ñòð. 9 |

k Q); k 0:

Now k (z0 ) is associated to the imbedding

O!

k

j

,! P Q

P1

obtained by composing P1 ! P(Q) with the canonical imbeddings P(Q) ,!

Nk Q). This means that j is dened by:

P(

O

k

(xn 0 + xn 1 y1 + + yn n)

(x; y) 7 !

= xkn0 + xkn 1y1 + + yknkn

N

where kn 2 k Q. Then by a similar argument, using the criterion of De

Concini and Processi, it follows that Z2 is the wonderful compactication of

PGL(n)

245

Degenerations of moduli spaces

6 Concrete descriptions of the moduli spaces and

applications

A vector bundle V on X0 can be described by the following datum on the

normalisation X of X0 , namely by W = (V ) ( : X ! X0 ) and an

isomorphism j : Vp1 ! Vp2 of the bres of V at fpi g, i = 1; 2. One would

therefore expect to describe a torsion free sheaf on X0 as some limits of the

isomorphisms j. This leads to the notion of a generalized parabolic bundle

(GP B) due to U. Bhosle.

Unless otherwise stated we will hereafter restrict to the case of rank 2

vector bundles.

Denition 6.1. A GPB on X is a vector bundle E on X together with

an element of Gr (Ep1 Ep2 ; 2) i.e. a quotient

8

> Ep Ep ! Q ! 0; dim Q = 2

< 1 2

> or equivalently 0 ! N ! Ep Ep , dim N = 2

: (0 ! N ! Ep Ep ! Q ! 0).

1 2

1 2

We denote this GPB by (E; Q) (or (E; N)).

To (E; Q) we can canonically associate a torsion free sheaf F on X0 as

follows. Now Q need not have an OX -module structure. However it has an

OX0 -module structure (as a sky-scraper sheaf with support at p) and F is

dened by:

0 ! F ! (E) ! Q ! 0:

The important point is that deg F = deg E, since we have

(F) = ((E)) 2 = (E) 2 i:e:

deg F 2(g 1) = deg E 2[(g 1) 1] 2 = deg E 2(g 1):

Now any torsion free sheaf F on X0 can be represented by a GPB (E; Q)

in this manner. However, this representation is not unique, as we shall see

below.

Let E be a vector bundle on X and M a linear subspace of Ex , x 2 X.

Then we have two well-dened vector bundles E 0 ; E 00 on X called Hecke

246

modications dened by homomorphisms

j j

E 0 ! E; Im Ex = M; E ! E 00 ; Ker jx = M:

0

We can of course dene a Hecke modication dened at several points of X.

Proposition 6.1. Let F be a torsion free sheaf on X0. Then we have the

following:

(i) if (E; Q) is a GPB which represents F, then we have a homomorphism

( (F ) mod. torsion) ! E which is a Hecke modication at p1 ; p2

(ii) if F is a vector bundle, then the representation (E; Q) is unique, E =

(F ) and N is the graph of an isomorphism Ep1 ! Ep2

(iii) let F be of type 1 (i.e. F = m O at p). Then there are precisely two

GPB's (E; Q) and (E 0 ; Q0 ) which represent F and N (resp. N 0) is the

graph of a rank 1 homomorphism

000

Ep1 ! Ep2 (resp: Ep2 i! Ep1 ):

i

We have Hecke modications

(

( (F ) mod torsion) ! E (only modication at p2 associated to Im i)

( (F ) mod torsion) ! E 0 (only modication at p1 associated to ker i).

(iv) Let F be type 2 i.e. F = mm at p. In this case, there are an innite

number of GPB's which represent F and they are given as follows:

(a) (E1 ; Q1 ) and N1 is the graph of the 0-map (E1 )p1 ! (E1 )p2 .

(b) (E2 ; Q2 ) and N2 is the graph of the 0-map (E2 )p2 ! (E2 )p1 .

(c) Consider (E; Q) with N = K1 K2 , dim Ki = 1 and K1 ,! Ep1 ,

K2 ,! Ep2 so that Q = Q1 Q2 , Qi = Epi =Ki, i = 1; 2. We

denote by E 0 the Hecke modication of E, E 0 ! E such that

Im Epi in Epi is Ki , i = 1; 2. Then all (E; Q) such that E 0 =

0

(F ) mod torsion, represent F. We see that all these (E; Q) are

parametrized by P1 P1 .

247

Degenerations of moduli spaces

Remark 6.1. For the objects in (a) and (b) of (iv), we have the relation

(

E1 ( p2 ) = E 0 = E2 ( p1 ) or E2 = E1 (p1 p2 )

E 0 = (F ) mod torsion:

Note that if F is of rank 1 and type 1, then if (E1 ; Q1 ) and (E2 ; Q2 ) represent

F, we have

E2 = E1 (p1 p2 ):

Denition 6.2. A GPB (E; Q) on X is semi-stable (resp. stable) if for

every (resp. proper = 0) subsheaf E 0 of E, we have

6

8

> deg E 0 dim QE0 deg E dim Q

< (resp: <);

rk E 0 rk E

() > E0

: Q = the image of E 0 in Q.

Note that E 0 is a vector bundle and the canonical map E 0 ! E need not be

injective everywhere but only generically injective; it could even be a generic

isomorphism.

Remark 6.2.

(a) Let (E; Q) represent the torsion free sheaf F on X0 . Then one has (see

[NR], [Su]):

Fsemi-stable () (E; Q) semi-stable:

However, if F is stable, (E; Q) need not be stable. If (E; Q) is as in

(iv)(c) of Prop. 6.1 (which implies F is type 2), consider the Hecke

modication E 0 ! E which maps Epi onto Kpi and an isomorphism

0

outside fp1 ; p2 g. Then we see that the inequality in () is, in fact, an

equality. A similar argument works for (iv) (a), (b) of Prop. 6.1. Thus

for example, if F is semi-stable and torsion free of degree 1, it is stable;

however, say it is of type 2, then any (E; Q) which represents F is only

semi-stable and not stable.

(b) We can dene families of GPB's and then we obtain a functor

(GP B) ! (Torsion free sheaves on X0 ):

248

(c) There is a canonical moduli space associated to the semi-stable GPB's

of rank n and degree d of X. We denote this by GPBX (n; d); in par-

ticular GPBX (2; 1). It has a structure of a normal projective variety.

Further if (E; Q) is a semi-stable GPB and F is the semi-stable torsion

free sheaf on X0 associated to (E; Q) (see (a) of Remark 6.2 above),

the map (E; Q) 7 ! F denes a morphism GPBX (n; d) ! UX (n; d);

in fact GPBX (n; d) identies with the normalisation of UX (n; d) (see

[NR] and [Su]). Note that GPBX (2; 1) is not the set of isomorphism

classes of semi-stable GPB's. We have to have an equivalence relation.

(d) We have also a forget functor:

(GP B) ! (vector bundles on X):

This is not well-behaved with respect to semistability. If (E; Q) is

semi-stable (as a GPB), the underlying vector bundle need not be

semi-stable; further if E is semi-stable, (E; Q) need not be semi-stable.

Remark 6.3. Take two copies V1; V2 of a 2-dimensional vector space. Then

we have identications Aut Vi ' GL(2) and Isom (V1 ; V2 ) ' GL(2). Imbed-

ding Isom (V1 ; V2 ) by their graphs in Gr(V1 V2 ; 2), we get a compacti-

cation of GL(2). However, this is not a \good compactication" as the

complement of GL(2) is not a variety with normal crossings. Let v1 ( resp.

v2) represent the points of Gr(V1 V2 ; 2) corresponding to the zero homo-

morphism V1 ! V2 (resp. V2 ! V1 ) represented by its graph. We blow

up (Gr(V1

V2 ; 2) at these points v1 ; v2 and obtain a variety H with ex-

ceptional divisors H1 and H2 mapping to v1 ; v2 respectively. We see that

H ! Gr(V1

V2; 2) is Aut V1 Aut V2 equivariant. Further, this compact-

ication H of GL(2) ' Isom(V1 ; V2 ) is \good" as we check easily that its

complement has normal crossing singularities, i.e. H is a \good" compacti-

cation of GL(2) unlike Gr(V1

V2 ; 2).

Denition 6.3.

(a) An H-structure on a vector bundle V on X is just giving a point of H

where H is the blowing up of Gr(Vp1

Vp2 ; 2) as dened above, or we

can say that an H-bundle on X is a pair (V; h), where V is a vector

bundle, h 2 H; H being the blowing up of Gr(Vp1

Vp2 ; 2) as indicated

above. We have functors

(H bundles ) ! (GP B) ! ( Torsion free sheaves on X0 ):

249

Degenerations of moduli spaces

(b) We say that an H-bundle is stable if the corresponding torsion free

sheaf on X0 is stable (recall that we are in the rank 2 case).

Remark 6.4. We have a canonical morphism : G(2; 1) ! UX0 (2; 1). As

mentioned in Remark 6.2(c), GPBX (2; 1) is the normalisation of UX0 (2; 1).

Hence induces a canonical morphism

() : G(2; 1) ! GPBX (2; 1)

^

where G(2; 1) denotes the normalisation of G(2; 1). This map can be more

^

concretely seen as follows.

Let E be a vector bundle on Xk (of rank 2 and degree 1) such that EjR

is strictly positive and (E) is torsion free (which implies EjR is strictly

standard and restricts the possibilities as below). Then to E we shall now

associate, upto a nite number of choices, a GPB such that the underlying

bundle is of rank 2 and degree 1. This leads to the canonical morphism ()

above.

(i) k=0 i.e. E is a bundle on X0 . Then E is given by an element Isom

(Ep1 ; Ep2 ); E 0 = (E) and hence we associate this GPB structure on

00

E0

(ii) k = 1; EjR = O O(1)

X1

X O O(1)

We set E 0 = EjX . Let Li be the 1-dimensional subspace of Epi (= Epi )

0

dened by O(1)pi . Let E1 (resp. E2 ) be the Hecke modication:

(

E 0 ! E1 (resp. E2 );

0 0

Ker (Ep2 ! (E1 )p2 )(resp. Ker (Ep1 ! (E2 )p1 ) is L2 (resp. L1 )

250

Then we have well-dened rank 1 homomorphisms:

(E1 )p1 f! (E1 )p2 (resp. (E2 )p2 f! (E2 )p1 ):

1 2

Let N1 (resp. N2 ) be the graph of f1 (resp. f2 ). Then we have exact

sequences

0 ! N1 ! Ep1 Ep2 ! Q1 ! 0

0 ! N2 ! Ep1 Ep2 ! Q2 ! 0

and (E1 ; Q1 ); (E2 ; Q2 ) are the two GPB's which we associate to E.

They are the two GPB's which represent (E) (see (iii) of Prop.

6.1). One sees that E 0 = (F )mod torsion.

(iii) k = 1; EjR = O(1) O(1) We set E 0 = EjX . We check E 0 = (F )mod

torsion. Let E 0 ! Ei (i = 1; 2) be the Hecke modications de-

ned by Ei = E 0 (pi ), (i = 1; 2) and (E1 ; Q1 ) (resp. (E2 ; Q2 )) be

the GPB dened by the zero homomorphism (E1 )p2 ! (E1 )p1 (resp.

(E2 )p1 ! (E2 )p2 ). We assign to E these two GPB's. These two

GPB's represent (E).

p1 X1

X O(1) O(1)

p2

Given E 0 on X, we can extend E 0 to E on X1 with EjR ' O(1)O(1)

by giving isomorphisms:

0

g1 : Ep1 ! (O(1) O(1))p1 ;

0

g2 : Ep2 ! (O(1) O(1))p2

Now we can modify gi by an automorphism of O(1) O(1). Since gi

can be identied with elements of GL(2), we can suppose that g2 =

identity. Since there are two components, we can modify g1 by an

arbitrary scalar. Thus the number of ways of extending E 0 to E on X1

(with EjR = O(1) O(1)) depends upto an element of PGL(2).

251

Degenerations of moduli spaces

(iv) k = 2; EjRi = O O(1) We set E 0 = EjX and check that E 0 ' (F )mod

torsion.

X2

p1

O O(1)

X

O O(1)

p2

0

Let Ki Epi be the 1-dimensional subspace by the bre at pi ('

O(1)p1 ) of the canonical sub-bundle of EjRi . Let E 00 be the Hecke

modication E 0 ! E 00 such that Ker (Epi ! Epi ) = Ki ; i = 1; 2.

0 00

Let Ni = Im Epi , Qi = Epi =Ni , Q = Q1 Q2 . Then (E 00 ; Q) is the

0 00

GPB assigned to E.

The above discussion, especially of cases (iii) and (iv) indicates that

a bre of the morphism () is either a point or P3 (compare also the

general result in Remark 5.2). For the morphism H ! Gr(V1 V2; 2),

the bre over v1 (resp. v2 ) is P3 . Thus we may expect the following

result which is an important step in the concrete determination of the

Gieseker moduli space.

Theorem 6.1. On the set of isomorphism classes of stable H-bundles on

X of rank 2 and degree 1 (i.e. the underlying vector bundles have these

properties), there is a canonical structure of a smooth projective variety,

which can be identied with the normalisation G(2; 1) of the Gieseker variety

^

G(2; 1). We denote G(2; 1) by SII .

^

Proof. We do not give a formal proof but carry it out in a test case. In the

following considerations, compare Remark 6.4. Let us take a 1-parameter

Gieseker family of the following type : We have a T-morphism

X X0 T

!

252

where T is a smooth curve and we x a point t0 2 T (e.g. T = Spec . of a

d.v.r. and t0 is the closed point of T. We suppose that for t 6= t0 ; t : X !

X0 t X0 is an isomorphism. The curve Xt is the form Xk (0 k 2).

We have a vector bundle V on X which denes a Gieseker family of rank

2 and degree 1. A test case for the theorem is that one should be able to

associate to this Gieseker family a canonical 1-parameter family of stable

H-bundles on X. We can suppose that 1 k 2.

Let Z denote the normalisation of X. Then we have a canonical mor-

phism.

: Z ! X T (normalisation of X0 T):

Then we have the following possibilities:

(i) Z is obtained by blowing up either (p1 ; t0 ) or (p2 ; t0 ). Then the excep-

tional bre is D ' P1 (k = 1)

(ii) Z is obtained by blowing up both the points (p1 ; t0 ) and (p2 ; t0 ). Then

the exceptional bres are D1 ; D2 ' P1 and D1 \ D2 = ; (k = 2)

(iii) Z is obtained by rst blowing up say (p2 ; t0 ) and then blowing up a

point on the exceptional bre. Hence the exceptional bre is D1 [ D2 ,

such that Di ' P1 , the intersection D1 \D2 reduces to a point and the

exceptional bre maps to (p1 ; t0 ) or (p2 ; t0 ).

If T0 = Tnt0 , we can identify p T0 (p singular point of X0 ) as a sub-

scheme of XnXt0 . The closure S of p T0 is a section of X over T and S

meets one of the singular points Xt0 ' Xk . The above possibilities (i), (ii)

and (iii) depend upon the way S meets the singular point of Xk .

Let C; T1 ; T2 denote respectively the proper transforms of X t0 ; p1 T

and p2 T in Z. Let W denote the vector bundle on Z obtained as the

inverse image of V by the canonical morphism Z ! X. In the case (i)

above the restriction of W to D is either O O(1) or O(1) O(1). In all

the other cases the restriction of W to Di is O O(1).

Case k = 1 and WjD ' O O(1) Now D maps to (p1 ; t0 ) or (p2 ; t0 ), say

(p2 ; t0 ). Then we see that T2 does not meet C. We see that T2 meets D; T1

meets C and C meets D.

253

Degenerations of moduli spaces

T1 q1

q1 = C \ T1

C

q2 = D \ T2

T2

q2

D

We have a canonical isomorphism : WjT1 ! WjT2 , which denes the

vector bundle V (the non-normal variety is obtained by identifying T1

and T2 ). Let F be the direct image (W ) which is a torsion free sheaf on

X T and E = F - the double dual of F. Then E is a vector bundle on

X T. It is not dicult to see that we have a canonical homomorphism

j : W ! (E) such that when restricted to D, the trivial quotient line

bundle of WjD maps onto a line sub-bundle of (E)jD i.e. (E) is the

\push-forward Hecke modication" of W along D by the trivial quotient line

bundle of WjD . Then jjT2 : WjT2 ! (E)jT2 is the Hecke modication

(on T2 ) of WjT2 by the canonical 1-dimensional quotient of the bre of WjT2

at q2 (namely the bre of the canonical trivial quotient line bundle of WjD

at q2 ). We have identications

WjT1 ' (E)jT1 and (E)jTi ' Epi T (i = 1; 2):

Hence if we set = (jjT2 ), denes a homomorphism : Ep1 T ! Ep2 T

such that t (t 6= t0 ) is an isomorphism and 0 = t0 is of rank one. Now

denes a family of H-bundles, in fact a family of GPB's. This is the required

1-parameter family of H-bundles.

Case k = 2 and Z is obtained by blowing up (p1 ; t0 ) and (p2 ; t0 )

In this case we have:

254

q1

D1

T1

C

T2 D2

q2

We see that C meets D1 and D2 , T1 meets D1 at q1 ; T2 meets D2 at q2

and T1 ; T2 do not meet C. We dene E as above and then we get a canonical

homomorphism W ! (E). We have also a canonical isomorphism :

WjT1 ! WjT2 inducing an isomorphism 0 : Wq1 ! Wq2 . We see that the

kernel Ki of Wqi ! (E)qi identies with the bre of the line subbundle

of WjDi (= O(1)O) isomorphic to O(1), so that by Prop. 3.4 (and Remark

3.1) we have 0 (K1 ) \ K2 = (0). From this we deduce that under the

canonical homomorphism

Wq1 Wq2 ! (E)q1 (E)q2

the graph of 0 maps isomorphically onto the image of the above homomor-

phism, which is N1 N2 , Ni = Wqi =Ki . This means that the family of

GPB's represented by the isomorphisms t : Ep1;t ! Ep2 ;t (t 6= t0 ) special-

izes to the GPB t0 represented by (Et0 ; N1 N2 ). Thus t denes a family

of H-bundles, in fact a family of GPB's. This is the required 1-parameter

family of H-bundles.

Case k = 1 and WD = O(1) O(1). In this case the gure is as in the rst

case (k = 1) considered above. We dene E as before. We have canoni-

cal isomorphisms : WjT1 ! WjT2 , 0 : Wq1 ! Wq2 and a canonical

homomorphism j : W ! (E). We observe that (E) = W(D) and

j is a \push-forward Hecke modication" of W along D inducing the zero

homomorphism when restricted to D. We see that jjT2 : WjT2 ! (E)jT2

is the Hecke modication (on T2 ) inducing the zero map on Wq2 - the bre

at q2 . Then = (jjT2 ) denes a homomorphism : Ep1 T ! Ep2 T

such that t (t 6= t0 ) is an isomorphism and the local expression for t at

255

Degenerations of moduli spaces

t0 takes the form t (t - local coordinate at t0 ) such that (essentially ):

Ep1 ;t0 ! Ep2 ;t0 is an isomorphism. We get a GPB on Et0 by taking the

zero homomorphism Ep1 ;t0 ! Ep2 ;t0 and we endow it with an H-structure

by taking the image of in P(Hom (Ep1 ;t0 ; Ep2 ;t0 ) (which is canonically the

bre of H ! Gr(Ep1 ;t0 Ep2 ;t0 ; 2) over the point represented by the zero

homomorphism Ep1 ;t0 ! Ep2 ;t0 ). We check that t (t 6= t0 ) together with

this H-structure on Et0 denes a family of H-bundles. This is the required

1-parameter family of H-bundles.

Case k = 2 and as in (iii) above. In this case we have the gure

q1

T1

r = D1 \ D2

C

q2 T2

D2

r D1

The rst blow up gives D1 and then the next blow up gives D2 . Let E1 be

the push-forward Hecke modication of W along D2 for the canonical quo-

tient line bundle of WjD2 . We have a canonical homomorphism W ! E1

and canonical isomorphisms : WjT1 ! WjT2 and 0 : Wq1 ! Wq2 .

We claim that E1 jD1 ' O(1) O(1). To prove this observe rst that

WjD1 ! E1 jD1 is a Hecke modication. Further, for this homomorphism,

the kernel of the bre of WjD1 (' O O(1)) at r = D1 \ D2 , is not the

bre of the canonical line sub-bundle (' O(1)) of WjD1 . This is an easy

consequence of the fact that the bre at r of the canonical sub-bundles of

WjD1 and WjD2 generate the bre of W at r. Then the claim follows easily.

The homomorphism W ! E1 induces a rank one homomorphism Wq2 !

(E1 )q2 ' (E1 )r (the restriction of E1 to D2 is trivial). Then composing with

the isomorphism 0 : Wq1 (' (E1 )q1 ) ! Wq2 , we get a canonical rank one

homomorphism : (E1 )q1 ! (E1 )q2 ' (E1 )r .

Let Z1 denote the variety which gives the rst blow up D1 so that we

have the factorisation of the morphism :

Z ! Z1 ! X T:

256

Then E1 goes down to a vector bundle on Z1 and we denote this by the

same letter E1 . We have E1 jD1 ' O(1) O(1) and we are in a situation

similar to the case considered above (k = 1 and WjD = O(1) O(1)).

Set E2 = E1 (D1 ). Then the restriction of E2 to D1 is trivial and E goes

down to a vector bundle E on X T. Then we see easily that we have a

homomorphism : Ep1 T ! Ep2 T such that t (t 6= t0 ) is an isomorphism

and the local expression for t at t0 takes the form t (mod t2 ) such that

(essentially the above : Ep1 ;t0 ! Ep2 ;t0 is a rank one homomorphism.

By taking the image of in P (Hom (Ep1 ;t0 ; Ep2 ;t0 )), we get a family of

H-bundles as we did in the case above. This is the required family of H-

bundles.

One can give a structure of a smooth projective variety on the isomor-

phism classes of H stable bundles of rank 2 and degree one on X. We shall

outline a proof later (see Remark 6.6). Call this moduli space H(2; 1). The

preceding considerations show that we have a morphism SII = G(2; 1) !

^

H(2; 1). This is certainly birational. It is not hard to show that the bres

of this morphism are nite. This would show that it is an isomorphism.

Remark 6.5.

(a) Consider the moduli space UX = UX (2; 1) of stable vector bundles of

rank 2 and degree 1 on X. Let P be a Poincar bundle on XUX (2; 1).

e

Then Grass (Pp1 Pp2 ; 2) is a Grassmann bundle on UX . Then if P1 ; P2

are the principal GL(2) bundles associated to the vector bundles Ppi

on UX . Then we can construct a bundle over UX with bre H as a

bundle associated to P1 P2 . We call this variety SI . This variety is

\inductively good" with respect to the genus g i.e. it is a bre space

with bre H over the moduli space UX with X of genus (g 1). We

see we have a diagram with morphisms j1 ; j2 :

SI j! Gr(Ppi Pp2 ; 2) j! VX ;

1 2

We have of course a family of GPB's on X parametrized by Gr(Pp1

Pp2 ; 2) which is the set of all GPB's over the moduli space UX . We

get then a family of GPB's parametrized by SI .

(b) If V is a vector bundle on X, the set of all GPB's on V is parametrized

by Gr(Vp1 Vp2 ; 2) and those which yield vector bundles on X0 corre-

spond to the open subset Isom (Vp1 ; Vp2 ). One sees that the motivation

257

Degenerations of moduli spaces

for dening H bundles is to have a better compactication for the mod-

uli space of vector bundles on the singular curve X0 .

(c) We see that SI and SII are birational. In fact, we have the follow-

ing. Let 1 ,! SI be the closed subscheme such that the underlying

GPB's are not semi-stable or equivalently the corresponding tension

free sheaves on X0 are not stable. Similarly let 2 be the closed sub-

scheme of SII such that the underlying vector bundles on X are not

semi-stable (or equivalently not stable since we are in the degree 1

case). We see that

SI n1 ' SII n2 :

We have the following result essentially due to Gieseker (see [G]).

Theorem 6.2. The varieties 1; 2 are smooth. Let S be the blow up SI

along 1 and the inverse image of 1 in S. Then we have a canonical

morphism : S ! SII so that we have a diagram: ( : S ! SI canonical

map).

S

SI SII = G(2; 1)

^

j1

G(2; 1)

Pp2 ; 2

GrPp1

j2

UX UX0 (2; 1)

Further S is the blow up of SII along 2 and is the canonical morphism

S ! SII . Besides, we have the following properties of 's.

The variety of 1 ' PJ0 J1 (F1 ), where Ji = Jacobian of degree i on X

and F1 is the vector bundle on J0 J1 representing the space of all extensions

0 ! L0 ! E ! Li ! 0; Li Ji :

258

In fact 1 maps onto its image in UX (we see that these extensions

which are non-trivial dene stable bundles of degree 1 on X. These acquire

canonical GPB's dened by the quotients Ep1 Ep2 ! (L1 )p1 (L1 )p2 .

Similarly, one has a vector bundle F2 on J0 J1 such that 2 = PJ0 J1 (F2 ).

We have

= PJ0 J1 (F1 ) J0 J1 PJ0 J1 (F2 ):

1 2

J0 J1

Remark 6.6. We shall now indicate how SI and SII can be viewed as GIT

quotients for two dierent polarizations on the same variety. This gives

also the required construction of the variety structure on H(2; 1) (' SII )

stated in the proof of Theorem 6.1. One knows that we have a process of

blowing up and blowing down when comparing GIT quotients for dierent

polarisations (as exploited by Thaddeus [T]). This should clarify the blowing

up and blowing down process to obtain SII from SI .

We call a GPB (E; Q) on X -semi-stable (resp, -stable) if for every

(resp. proper 6= 0) subsheaf E 0 of E, we have

deg E 0 dim QE 0 deg E dim QE (resp: <):

rk E 0 rk E

In Def. 6.2 we had taken = 1.

We shall now relate this to GIT stability (see [U]) and ([NR]). We restrict

to the case of rank 2 and degree 1.

Let Q be the Quot scheme of quotients of the trivial bundle of rank

n(n >> 0) with Hilbert polynomial of a rank two vector bundle of suciently

259

Degenerations of moduli spaces

large odd degree. We denote by E the universal quotient on X Q. For a

suitable polarization on Q, we know that the GIT quotient Qs mod PGL(n)

e

is the moduli space UX (2; 1). Let Q denote the Q-scheme Gr(Ep1

Ep2 ; 2)-

the Grassmannian of 2 dimensional quotients of Ep1 = Ejpi Q. Then we have

e

a family of polarisations L parametrized by , 0 < < 1, on Q such that

(

GIT stability (resp. semi-stability) for L

() stability (resp. semi-stability) as above for GPB's.

This implies, in particular, that if Eq is the quotient sheaf of the trivial

e

rank n vector bundle corresponding to a point q 2 Q which is GIT semi-

stable for L , then Eq is locally free. The above assertion is in [U], it follows

also from the arguments in [NR]. The notion of -stability does not gure

in [NR] but only in [U]. It is also shown in [NR] that

(

1 stability (resp. semi-stability) i.e. as in Def. 6.2

) GIT stability (resp. semi-stability) for L1.

The converse is not true as there are GIT semi-stable objects for L1

which are not locally free. However, in each GIT semi-stable equivalence

class, there are vector bundles and the converse is true for vector bundles (see

[NR]). The GIT quotient for the polarisation L1 is the variety GPBX (2; 1)

(see Remark 6.2 (c)).

We see easily that stable (resp. semi-stable) GPB's remain the same,

respectively in the intervals 0 < < 2 and 2 < < 1 and consequently the

1 1

GIT quotients behave the same way. We denote the GIT quotients respec-

tively by GPB(0 ) and GPB(1 ). We see also that in these cases stable ()

semi-stable, so that GPB(1 ) (resp. GPB(0 )) is a smooth projective variety

and its underlying set is the set of isomorphism classes of -stable GPB's

with 2 < < 1 (resp. 0 < < 1 ). We denote the GIT quotients for L1

1

2 2

and L1 respectively by GPB( 2 1 ) and GPB(1)(= GPB (2; 1) as mentioned

X

above). By the general process of blowing up and blowing down for GIT

quotients for varying polarisations, we get canonical birational morphisms.

260

GPB(1 )

GPB(0 )

GPB(1)

GPB( 2 )

1

We observe that for 1 < < 1, we have:

2

(

GPB -stable ) GPB 1-semi-stable

GPB 1-semi-stable ) GPB stable.

From this it follows that GPB(1 ) (rather its underlying set) is, in fact,

the set of isomorphism classes of GPB's which dene stable torsion free

shears rank 2 and deg 1 on X0 , whereas GPB(1) = GPBX (2; 1) is a further

quotient of GPB(1 ). It is also easily seen that GPB(0 ) is a Grassmann

bundle Gr(Pp1 Pp2 ; 2) over UX (see Remark 6.5).

The maps (blow ups and blow downs) in (1) above result from the follow-

ing observation in GIT. Let us x a polarisation L (on a projective variety

with an action of a reductive group etc.). Then if L0 is a polarisation which

is suciently close to L, we have

(a) L0 semi-stability =) L semi-stability.

(b) L stability =) L0 stability.

(Note the slightly more general observation prior to Def.4.2 which is used

for the construction of the variety structure on G(n:d).) The set of L0 semi-

stable points is contained in that of L-semi-stable points so that we get a

canonical morphism.

L0 GIT quotient ! L GIT quotient.

Further, if the set of L stable points is non-empty, then the above is a

(proper) birational morphism. This gives the maps in (1) above. Further,

from general considerations, it follows that we can divide the ample cone

(rather the one with the group action) into nice regions in each of which

stable and semi-stable points remains the same. This is the general phe-

nomenon behind stable and semi-stable GPB's remaining the same in the

intervals 0 < < 2 and 1 < < 1.

1

2

261

Degenerations of moduli spaces

Now to get the H-stable moduli, we construct a scheme Q0 which is proper

e

over the open subset of Q corresponding to quotients which are locally free,

and corresponds to the blowing up of H of the Grassmannian. We can take

a suitable Q which contains Q0 as an open subset, Q being proper over

Q and we can suppose that the actions of the group lift to Q0; Q etc. Fix

a in such a way that 0 < < 2 . Let M be a polarisation on Q which

1

is relatively ample with respect to Q Q. Then (as we saw prior to

!

Def.4.2) if a is suciently small (a > 0) and < 1, we have

(a) q 2 Q is (L + aM) semi-stable then (q) is L semi-stable.

(b) q 2 Q is L stable, then 1 (q) is (L + aM) stable.

Now it is not dicult to see that the variety SI (see Remark 6.5) is the

GIT quotient for the polarisation for L +aM with < 2 and that SII is

1

the GIT quotient for L +aM with 2 < < 1. For all these polarisations we

1

have stability () semi-stability, so that we get, in particular, a canonical

structure of a projective variety, namely SII , on the set of isomorphism

classes of H-stable vector bundles on X of rank 2 and deg. 1 (mentioned in

the proof of theorem 6.1).

Let S 0 be the GIT quotient for L1 + aM. Then we get as in (1) above

2

(by general considerations explained above) canonical birational morphisms.

SII

SI

S0

The variety S in Theorem 6.2 seems to be the bre product SI S 0 SII .

Remark 6.7. Vanishing of Chern classes for the moduli space on a smooth

projective curve of genus g.

We shall now very brie

y outline Gieseker's proof of the conjecture of New-

stead and Ramanan, namely that

ci(

U2;1Y ) = 0; i > 2g 2;

262

where Y is a smooth curve of genus g, which can be taken as the generic

bre of X ! S.

One shows that there is a vector bundle

on G(2; 1)S such that the

restriction

of this bundle to the generic bre of G(2; 1)S over S is the

cotangent bundle and the restriction

0 to the closed bre is

G2;1 (log D0 ),

where D0 is the singular locus of G(n; d). This uses the fact that G(n; d)S

is regular and its closed bre is a divisor with normal crossings. Then one

shows that it suces to prove that ci (

0 ) = 0 for i > 2g 2 and in fact

e

that it suces to prove this vanishing for

0 the pull-back of

0 on the

e ee

normalisation G(2; 1) of G(2; 1). It is seen that

0 =

G2;1 (log D0 ), D

^

^

being the inverse image of D0 in G(2; 1). Then the problem reduces to

^

proving the vanishing of Chern classes of the pull-back of this bundle on

the variety S (see Theorem 6.2). One has an explicit hold on this bundle.

For the vanishing of Chern classes one uses the factorisation: S ! SI !

Gr(Pp1 Pp2 ; 2) ! UX . The proof is by induction on the genus and one

can start the induction process, since for g = 1, G(2; 1) is the curve X0 itself.

Hence we can suppose that the vanishing result holds for the moduli space

UX (2; 1) = UX on X. Then the required vanishing result follows by this

inductive argument.

7 Comments

(I) It should be possible to work out generalisations of Gieseker moduli

spaces for (n; d) 6= 1 say for X0 . For obvious reasons one cannot expect

normal crossing singularities since the quotients are not by free actions.

Semi-stability has to be more carefully dened. We have to add more

conditions besides the condition that the direct image by is torsion

free and semi-stable.

If there is more than one ordinary double point (say the curve is irre-

ducible), even if (n; d) = 1 the singularities for the generalized Gieseker

moduli spaces are not normal crossings since they are only products of

normal crossings.

(II) It should be possible to work out generalisations of Gieseker moduli

spaces for any family, say for stable curves over general base schemes,

263

Degenerations of moduli spaces

for any rank and degree.

(III) The method of Gieseker for constructing the moduli space G(2; 1) is by

giving criteria for semi-stability of the points in the Hilbert scheme rep-

resented by imbeddings of curves into a Grassmannian (of two planes).

This is a very natural method but seems complicated and therefore

dicult to generalize for arbitrary rank. It would be interesting to see

if our method would imply some results on the \Hilbert stability" of

imbeddings of curves into Grassmannians of n-planes.

(IV) The moduli space G(n; d) can be thought of as a moduli problem of

vector bundles on the curve Xn (we need not take all k, k < n) modulo

the action of the automorphism group of Xn which is identity on X.

One knows how to construct the moduli of semi-stable torsion free

sheaves on Xn (now thanks to Simpson for general projective schemes).

It would be interesting to construct the moduli spaces G(n; d) more

directly as quotients of the moduli spaces of vector bundles or torsion

free sheaves on Xn .

(V) In the proof of properness, one saw that a torsion free sheaf F on the

isolated normal singularity represented by C is the invariant direct

image of a vector bundle on a nite covering represented by a disc. In

fact, it is easily checked that a torsion free sheaf on X0 is, locally at

p, an invariant direct image of -vector bundle on a ramied Galois

covering with Galois group ( -cyclic group). This may suggest a

good denition for G-objects on a nodal singularity for a semi-simple

algebraic group G.

(VI) These moduli spaces should be considered as solutions of the moduli

problem associated to the following objects over X0 :

f(; E); proper map X 0 ! X0 , E a vector bundle on X0

such that (E) is torsion free and is an isomorphism over X0 nfpg:

We have of course to x invariants. It is tempting to ask for generali-

sations when X0 is replaced by a higher dimensional variety, say even

a smooth surface. We may get compact moduli spaces only with the

use of vector bundles but over varying varieties dominating the quasi

variety.

264

Acknowledgements. This is a revised version of the hand-written notes dis-

tributed during the time of the lectures (School on algebraic geometry, ICTP,

July-August 1999). The author thanks D.S. Nagaraj for all his help in this

work.

265

Degenerations of moduli spaces

References

[D-P] C. De Concini nad C. Processi - Complete symmetric varieties, Lecture

Notes in Mathematics, 996, Springer-Verlag.

[F] G. Faltings - Moduli stacks for bundles on semistable curves, Math.

Ann. 304, (1996), No. 3, p. 489-515.

[G] D. Gieseker - A degeneration of the moduli space of stable bundles, J.

Di. Geometry 19, (1984), p. 173-206.

[K] Ivan Kausz - On a modular compactication of GLn, math.

AG/9910166.

[NR] M.S. Narasimhan and T.R. Ramadas - Factorisation of generalised

theta function I, Invent. Math. 114, (1993), p. 565-623.

[NS] D.S. Nagaraj and C.S. Seshadri - Degeneration of the moduli spaces

of vector bundles on curves II (Generalized Gieseker moduli spaces),

Proc. Indian Acad. Sci. (Math. Sci.), 109, (1999), p. 165-201.

[S] C.S. Seshadri - Fibrs vectoriels sur les courbes algbriques, Astrisque

e e e

96 (1982).

[St] E. Strickland - On the canonical bundle of the determinantal variety,

J. Alg. 75, (1982), p. 523-537.

[Su] Xiaotao Sun - Degeneration of moduli spaces and generalized theta

functions - to appear.

[Te] M. Teixidor i Bigas - Compactication of (semi) stable vector bundles:

two points of view (Preprint).

[T] Michael Thaddeus - Geometric invariant theory and

ips, J. AMS, Vol.

9, (1996), p. 691-723.

[U] Usha Bhosle N. - Generalized parabolic sheaves on an integral projec-

tive curve, Proc. Indian Acad. Sci. (Math. Sci.), 102, (1992), p. 13-22,

- Generalized parabolic bundles and applications,

Proc. Indian Acad. Sci. (Math. Sci.), 106, (1996), p. 403-420.

A minicourse on moduli of curves

Eduard Looijenga

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

looijeng@math.ruu.nl

Abstract

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

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

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

algebraic classes that naturally live on these spaces; these can be thought of

as the characteristic classes for bundles of curves.

Contents

1. Structures on a surface 271

2. Riemann's moduli count 273

3. Orbifolds and the Teichmller approach

u 274

4. Grothendieck's view point 275

4.1. Kodaira-Spencer maps 275

4.2. The deformation category 276

4.3. Orbifold structure on Mg . 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. Stratication of Mg;n 285

7. Tautological classes 286

7.1. The Witten classes 286

7.2. The Mumford classes 286

7.3. The tautological algebra 287

7.4. Faber'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 positive denite inner product ( )

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

is lost, but we retain the notion of angle, for if v1 ; v2 2 T are independent,

then

(v v

\(v1 ; v2 ) := kv 1kkv2 )k

1 2

does not change if we multiply ( ) with a positive scalar. A complex structure

on T is a linear automorphism J of T such that J 2 = 1V ; p makes T

this

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

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

2

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

we have an orientation prescribed by the condition that (v; Jv) is oriented

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

product preserved by J.

Let S be an oriented C 1 surface. A conformal structure on S is given

by a smooth Riemann metric on S given up to multiplication by a positive

C 1 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 2 = 1TS , that is compatible with the given orientation.

Given such a structure, then we have a notion of holomorphic function: a C 1

function f : U ! C on an open subset U of S open is said to be holomorphic

p

if for all p 2 U, dfp Jp = 1dfp : TpS ! C . This generalizes the familiar

notion for if S happens to be C with its standard almost-complex structure,

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

is a special property of dimension two that S admits an atlas consisting of

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

metric on a neighborhood of p 2 S we can nd local coordinates x; y at p

such that the metric takes the form c(x; y)(dx2 + dy2 ).) Coordinate changes

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

nd that S has actually a complex-analytic structure.

A surface equipped 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-Roch theorem and Serre-duality and the notions that enter here.

These give us for instance

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

complex-analytically embedded in Pg+1 such that the image is a nonsingular

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

thus receives is canonical.

The proof may be sketched as follows. Choose p 2 C. Then the complete

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

complex-analytic embedding of C in Pg+1 of degree 2g + 1. A theorem of

Chow asserts that a closed analytic subvariety of a complex projective space

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

the algebraic structure is unique: if C is complex analytically embedded

into two projective spaces as C1 Pk and C2 Pl , then consider the

diagonal embedding of C in Pk Pl , composed with the Segre embedding of

Pk Pl in Pkl+k+l ; by Chow's theorem the image is a complex projective curve

C3. The curves C1 and C2 are now obtained as images of C3 under linear

projections. These are therefore (algebraic) morphisms that are complex-

analytic isomorphisms. Such morphisms are always algebraic isomorphisms.

The above theorem shows in particular that a compact Riemann surface

of genus zero resp. one is isomorphic to P1 resp. to a nonsingular curve in

P2 of degree 3.

What can we say about the automorphism group of a compact Riemann

surface C of genus g? If g = 0, then we can assume C = P1 and Aut(P1 ) is

then just the group of fractional linear transformations z 7! (az+b)(cz+d) 1

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 be based on the uniformization theorem, another on the fact

that Aut(C) acts faithfully on H1 (C; Z) (any automorphism acting trivially

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

hence must be the identity) and preserves a positive denite Hermitian form

on H1 (C; C ).

We denote by Mg 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 Mg when g 2. We will discuss four approaches

to this:

A minicourse of moduli of curves 273

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

y.

The approach through Teichmller theory. Actually there are several

u

of this type, but we mention just one. More is said about this in Hain's

lectures.

The introduction of an orbifold structure on Mg in the spirit of Grothen-

dieck's formalization and generalization of the Kodaira-Spencer theory.

The introduction of a quasi-projective structure on Mg by means of

geometric invariant theory.

The last two approaches lead us to consider a compactication of Mg as

well.

2. Riemann's moduli count

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

Choose a point p 2 C. By Riemann-Roch the linear system jd(p)j has

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

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

assumption ensures that this pencil has no xed points. Choose an ane

coordinate w on L such that w = 1 denes d(p). We now have a nite

morphism C ! P1 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):

X

x (f) = 2g 1 + d;

x2C fpg

where x (f) is the ramication index of f atP (= the order of vanishing

x

of df at x). The discriminant divisor Df is x2C fpg x (f)(f(x)) (so the

coecient of w 2 C is the sum of the ramication indices of the points of

f 1 (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 ! P1 (up to isomorphism) with nite

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

dimensional projective space of eective degree 2g 1 + d divisors on P1 the

discriminant divisors make up a Zariski open subset. We now count moduli

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) (d g 1

parameters) and the ane coordinate w (2 parameters). Hence the number

of parameters remaining for C is

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

274

This suggests that Mg is like a variety of dimension 3g 3.

3. Orbifolds and the Teichmuller approach

We begin with a modest discussion of orbifolds. Let G be a Lie group

acting smoothly and properly on a manifold M. Proper means that the map

(g; p) 2 G M ! (g(p); p) 2 M M is proper; 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

smooth manifold: for every p 2 M, choose a submanifold S of M through p

such that TpS 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 denes a smooth atlas for GnM, making it a

manifold.

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

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

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

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

injection GpnS ! GnM. This is in fact an open embedding and hence GnM

is locally like a manifold modulo a nite group. It is often very useful to

remember the local genesis of such a space, because this information cannot

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

roots of unity on a one dimensional complex vector space has orbit space

isomorphic to R2 , 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 smooth manifold on which a nite group G acts, and h is an

open embedding of the orbits space U nG in X. The images of these open

embeddings must cover X and there should be compatibility relations on

overlaps. It is understood that two such atlasses whose union is also atlas

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

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

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

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

the atlas is closed under the formation of bered products (`intersections'):

U X U with its G G -action and the identication 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 eective. I leave it to you 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 rened notion of orbifold (a `stack'), but

we will not go into this now.

The case that concerns us is in innite dimensional analogue of the above

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

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

preserving dieomorphisms take the role of G. It is understood here that

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

dimensional Lie group on an innite dimensional space. This action turns

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

codimension 6g 6 and so it is at least plausible that Mg has the structure

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

justied 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 purpose, for the methods and notions that we need

come up in virtually all deformation problems.

4.1. Kodaira-Spencer maps. Suppose : C ! B is a proper (holomor-

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

bration theorem, is then locally trivial in the C 1-category, that is,

for every b 2 B we can nd an open U 3 b and a smooth retraction

˜

h : CU = 1 U ! Cb = 1 (b) such that h = (h; ) : CU ! Cb U is

a dieomorphism. In particular, when B is connected, then all bers of

are mutually dieomorphic. If both B and the bers 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 bers Cb are mu-

tually isomorphic as complex manifolds. Suppose that the family is trivial

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

˜

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

˜

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

Suppose now the converse, namely, that every holomorphic vector eld at

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

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

open U 3 b (always obtainable by means of a coordiante chart) that lifts

holomorphically to a vector eld on CU . The properness of ensures that

276

this lift is integrable to a holomorphic

ow on CU . The

ow surfaces (=

complex

ow lines) produces the desired retraction h : CU ! Cb . The case

of a higher dimensional base goes 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 begin with the

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

have an exact sequence

0 ! Tx Cb ! Tx C ! Tb B ! 0; b := (x):

This sheaes as an exact sequence of OC -modules

0 ! C=B ! C ! B ! 0

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

subsheaf of C of vector elds that are tangent to the bers of ). Now take

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

0 ! C=B ! C ! B ! 0:

This sequence diplays our lifting problem: an element of B is holomorphi-

cally liftable i it is in the image of C=B . But the sequence may fail to be

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

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

0 ! C=B ! C ! B !R1 C=B ! :

The sheaf R1 C=B is a coherent OB -module, whose value at b is equal to

the cohomology group H 1 (Cb ; Cb ). So an element of B is holomorphically

liftable i its image under vanishes. Hence gives us a good idea of how

nontrivial the family at a point b is: the family is locally trivial at b i is

zero in b. Both and its value at a point b, (b) : Tb B ! H 1 (Cb ; Cb ), are

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; b0 ) is given by a proper

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

point b0 2 B and an isomorphism : C Cb0 , with the understanding

=

that replacing by its restriction to a neighborhood of b0 in B denes the

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

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

of complex structure on C parametrized by the manifold germ (B; b0 ).

A minicourse of moduli of curves 277

The deformations of C are objects of a category: a morphism from (;0 0 )

˜

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

C0 ! C

˜

? ?

? ?

0 y y

(B 0 ; b00 ) ! (B; b0 )

˜

such that the square is cartesian (this basically says that sends the ber Cb0 0

˜

isomorphically to the bre C(b0 ) ) and 0 = . So (; ) is more interesting

than (;0 0 ) as every ber of the latter is present in the former. In this sense,

the most interesting object would be a nal object, if it exists (which is often

not the case). A deformation (; ) is said to be universal if it is a nal object

for this category. So this means that for every deformation (0 ; 0 ) of C there

is a unique morphism (0 ; 0 ) ! (; ). A universal deformation is unique

up to unique isomorphism (a general property of nal objects). We also

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

then (; g 1 ) is another deformation of C and so there is a unique morphism

˜ ˜ ˜˜

(g ; g )) : (; g 1 ) ! (; ). The uniqueness implies that gh = g h and

˜

that 1 is the identity. So the action of Aut(C) extends to (C; Cb0 ). Similarly,

Aut(C) acts on (B; b0 ) such that is equivariant.

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

be inconvenient. The custom is to allow B to be singular. The submersivity

requirement for is then replaced by the condition that be locally trivial

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

˜

(Cbo ; 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 smooth base was

only for didactical purposes: a universal deformation is always understood

to be the nal object of this bigger category (and therefore need not have a

smooth base).

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

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

is that it be

at, which is an algebraic way of saying that the map must

be open. In contrast to the situation considered above, the topological type

of C can now change (simple example: take the family of conics in P2 with

ane equation y2 = x2 +t). The Kodaira-Spencer theory has to be modied

as well. For example, H 1 (C; C ) must be replaced by Ext1 (

C ; OC ).

278

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

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

˜

deformation (; 0 ) there exists a morphism (; ) : (0 ; 0 ) ! (; ) 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 Mg . 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

Tb0 B ! H 1 (Cb0 ; Cb0 ) H 1 (C; C ) is an isomorphism.

=

(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 Cb1 ! Cb2 is the restriction of the

action of an automorphism of C.

So the universal deformation of C is smooth of dimension h1 (C ). A

simple application of Riemann-Roch shows that this number is 3g 3.

A universal deformation as in (iii) denes a map B ! Mg that factorizes

over an injection Aut(C)nB ! Mg . We give Mg the nest topology that

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

theorem that with this topology the maps Aut(C)nB ! Mg become open

embeddings. It is harder to prove that the topology is Hausdor. Thus Mg

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

3g 3.

Remark 4.4. The orbifold structure on Mg has the property that every orb-

ifold chart (U; G; h : GnU ! Mg 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 hyperelliptic). For this reason, the

orbifold structure as dened here is not quite adequate and we have to resort

to a more sophisticated version: ultimately we want only charts that support

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

of families.

The space Mg is not compact. The reason is that one easily denes

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

with monodromy of innite order. A classic example is that of degerating

A minicourse of moduli of curves 279

family of genus one curves given by the ane equation y2 = x3 +x2 +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 x3 by

x2g+1. Such a family denes an analytic map f0g ! Mg such that the

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

suppose the opposite. One then shows that the map f0g ! Mg must

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

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

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

monodromy. This contradicts our assumption. The remedy is simple in

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

as possible). This leads us to the next topic.

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

is, a connected complex projective whose singularities are normal crossings

(nodes), analytically locally isomorphic to the union of the two coordinate

axes in C 2 at the origin) such that

the euler characteristic of every connected component of the smooth

part of C is negative.

The genus of such a curve can be dened algebro-geometrically as h1 (OC )

or topologically by the formula 2 2g = e(Creg ). So it has to be 2. The

itemized condition is equivalent to each of the following ones:

g 2 and Creg has no connected component isomorphic to P1 f1g

or P1 f0; 1g,

Aut(C) is nite,

C has no innitesimal automorphisms.

Topologically a stable genus g curve is obtained as follows. Let Sg be a

closed oriented genus g surface. Choose on S a nite collection of embedded

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

sense that it bounds a disk. Then the space obtained by contracting each of

the circles underlies a stable curve and all these topological types are thus

obtained. (Note that removal of an embedded circle from a surface does not

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

which is almost as good as if the curve were smooth:

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 Cb1 ! Cb2 is the restriction of the

action of an automorphism of C,

(ii) all bers are stable genus g curves,

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

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

and the (x )x2Csing cross normally.

So the complement of the normal crossing hypersurface := [x2Csing x

in B parametrizes smooth genus g curves. If there is just one singular point,

then you may picture the degeneration from a smooth curve Cb to the sin-

gular curve Cb0 C in metric terms as by letting the circumference of the

=

embedded circle on Sg that denes the topological type of C go to zero. The

monodromy around Bx is in this picture the Dehn twist along that circle

(see the lectures by Hain).

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

above 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 Mg of dimension

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

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

This is why Mg is called the Deligne-Mumford compactication of Mg . A

generic point of the boundary divisor corresponds to a stable curve C with

just one singular point. The underlying topological type of such a curve is

determined by a nontrivial isotopy class of an embedded circle on Sg . The

following cases occur:

0 : C is irreducible (Sg is connected) or

fg0 ;g00 g : C is the one point union of two smooth curves of positive genera g0 ; g00

with sum g0 + g00 = g (Sg disconnected with components punctured

surfaces of genera g0 and g00 ).

These cases correspond to irreducible components of @Mg . We denote them

by 0 and fg0 ;g00 g .

5. The approach through geometric invariant theory

Perhaps the most appealing way to arrive at Mg and its Deligne-Mumford

compactication 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 projective category. The disadvantage is that we

do not know a priori what objects we are parametrizing.

A minicourse of moduli of curves 281

Let us begin with a minimalist discussion of the general theory. Let a

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

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

called semistable if it is the projection of a G-orbit in C r+1 f0g that does

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

contained in X is a subvariety of X. This subvariety need not be closed and

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

results of Hilbert and Mumford are as follows:

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

orbit that is closed in X ss .

2. There exists a positive integer N such that the semistable orbits that

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

polynomials of degree N.

3. If RX stands for the homogeneous coordinate ring of X, then the sub-

G

ring of its G-invariants RX is noetherian.

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

Property 1 implies that there is a natural quotient map X ss ! GnnX. If

G happens to act properly on X ss, then every orbit in X ss is closed in X ss

and so GnnX will be just the orbit set GnX ss. From property 2 it follows

G

that if f0 ; : : : ; fm is a basis of the degree N part of RX , then the map

[f0 : : fm ] : X ss ! Pm is well dened and factorizes over an injection

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

thus GnnX acquires the structure of a projective variety. (A more intrinsic

G

way to give it that structure is to identify it with Proj(RX ).)

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

to know what the dualizing sheaf !C of a nodal curve C is: it is the coherent

subsheaf of the sheaf of meromorphic dierentials characterized by the prop-

erty that on Creg it is the sheaf of regular dierentials, whereas at a node p

we allow a local section to have on each of the two branches a pole of order

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

always a line bundle (as opposed to

C ) and that its degree is 2g(C) 2. It

k

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

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

Serre duality. But that property 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,

k

h0 (!C ) = (2k 1)(g 1). Let us x for each k 2 a complex vector space

282

k

Vk of dimension dk := (2k 1)(g 1) (k 2). Choose k 3. Since !C is

k

very ample, we have an embedding of C in P(H 0 (C; !C ) ). The choice of

an isomorphism : H 0 (C; !C ) Vk allows us to identify C with a curve

k =

C P(Vk ). It is a standard result of projective geometry that for m large

enough,

1. the degree m hypersurfaces in P(Vk ) induce on C a complete linear

system,

2. C is an intersection of degree m hypersurfaces.

In other words, the natural map

Symm (Vk ) Symm H 0 (C; !C ) ! H 0 (C; !C )

k

mk

=

is surjective (property 1) and its kernel denes C (property 2). So the

image W Symm Vk of the dual of this map is of dimension dmk and

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

and regard it as a point of P(^dmk Symm Vk ). Now let Xk;m be the set of

points in P(^dmk Symm Vk ) 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(Vk )-

action on P(^dmk Symm Vk ). One can show that SL(Vk ) acts properly on

Xk;m . It is clear from the construction that as a set, SL(Vk )nXk;m may be

identied with Mg . The fundamental result is

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

cus of the closure of Xk;m in P(^dmk Symm Vk ) is Xk;m itself.

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

taining Mg as an open dense subvariety

In particular, Mg acquires a quasi-projective structure. As one may ex-

pect, the structure of projective variety Mg is compatible with the analytic

structure dened before. Incidentally, geometric invariant theory also allows

us to put the orbifold structure on Mg , 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 pointed curves. If n is a nonnegative integer, then an n-pointed curve

is a curve C together with n numbered points x1 ; : : : ; xn on its smooth part

Creg . If (C; x1 ; : : : ; xn) is an n-pointed smooth projective genus g curve, then

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 way as for Mg one shows that the set of

isomorphism classes Mg;n of n-pointed smooth projective genus g curves has

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

for Mg , we compactify Mg;n by allowing mild degenerations. The relevant

denition is as follows:

An n-pointed curve (C; x1 ; : : : ; xn ) is said to be stable if C is a nodal curve

C such that

the euler characteristic of every connected component of

Creg fx1 ; : : : ; xng is negative,

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

2g 2 + n > 0 (where g is the genus dened as before) and Creg has

no connected component isomorphic to P1 f1g or P1 f0; 1g,

Aut(C; x1; : : : ; xn) is nite,

(C; x1; : : : ; xn) has no innitesimal automorphisms.

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

type (g; n). The underlying topology is obtained as follows: x p1 ; : : : ; pn

distinct points of our surface Sg and choose on Sg fp1 ; : : : ; pn g a nite

collection of embedded circles (e )e2E in distinct isotopy classes relative to

Sg fp1; : : : ; png such that none of these bounds a disk on Sg containing at

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

way of describing the topological type that we will use later. It is given by

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