable p ose

g;n

curves of typ e (g; n) with xed top ological typ e (and k singular p oints). You

maycheck that distinct strata corresp ond to distinct top ological typ es.

We next show that the closure of every stratum is naturally covered

by a pro duct of mo duli spaces of stable curves. Let Y be a stratum.

If (C ; x ;::: ;x ) represents a p oint of Y , then consider the normaliza-

1 n

^ ^

tion n : C ! C and the preimage of the set of sp ecial p oints X :=

^ ^

n (C [fx ;::: ;x g). The connected comp onents of the pair (C; X )

sing 1 n

are stable curves, at least after suitably (re)numb ering the p oints of X ,

^ ^

for every comp onent of C X maps homeomorphically to a comp onent of

C fx ;::: ;x g, hence has negative euler characteristic. So if the typ es of

reg 1 n

the stable curves are (g ;n ) , then we nd an elementof M . Con-

i i i2I g ;n

i i

i2I

versely if we are given a nite collection of smo oth curves of typ e (g ;n ) ,

i i i2I

then by identifying some pairs of p oints we nd a stable curveoftyp e (g; n).

In terms of stable graphs: the rst pro cedure amounts to cutting all the

b onds in the middle of the stable graph asso ciated to Y so that we end

up with a nite set of stars, whereas the second builds out of stars a sta-

ble graph by identifying certain pairs of edges. This recip e de nes a map

286

M ! Y of which it is not dicult to see that it is a nite surjec-

g ;n

i i

tive morphism. The same recip e can b e used to glue stable (not necessarily

smo oth) curves of typ e (g ;n ), so that the map extends to

i i

f : M ! M :

g ;n g;n

i i

It is easy to verify that this is a nite morphism with image the closure of

Y . (It is in fact an orbifold cover of that closure.) The section of the

universal curve is a sp ecial case of this construction.

7. Tautological classes

Throughout the discussion that follows we take rational co ecients for

our cohomology groups. One reason is that the space underlying an orbifold

is a rational homology manifold, so that it satis es Poincare duality when

oriented (which is always the case in the complex-analytic setting). We shall

be dealing with complex quasi-pro jective orbifolds and the coholomology

classes that we consider happ en to have Poincare duals that are Q -linear

combinations of closed subvarieties. Such classes are called algebraic classes.

7.1. The Witten classes. Given a stable curve (C ; x ;::: ;x ), then for

1 n

a xed i 2f1;::: ;ng, we may asso ciate to it the one dimensional complex

vector space T C . This generalizes to families: if (C ! B : ( : B !

C ) ) is a family of stable curves, then for a xed i, the conormal bundle

i=1

M ! of de nes a line bundle over B . In the universal example (

g;n+1 i

M ; ;::: ; ) this pro duces an orbifold line bundle over M . Its rst

g;n 1 n g;n

Chern class, denoted 2 H (M ), is called the ith Witten class. Since

i g;n

the notation wants to travel lightly, it is a little ambiguous. For example,

M is the ith Witten class it is not true that the image of in H (

g;n+1 i

of M . The euler class of the normal bundle of a parametrization f :

g;n+1

M ! M of a closed stratum is a pro duct of Witten classes of

g ;n g;n

i i

the factors. This illustrates a p ointwe are going to make, namely, that all

classes of interest app ear to b e obtainable from the Witten classes.

M b efore 7.2. The Mumford classes. Mumford de ned these classes for

the Witten classes were considered; Arbarello-Cornalba used the Witten

classes to extend Mumford's de nition to the p ointed case. It go es as follows:

if denotes the last Witten class on H ( M ), and r is a nonnegative

n+1 g;n+1

integer, then the r th Mumford class is

r +1 2r

:= ( ) 2 H ( M ):

r g;n

n+1

A minicourse of moduli of curves 287

(It is an interesting exercise to check that =2g2+n.) Arbarello and

Cornalba showed that is the rst Chern class of an ample line bundle.

They also proved that H ( M ) is generated by ;::: ; ; and the

g;n 1 n 1

Poincare duals of the irreducible comp onents of the b oundary @ M and

g;n

that these classes form a basis when g 3.

7.3. The tautological algebra. It is convenient to make an auxiliary

de nition rst: de ne the basic algebra B ( M ) of M as the subal-

g;n g;n

gebra of H (M ) generated by the Witten classes ( ) and the Mum-

g;n j j

ford classes ( ) . Then the tautological algebra of M , R(M ), is

r r 0 g;n g;n

by de nition the subalgebra of H ( M ) generated by the direct images

g;n

even

f ( B (M ) H (M ), where the maps f : M ! M

i g ;n g;n g ;n g;n

i i i i

run over the parametrizations of the strata. Since the tautological algebra

is made of algebraic classes we grade it by half the cohomological degree:

k 2k

R ( M ) H (M ). It is clear that the tautological algebra and the

g;n g;n

basic algebra have the same restriction to M ; we denote that restriction

g;n

by R(M ) and call it the tautological algebra of M .

g;n g;n

M are in the It is remarkable that all the known algebraic classes on

g;n

tautological subalgebra. We illustrate this with two examples.

Example 7.1 (The Ho dge bundle). If C is a stable curve of genus g 2,

then H (C; ! ) is a g -dimensional vector space. On the universal example

this gives a rank g vector bundle E over M called the Hodge bund le. (Since

H (C; ! ) only dep ends on the (generalized) Jacobian of C , E is the pull-

back of a bundle that is naturally de ned on a certain compacti cation of

the mo duli space of principally p olarized ab elian varieties of dimension g .)

M ) as an element Mumford [3 ] expressed the Chern class := c (E ) 2 H (

g i i

M ). of R (

Example 7.2 (The Weierstra lo ci). Supp ose C is a smo oth, connected pro-

jective curve C of genus g 2, p 2 C , and an l a p ositiveinteger. It is easy

to see that the following conditions are equivalent:

The linear system jl (p)j is of dimension 1.

There exists a nonconstant regular function on C fpg that has in p

with a p ole of order l .

There exists a nite morphism C ! P of degree l that is totally

rami ed in p.

By Riemann-Ro ch these conditions are always ful lled if l g +1. If l =2,

then the morphism f app earing in the last item must have degree 2 so that

C is hyp erelliptic and p is a Weierstra p oint.

288

These equivalent conditions de ne a closed subvariety W of the universal

smo oth curveofgenus g , C . Arbarello, who intro duced these varieties, noted

that W is irreducible of co dimension g +1l in C . Mumford expressed

2(g +1l ) g+1l

the corresp onding class in H (C ) as an elementofR (C ).

g g

The validity of Grothendieck's standard conjectures implies that the al-

even

gebraic classes on M make up a nondegenerate subspace of H (M )

g;n g;n

with resp ect to the intersection pro duct. Since we do not knowany algebraic

class that is not tautological, we ask:

Question 7.3. Do es R( M ) satisfy Poincare duality?

g;n

Since all tautological classes originate from Witten classes, this question

could, in principle, be answered for a given pair (g ;n ) if we would know

0 0

all the intersection numb ers

2 Q;

Mg;n

(where it is of course understo o d that this number is zero if the degree

k + + k of the integrand fails to equal 3g 3+n) for all (g; n) with

1 n

2g + n 2g + n . A marvelous conjecture of Witten (which is a conjecture

0 0

no longer) predicts the values of these numb ers. We shall state it in a form

that exhibits the algebro-geometric content b est (this formulation is due

to Dijkgraaf-E. Verlinde-H. Verlinde). For this purp ose it is convenient to

renormalize the intersection numb ers as follows:

;

[ ] := (2k + 1)!!(2k + 1)!! (2k + 1)!!

g 1 2 n

k k k

1 2

Mg;n

where (2k + 1)!! = 1:3:5: :(2k + 1). We use these to form the series in the

variables t ;t ;t ;::: :

0 1 2

X X

F := [ ] t t t :

g g

k k k k k k

n n

1 2 1 2

n=1

k 0;;k 0;:::;k 0

1 2

It is invariant under p ermutation of variables. The Witten conjectures assert

that these p olynomials satisfy a series of di erential equations indexed by

the integers 1. For index 1 this is the string equation:

@F @F

g g

= + t ; (2m +1)t

0;g m

@t @t

0 m1

A minicourse of moduli of curves 289

for index 0 the dilaton equation:

@F @F

g g

(2m +1)t = + t

m 1;g

@t @t

1 m

and for k 1we get:

@F @F

g g

(2m +1)t =

@t @t

k+1 m+k

> +

0 00

@t @t

m m

0 00

m +m =k 1

X X

0 00

@F @F

g g

: +

0 00

@t @t

m m

0 00 0 00

m +m =k 1 g +g =g

You mayverify that these equations determine the functions F completely

(note that F has no constant term). The string equation and the dilaton

equation involve a single genus only and were veri ed by Witten using stan-

dard arguments from algebriac geometry. But the equations for k 1were

proved in an entirely di erent manner: Kontsevich gave an amazing pro of

based on a triangulation of Teichmuller space on which the intersection num-

b ers app ear as integrals of explicitly given di erential forms. Yet there are

reasons to wish for a pro of within the realm of algebraic geometry. The form

M , of the equations is suggestive in this resp ect: the rst line involves the

g;n

but the second and third seem to b e ab out intersection numb ers formed on

irreducible comp onents of @ Mg; n ( and the 0 0 00 00 resp ectively).

f(g ;I );(g ;I )g

Consider this a challenge.

7.4. Fab er's conjectures. These concern the structure of R(M ). From

the de nition it is clear that these are generated by the restrictions of the

Mumford classes. Let us for convenience denote these classes instead of

jM . Fab er made the essential part of his conjectures around 1993. We

r g

shall not state them in their most precise form (we refer to [4] for that).

1. R is zero in degree g 1 and is of dimension one in degree g 2

and the cup pro duct

i g2i g2

R (M)g ) R (M ) !R (M ) Q

g g

is nondegenerate.

2. The classes ;::: ; generate R(M ) and satisfy no p olynomial

1 g

[g=3]

relation in degree [g=3].

290

Let us review the status of these conjectures. First, there is the evidence

provided Fab er himself: he checked his conjectures up to genus 15.

As to conjecture 1: the vanishing assertion was proved in a pap er of

g 2

mine where it was also shown that dim R (M ) 1. Subsequently Fab er

proved that the dimension is in fact equal to one. The rest of conjecture

1 remains op en. Conjecture 2 now seems settled: Harer had shown around

the time that Fab er made his conjectures that the kappa classes have no

p olynomial relations in degree [g=3] and Morita has recently announced

that ;::: ; generate R(M ).

1 g

[g=3]

Let me close with saying a bit more ab out Fab er's nonvanishing pro of.

2g 1

He observes that the class 2 R ( M ) restricts to zero on the

g g 1 g

b oundary @ M . This implies that for every u 2R ( M ), the intersection

g g

pro duct u only dep ends on ujM . So this de nes a `trace' t :

g g 1 g

R(M ) ! Q . Fab er proves that this trace is nonzero on . The unproven

g g 2

part of the conjecture can be phrased as saying that the asso ciated form

(u; v ) 2 R(M ) R(M ) 7! t(uv ) is nondegenerate. Fab er has also an

g g

explicit prop osal for the value of the trace on any monomial in the Mumford classes.

A minicourse of moduli of curves 291

References

[1] E. Arbarello, M. Cornalba, P.A. Griths, J. Harris: Geometry of Algebraic Curves,

Vol. I, Grundl. der Math. Wiss. 267, Springer 1985.

[2] J. Harris and I. Morrison: Moduli of Curves, Graduate texts in Math. 187, Springer

1998.

[3] D. Mumford Towards an enumerative geometry of the moduli space of curves, In

Arithmic and Geometry, II, Prgress in Math. 36, 271{326. Bikhauser 1983.

[4] Moduli of Curves and Abelian varieties, C. Fab er and E. Lo oijenga eds., Asp ects of

Mathematics E33, Vieweg 1999.