<< ñòð. 9(âñåãî 12)ÑÎÄÅÐÆÀÍÈÅ >>
k : Z2 ! Gr(kn + 1;
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 de ned by:
P(

O
k
(xn 0 + xn 1 y1 +    + yn n)
(x; y) 7 !
= xkn0 + xkn 1y1 +    + yknkn
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 compacti cation 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.

De nition 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
de ned 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-de ned vector bundles E 0 ; E 00 on X called Hecke
246

modi cations de ned by homomorphisms
j j
E 0 ! E; Im Ex = M; E ! E 00 ; Ker jx = M:
0

We can of course de ne a Hecke modi cation de ned 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 modi cation 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 modi cations
(
( (F ) mod torsion) ! E (only modi cation at p2 associated to Im i)
( (F ) mod torsion) ! E 0 (only modi cation at p1 associated to ker i).

(iv) Let F be type 2 i.e. F = mm at p. In this case, there are an in nite
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 modi cation 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 ):
De nition 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
modi cation 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 de ne 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 de nes a morphism GPBX (n; d) ! UX (n; d);
in fact GPBX (n; d) identi es 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 identi cations 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 compacti cation" 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-
i cation 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).
De nition 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 de ned 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
de ned by O(1)pi . Let E1 (resp. E2 ) be the Hecke modi cation:
(
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-de ned 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 modi cations de-
ned by Ei = E 0 (pi ), (i = 1; 2) and (E1 ; Q1 ) (resp. (E2 ; Q2 )) be
the GPB de ned 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 identi ed 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
modi cation 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 identi ed 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 de nes 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 de nes 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 dicult 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 modi cation" of W along D by the trivial quotient line
bundle of WjD . Then jjT2 : WjT2 !  (E)jT2 is the Hecke modi cation
(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 identi cations

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

Hence if we set  = (jjT2 ),  de nes a homomorphism  : Ep1 T ! Ep2 T
such that t (t 6= t0 ) is an isomorphism and 0 = t0 is of rank one. Now 
de nes 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 de ne 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 identi es 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 de nes 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 de ne 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 modi cation" of W along D inducing the zero
homomorphism when restricted to D. We see that jjT2 : WjT2 !  (E)jT2
is the Hecke modi cation (on T2 ) inducing the zero map on Wq2 - the bre
at q2 . Then  = (jjT2  ) de nes 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 de nes 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 modi cation 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 modi cation. 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 XUX (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 de ning H bundles is to have a better compacti cation 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 n1 ' SII n2 :
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
GrPp1


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 de ne stable bundles of degree 1 on X. These acquire
canonical GPB's de ned 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 di erent 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 di erent
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 suciently
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 de ne 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 suciently 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 suciently 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 dicult 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-
ci(
U2;1Y ) = 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
G2;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 suces to prove that ci (
0 ) = 0 for i > 2g 2 and in fact
e
that it suces 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 =
G2;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.

(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 de ned. 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
dicult 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 rami ed Galois
covering with Galois group ( -cyclic group). This may suggest a
good de nition 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 compacti cation 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 - Fibrs vectoriels sur les courbes algbriques, Astrisque
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 - Compacti cation 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 Teichmller 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. Strati cation 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 de nite 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 satis es 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
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 de nes 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 de nite 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 Teichmller theory. Actually there are several
u
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 compacti cation 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 ane
coordinate w on L such that w = 1 de nes 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 rami cation 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
coecient of w 2 C is the sum of the rami cation 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 e ective 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 ane 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 de nes 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
de ne 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 U nG ) 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 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 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 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 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 di eomorphisms take the role of G. It is understood 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 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
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 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 di eomorphism. In particular, when B is connected, then all bers of 
are mutually di eomorphic. 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 shea es 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 de nes 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
ane equation y2 = x2 +t). The Kodaira-Spencer theory has to be modi ed
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) de nes 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 dicult 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 de ned 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 de nes
families of smooth genus g curves over the punctured unit disk C !  f0g
with monodromy of in nite order. A classic example is that of degerating
A minicourse of moduli of curves 279

family of genus one curves given by the ane equation y2 = x3 +x2 +t, with
t the parameter of the unit disk. I admit that we dismissed genus one, but
it is not dicult to generalize to higher genus: for example, replace x3 by
x2g+1. Such a family de nes 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 rami ed 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 de nition 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 de ned 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 in nitesimal 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 de nes 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 compacti cation 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
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 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 de ned 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 di erentials characterized by the prop-
erty that on Creg it is the sheaf of regular di erentials, 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 de nes 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
identi ed with Mg . The fundamental result is
Theorem 5.1 (Gieseker). For k and m suciently 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 de ned 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
de nition 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 de ned 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 in nitesimal 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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