<< ñòð. 8(âñåãî 12)ÑÎÄÅÐÆÀÍÈÅ >>

Lecture given at the
School on Algebraic Geometry
Trieste, 26 July { 13 August 1999

LNS001005

 css@smi.ernet.in
Contents
1 Introduction 209
2 Review of basic facts of the moduli space U(n; d)
on X0 210
3 Vector bundles on the curves Xk 212
4 The moduli space 221
5 Properness and specialization 229
6 Concrete descriptions of the moduli spaces and
applications 245
References 265
209
Degenerations of moduli spaces

1 Introduction
Let Y be a smooth projective curve of genus g and UY = UY (n; d) the
moduli space of (semi-stable) vector bundles on Y of rank n and degree
d. One strategy for studying the variety UY in depth is by the method of
degeneration (or specialization), namely one specializes Y to a curve X0 ,
say with only one singularity which is an ordinary double point. One would
have a moduli object UX0 on X0 such that UY specializes to UX0 and one
expects a close relationship between UX0 and the moduli space UX on the
normalisation X of X0 . Since the genus of X is (g 1), one would then obtain
a machinery for studying UY , especially its properties which are amenable
to specialization, by induction on g.
This strategy was employed by Gieseker to prove a conjecture of New-
stead and Ramanan for moduli spaces in rank 2, namely that the Chern
classes ci of the smooth projective variety UY (2; 1) vanish for i > 2 dim UY (2; 1),
1
i.e. i  (2g 1), since dim UY (2; 1) = (4g 3) (see [G]). A similar one was
employed by M.S. Narasimhan and T.R. Ramadas to prove what is called the
factorisation rule in the rank two case and recently it has been generalized
to arbitrary rank by Xiaotao Sun (see [NR] and [Su]).
Gieseker constructed a moduli object GX0 on X0 (we do not denote this
by UX0 since this will stand for a moduli space of torsion free sheaves on X0 )
such that it has nice singularities and UY (2; 1) specializes to GX0 . Further,
he gave a concrete realization of GX0 via the moduli space UX (2; 1), which
helps in solving the conjecture of Newstead and Ramanan by induction on
the genus.
Recently, in collaboration with D.S. Nagaraj, we have been able to gen-
eralize Gieseker's construction of GX0 for arbitrary rank (see [NS]). A good
part of these lectures is devoted to outlining this construction. Our method
for the global construction is quite di erent from that of Gieseker; it consists
in relating the Gieseker moduli space to that of torsion free sheaves on X0 ,
an aspect which does not gure in Gieseker's work.
We give also a brief sketch of the proof of the conjecture of Newstead
and Ramanan in the rank two case. This is essentially on the lines as done
by Gieseker ([G]). However, our proof for the concrete realization of the
Gieseker moduli space (via the moduli space UX (2; 1)) connects it with the
moduli space of GPB's (generalized parabolic bundles) and shows a close
210

relationship with good compacti cations of the full linear group. In this
sense it appears more conceptual and suggests a natural candidate for the
concrete realization in arbitrary rank.
For related work see [Te] and especially the very recent one [K].

2 Review of basic facts of the moduli space U n; d

on X0

We work over an algebraically closed base eld K, which we can take to be
C - the eld of complex numbers, as the emphasis is not on the characteristic
of K.
Let X0 be a projective, irreducible curve with only one singularity at
p 2 X0 , which is an ordinary double point. We x an ample line bundle
OX0 (1) on X0 . Unless otherwise stated we assume g = arithmetic genus of
X0  2.
Let F be a torsion free (coherent) OX0 -module on X0 . We have the
notion of the degree of F (say de ned by deg F = (F) rk F  (O )).
Following Mumford, we say F is semi-stable (resp. stable) if for every (resp.
proper) subsheaf G of F, we have
deg G  deg F (resp. <):
rk G rk F
We have a natural structure of a projective variety (irreducible and reduced)
on certain equivalence classes of semi-stable torsion free sheaves on X0 of
rank n and deg. d. We this by U(n; d) = UX0 (n; d). If (n; d) = 1,
U(n; d) is, in fact, the set of isomorphism classes of stable torsion free sheaves
of rank n and degree d.
The variety U(n; d) has good spe0ialization properties. This means, for
example, the following: Let S = Spe0A, where A is a d.v.r. (discrete val-
uation ring, in fact wecaneven take A as a K-algebra) with residue eld
the ground eld K. Let X ! S be a
at family of projectivecurvs such e
that the closed bre Xs0 ' X0 and the generic bre X is smooth (of genus
g), s0 (resp. ) being the closed point (resp. generic point) of S. (One may
have to x a section passing through the smooth points of the bres). Then
v ell-de ned moduli s0heme U(n; d)S which is
at and projective
wehaea w
211
Degenerations of moduli spaces

over S such that its generic bre U(n; d) can be identi ed with the moduli
space of vector bundles on X and we have a bijective morphism
UX0 (n; d) ! U(n; d)s0
(i)
U(n; d)s0 being the closed bre of U(n; d)S ! S. All these follow by
appealing to GIT over arbitrary base. We do not know whether, in general,
the map (i) is an isomorphism, even if we suppose that X ! S is smooth
i.e. the closed bre is smooth. However, if char K = 0, it is easily seen that
(i) is an isomorphism. Also if (n; d) = 1 the map (i) is an isomorphism.
The facts that UX0 (n; d) is a variety (in particular reduced) and that it
has good specialization properties are not very general results. For example,
if Y0 is a projective curve whose singularities are not ordinary double points,
the moduli space on Y0 may have irreducible components whose dimensions
are bigger than the expected dimension (this happens even in the rank one
case). To show that UX0 (n; d) is reduced (scheme theoretically) one has to
have a closer study of deformations of torsion free sheaves. It is known that,
locally at the singular point `p' of X0 , a torsion free sheaf F is of the form
! 0M 1
M
a b
m  @ OA ; m maximal ideal of O = OX ;p:
F' 0
i=1 j=1
We refer to `a' as the type of F (at p). It is known (see [S] and [F]) that the
miniversal deformation space of F (as an O-module) is formally smooth to
the singularity de ned by
XY = Y X = 0 (X; Y space of `a  a'-matrices):
(ii)
i.e. the analytic local ring at the origin of the closed subscheme of the 2a2
dimensional ane scheme, de ned by the equations in (ii). One knows that
this analytic local ring is reduced and Cohen -Macaulay (see [St.]). The
reduced nature of UX0 (n; d) can be deduced from this fact.
Let U(n; d)0 denote the subset of points of U(n; d) which represent vec-
tor bundles on X0 . Then U(n; d)0 is open in U(n; d) and if (n; d) = 1,
U(n; d)nU(n; d)0 = Sing U(n; d) (singular locus of U(n; d)). It can be seen
that the singularities of U(n; d) are not, in general, normal crossings. This
is a consequence of (ii) above.
One of the main goals of these lectures is to give a construction (see x4
and x5) of a variety G(n; d) (for (n; d) = 1) such that
212

(i) G(n; d) is projective and U(n; d)0 is open in G(n; d)
(ii) G(n; d) has only (analytic) normal crossing singularities and
Sing G(n; d) = G(n; d)nU(n; d)0
(iii) there is a canonical morphism  : G(n; d) ! U(n; d) which is an
isomorphism over U(n; d)0 .
(iv) G(n; d) has good specialization properties i.e. we have an S-scheme
G(n; d)S associated to a moduli problem on X ! S, which is
at and
projective over S. Besides, the generic bre G(n; d) of G(n; d)S over
S, identi es with U(n; d) and the closed bre with G(n; d).
This generalizes Gieseker's construction of GX0 , referred to above.
We study also the bres of  . They turn out to be the wonderful com-
pacti cations of the projective linear group (Rem. 5.2).
In x6 we give a concrete realization of G(2; 1) (Th. 6.1 and Th.6.2) and
sketch very brie
y how this helps in solving the conjecture of Newstead and
Ramanan in the rank two case.

3 Vector bundles on the curves Xk

De nition-Notation 3.1. We call a scheme R, a chain of projective lines if
S
R = n Ri , Ri ' P1 , Ri \Rj (for distinct i; j) is a single point if ji jj = 1
i=1
and otherwise empty. We call m the length of R. Let E be a vector bundle
L
of rank n on R. One knows that EjRi = n O(aij ), aij 2 Z. We say that
j=1
E is positive if aij  0 for all i; j. We say that E is strictly positive if it is
positive and for every i, there is a j such that aij > 0. We say E is standard
if it is positive and aij  1 for all i; j and strictly standard if, moreover, it is
strictly positive.

De nition-Notation 3.2. Let X0 be the curve as in x2 with an ordinary
double point singularity at `p'. Let  : X ! X0 be the normalisation of
X0 and  1 (p) = fp1 ; p2 g. Let Xk be the curves which are \semi-stably
equivalent to X0 " i.e. X is a component of Xk (k  1) and if  : Xk ! X0
denotes the canonical morphism, then  1 (p) is a chain R of projective lines
of length k, passing through p1 ; p2 i.e. they are the curves as follows:
213
Degenerations of moduli spaces

p1 p1
p1
---
X X
X
p
p2
p2 p2
X1 X2 X3
X0
Let Z be a scheme and E a vector bundle on Z of rank n such that H 0 (E)
generates E i.e. the canonical map H 0 (E) ! Ez ( bre of E at z 2 Z) is
surjective. Then we get a canonical morphism.
E =  : Z ! Gr(H 0 (E); n) (Grassmannian of n dimensional
quotients of H 0 (E))
such that E is the inverse image by  of the tautological quotient bundle on
Gr(H 0(E); n).
Proposition 3.1. Let R be a chain of projective lines. Then we have the
following:
(1) if E is a positive vector bundle on R, then H 0 (E) generates E and
H 1 (E) = 0. Moreover, if R1 is a subchain of R (in the obvious sense),
the canonical map
H 0 (R; E) ! H 0(R1 ; E)
is surjective.
(2) if E is strictly positive, the canonical morphism
: R ! Gr(H 0 (E); n)
is a closed immersion. Conversely, the pull-back of the tautological
quotient bundle by a closed immersion of R in a Grassmannian, is
strictly positive.
(3) (E) = deg E + n (deg E = total degree i.e. sum of deg EjRi , for
this claim E could be any vector bundle on R) so that if E is positive
h0 (E) = deg E + n.
215
Degenerations of moduli spaces

(ii) Ri  (E) = 0; i > 0
(iii) H i (Xk ; E) ' H i (X0 ;  (E))
(iv) If E is trivial on R, then  (E) is a vector bundle on X0 and E '
 ((E)).

Proof. The proof of (i) is immediate. For (ii) take an ane neighbourhood
V of p and set U =  1 (V ), V 0 = U \ X. Since V 0 is ane, the canonical
map EjV 0 ! Ep1  Ep2 is surjective so that we get
H i (EjU ) ' H i (EjV 0 )  H i (EjR ); i > 0:
The RHS is zero and then (ii) and (iii) follow. When EjR is trivial, we get
canonical isomorphisms of H 0 (EjR ) with Epi and hence a canonical identi -
cation  : Ep1 ! Ep2 . We see that H 0 (EjU ) identi es with the subspace of
H 0(EjV 0) consisting of elements `s' such that   s(p1) = s(p2). This shows
that  (E) identi es with the vector bundle on X0 de ned by EjX on the
normalisation X of X0 and the patching condition  : Ep1 ! Ep2 and then
(iv) follows.

Proposition 3.3. Let E be a vector bundle of rank n on Xk such that EjR
is strictly positive. If F = E
 (OX0 (l)), then for l  0, H 0 (F ) generates F
and the canonical morphism  : Xk ! Gr(H 0 (F ); n) is a closed immersion.
Further (for all l  0) H 1 (Xk ; F) = 0 so that by Prop. 3.2, we have
H 0 (Xk ; F) ' H 0 (X0 ;  (F )):
H i (Xk ; F) = H i (X0 ;  (F )) = 0; i > 0:
Note that EjR = FjR .

Proof. If  (OX0 (1)) were ample, this proposition would be immediate.
However, this is not the case (k  1) and the proof requires a little work
though it is not dicult. Observe that  (OX0 (1))jX is ample. From this
it follows that (for l  0) the canonical map H 0 (F jX ) ! Fp1  Fp2 is
surjective and H 1 (F jX ) = 0. Then from the patching exact sequence
0 ! F ! FjX  FjR ! T ! 0
216

we deduce that H 1 (F ) ' H 1 (F jX )  H 1 (F jR ), which implies that H 1 (F ) =
0. Then the last assertions follow from the previous Prop. 3.2.
Since H 0 (F jX ) ! Fp1 Fp2 is surjective, it follows that H 0 (Xk ; F) !
H 0(F jR ) is surjective. We cannot say that the canonical map H 0(Xk ; F) !
H 0(F jX ) is surjective, which causes the little complication. However sec-
tions of FjX which vanish at p1 ; p2 can be extended to the whole of Xk
(by putting zero on R) and one uses the sheaf Ip1 ;p2 FjX (where Ip1 ;p2 de-
notes the ideal sheaf of the closed subscheme fp1 ; p2 g of X). We can sup-
pose that for l  0, H 0 (Ip1 ;p2 FjX ) generates Ip1 ;p2 FjX and the canonical
morphism X ! Gr(H 0 (Ip1 ;p2 FjX ); n) is a closed immersion. We identify
H 0(X; Ip1 ;p2 FjX ) with the subspace of H 0 (Xk ; F) vanishing on R. Then
with these observations and the surjectivity of H 0 (Xk ; F) ! H 0 (Xk ; FjR ),
we see that H 0 (Xk ; F) generates F, that the canonical morphism  : Xk !
Gr(H 0(xk ; F); n) is injective (we see easily that given x; y, x 6= y, there exist
sections on Xk such that (s1 ^    ^ sn )(x) 6= 0 and (s1 ^    ^ sn)(y) = 0)
and that d is injective at all the points except fp1 ; p2 g. It is not dicult to
show that d is injective at fp1 ; p2 g (see [NS]) and the proposition follows.
Consider the vector bundles E on Xk such that EjR are strictly positive.
Our next aim is to characterize those E such that  (E) are torsion free.
This characterization involves only properties of EjR .
Remark 3.1.
(a) Let E be a strictly standard vector bundle on P1 . We have then a
well-de ned sub-bundle K of E, which we call the canonical subbundle
of E such that K is a direct sum O(1)'s and E=K = Q (called the
canonical quotient bundle) is free. Let x; y 2 P1 such that x 6= y and
Lx a linear subspace of Ex . Then we have a well-de ned subbundle F
of E such that K  F and Fx jKx = Image of Lx in Qx = Ex jKx . Then
if V is the linear subspace of H 0 (E) consisting of sections `s' such that
s(x) 2 Lx , then V  H 0 (F ) and the image of V in Ey in Fy . We say
that Fy is the subspace of Ey determined by Lx . We see that if Lx = 0,
then Fy = Ky . Note also that if s 2 H 0 (E) and s(x) = s(y) = 0, then
s is identically zero.
(b) Let E be a strictly standard vector bundle on a chain R of projective
lines of length m. Let Ki (resp. Qi ) be the canonical sub-bundle (resp.
quotient bundle) of EjRi (Ri the P1 -components of R). We denote by
217
Degenerations of moduli spaces

qi the points Ri \ Ri+1 1  i  (m 1) and q0 = p1, qm = p2 . Let us
take Lq0 = (0). We set
q0 = p1 q1
R1 R q2
2

qm 1

p2 = qm
Lq1 as the linear subspace of Eq1 determined by Lq0 for EjR1 . We
write Lq2 for the linear subspace of Eq2 de ned by Lq1 for EjR2 . In-
ductively, we thus de ne a linear subspace Lqi of Eqi . We write M for
the subspace Lqm  Ep2 = Eqm . Consider the condition:
dim M = rk K1 +    + rk Km :
()
We see that
() () dim Lqi = rk K1 +    + rkKi ; i  j  m:
Take for example m = 2. Then () means that
(K1 )q1 \ (K2 )q1 = (0):
Lemma 3.1. Let E be a strictly positive vector bundle on a chain R of
projective lines. Consider the property:
s 2 H 0 (E); s(p1 ) = s(p2 ) = 0 =) s  0:
()
Then () holds if and only if E is strictly standard and the property () of
Remark 3.1 holds.
Proof. Suppose that EjRi = O(aij ) and some aij  2. Then it is im-
mediate that there exists s 2 H 0 (EjR1 ) such that s(q0 ) = s(q1 ) = 0 and s
is not identically zero. Then we can extend s to a section of E vanishing
identically on all Ri , i  2. Thus () =) E is strictly standard.
Let us suppose, for simplicity, that the length m of R is 2. The proof
in the general case is quite similar (see [NS]). Suppose that () holds and
218

that (K1 )q1 \ (K2 )q1 = L 6= (0). Then it is clear that given l 2 L, l 6= (0),
there exist s1 2 H 0 (EjR1 ) and s2 2 H 0 (EjR2 ) such that s1 (q1 ) = l = s2 (q1 )
and s1 (q0 ) = 0 and s2 (q2 ) = 0. Hence s1 ; s2 de ne a section s of E such
that s(q0 ) = s(q2 ) = 0 and s is not identically zero. Hence we shall have
(K1 )q = (K2 )q = (0), which shows that () implies that () holds and E is
strictly standard.
Suppose on the other hand that E is strictly standard and () holds. Let
s 2 H 0 (E) such that s(p1 ) = s(p2 ) = 0. Then s(q1 ) is in (K1 )q1 as well as
(K2 )q1 . Since () holds, this implies that s(q1 ) = 0. Then we see that the
restriction of s to R1 as well as R2 is identically zero. Hence s is identically
zero and () holds.

Proposition 3.4. Let E be a vector bundle on Xk such that EjR is strictly
positive. Then we have the following:
(A)  (E) is torsion free on X0 if and only if the property () (of Lemma
3.1) holds. Then by Lemma 3.1, we see that  (E) is torsion free if
and only if E is strictly standard and the property () of Remark 3.1
holds. Note that () implies that
(
length of R = m  n = rk E; in fact
P
m  deg EjR = i deg EjRi  n:
(B) if  (E) is torsion free, then its type (at p) is deg EjR .

Proof. We have the following exact sequence of OXk -modules
0 ! IX E ! E 7 ! EjX ! 0:
(a)
IX being the ideal sheaf of X. Note that IX E can be identi ed with Ip1;p2 EjR
- the sheaf of sections of EjR vanishing at p1 ; p2 . Then we have the exact
sequence
0 !  (Ip1 ;p2 EjR ) !  (E) !  (EjX ):
(b)
Now  (EjX ) is torsion free on X0 and it is clear that  (Ip1 ;p2 EjR ) is a
torsion sheaf, in fact its support is at p. Hence it follows that the torsion
subsheaf of  (E) is precisely  (Ip1 ;p2 EjR ). It is clear that  (Ip1 ;p2 EjR ) is
219
Degenerations of moduli spaces

the sheaf determined by the vector space H 0 (R; Ip1 ;p2 EjR ) considered as an
OX0 ;p module (through its residue eld). From these remarks the assertion
(A) follows.
Now if  (E) is torsion free, continuing the exact sequence above we get
the exact sequence
0 !  (E) !  (EjX ) ! R1  (Ip1 ;p2 EjX ) ! 0
since R1  (E) = 0 by Prop. 3.2. Further, we see that R1  (Ip1 ;p2 EjR ) is
the \sky-scraper" sheaf with support at p and de ned by the vector space
H 1(Ip1 ;p2 EjR ).
Let us rst suppose that deg EjR = n, then deg Ip1 ;p2 EjR = n and
(Ip2 ;p2 EjR ) = 0. Since H 0 (Ip1;p2 EjR ) = 0, we see that H 1(Ip1 ;p2 EjR ) = 0
(see Prop 3.1.). Hence we get
 (E) '  (EjX )
(c)
and it is an easy exercise that the RHS is of type n at p. Thus in this
case the assertion (B) above follows. We have then to consider the case
deg EjR < n. Then by the considerations in Remark 3.1, it is not dicult to
see that EjR has a direct summand which is a trivial vector bundle of rank
t = n deg EjR . Then we see that we can choose a suitable neighbourhood
V of p such that in  1 (V ), E has a trivial direct summand of rank t. Thus
we see that to prove (B) we are reduced to the case deg EjR = n.
Remark 3.2.
(a) Let E be a vector bundle on Xk such that EjR is strictly standard and
(E) is torsion free. Then we have
mX0 ;p( (E)p ) =  (Ip1 ;p2 EjX )p , or equivalently (mX0 ;p denotes the
maximal ideal of OX0 ;p,  (E)p the stalk of  (E) at p)
IX0 ;p( (E)) =  (Ip1 ;p2 EjX )
(b) if  : X ! X0 is the normalisation map, the functor  : (Vector
bundles on X) ! (Torsion free sheaves on X0 ) is faithful i.e.
Hom (V1 ; V2 ) ' Hom ( (V1 );  (V2 ))
in particular V1 ' V2 ()  (V1 ) '  (V2 ).
220

(c)  (E) determines EjX i.e. if E1 ; E2 on Xk (possibly for di erent k
with EjR strictly positive) are such that  (Ei ) are torsion free and
 (E1 ) '  (E2 ), then E1 jX ' E2 jX
(d) if we have a family of vector bundles fEg on Xk (possibly for di erent
k with EjR strictly positive) such that f (E)g is a bounded family of
torsion free sheaves on X0 , then for l  0 (independent of E) fEjX g
is a bounded family and for F = E
 (OX0 (l)), H 0 (F ) generates
F and the canonical morphism  : Xk ! Gr(H 0(F ); n) is a closed
immersion. Besides we have
(i) H 0 (Xk ; F) ' H 0 (X0 ;  (F ))
(ii) H i (Xk ; F) ' H i (X;  (F )) = 0; i > 1.
(i.e. the properties of Prop. 3.2 hold. In a sense, we may say that fEg
is a bounded family if  (E) is a bounded family).
Proof.
(a) Consider the exact sequence
0 ! IR E ! E ! EjR ! 0:
We see that IR E ' Ip1 ;p2 EjX . Then we get the following exact se-
quence of OX0 ;p -modules
0 !  (Ip1 ;p2 EjX )p !  (E)p !  (EjR )p ! 0
since R1  (Ip1 ;p2 EjX ) = 0 (X ! X0 being an ane morphism). We
see that  (EjR )p is the sky-scraper sheaf at p associated to the vector
space H 0 (EjR ). We have dim H 0 (EjR ) = deg EjR + n and we saw in
Prop. 3.4 above that deg EjR is the type `a' of  (E). On the other
hand we see that
dim( (E)p =mX0 ;p (E)p ) = a + n:
Then the assertion (a) follows.
(b) Let A = OX0 ;p and let B the semi-local ring of X at p1 ; p2 (integral
closure of A). Then Vi are represented by free B-modules and to
prove (b) is to show that an A-module homomorphism B m ! B n is
in fact a B-module homomorphism. This follows from the fact that
Hom A (B; B) ' B (multiplication by elements of B).
221
Degenerations of moduli spaces

(c) By (a) and (b), it follows that if  (E1 ) '  (E2 ), then Ip1 ;p2 E1 jX '
Ip1;p2 E2 jX and then by multiplying by Ip11 2 the assertion (c) follows.
;p
(d) Now if f (E)g is bounded, by (a) and (b) it follows easily that fEjX g
is a bounded family and all the other assertions also follow (see [NS]
for more details).

4 The moduli space
We shall now illustrate our construction of the moduli space for the case of
line bundles. This is quite simple but provides a motivation for the general
considerations.
The curves Xk and X0 are as in Def. 3.2. For a line bundle L of X1 , we
write L1 = LjX and L2 = LjR (R ' P1 ).

X1 P1
R

X
P2

The line bundle L is de ned by the following isomorphisms of 1-dimensional
spaces:
1 : (L1 )p2 ! (L2 )p1 ; 2 (L1 )p2 ! (L2 )p2 :
We represent L by the 4-tuple (L1 ; L2 ; 1 ; 2 ). If we modify the 4-tuple by
an automorphism of L1 , as well as an automorphism of L2 , the resulting
4-tuple represents a line bundle isomorphic to L. We see that
(
(L1 ; L2 ; 1 ; 2 )  (L1 ; L2 ; 1 ; 2 )
(i)
() 1 = a1 ; 2 = a2; a non-zero scalar.
Let P a;b denote the set of isomorphism classes of line bundle L on X1 such
that deg L1 = a and deg L2 = b. Then we see that the canonical morphism
P a;b ! Pic a (X) (L 7 ! L1 ) is a principal G m bration. Also note that
222

we have an identi cation
(
 : P a;0  Pic a (X0 )
!
(ii)
L 7 !  (L);  : X1 ! X0
since in this case L2 is trivial. In particular, if we write G(1; 0)0 = Pic 0 (X0 ),
we have G(1; 0)0 ' P 0;0 .
Let g be an automorphism of X1 such that it is identity on the component
X. If L 2 P a;0 we see that g (L) ' L. Suppose now that L 2 P a;1. Then
g (L)jR ' L2 = O(1). We have R ' P1 ' P(V ), where V  = H 0 (L2). Then
identifying p1 ; p2 with f0; 1g, the automorphism g on R is represented by
an automorphism of V as a diagonal matrix
 0
:
0
From these considerations, it follows that if L is represented by (L1 ; L2 ; 1 ; 2 ),
then g (L) is represented by a 4-tuple
(L1 ; L2 ; t1 1 ; t2 2 ); t1 6= 0; t2 6= 0:
Let us introduce an equivalence relation in P a;1 , namely L  M if L ' g (M)
for some g 2 Aut X1 such that g is identity on X. Let us take the case
a = 1 so that we deal with the case deg L (total degree of L) = 0. Let
G(1; 0)1 denote the set of equivalence classes in P 1;1. Then we see that
G(1; 0)1 identi es with Pic 1 X (by L 7 ! L1 ). We see also that if L  M
(equivalence relation) then  (L) '  (M). In fact, we have a bijection
 : G(1; 0)1 ' P1
(iii)
where P1 is the set of isomorphism classes of torsion free sheaves of rank
one, degree zero and type one on X0 .
If U(1; 0) is the moduli space of isomorphism classes of torsion free
sheaves of rank one and degree zero on X0 , then one knows that
U(1; 0)nPic 0 (X0 ) ' P1:
Let G(1; 0) denote the disjoint union
a
G(1; 0) = G(1; 0)0 G(1; 0)1 ; G(1; 0)0 = Pic 0 (X0 ):
223
Degenerations of moduli spaces

Thus we get a bijection
 : G(1; 0) ! U(1; 0):
(iv)
As we shall see, we have a similar but more complicated phenomenon in
higher rank. The generalisation of  is no longer a bijection.
De nition 4.1.
(i) Let E be a vector bundle on Xk such that EjR is strictly positive. It
is said to be stable if  (E) is a stable (torsion free) sheaf on X0 . Note
that it has all the nice properties stated in Prop. 3.4, in particular EjR
is strictly standard.
(ii) We call two vector bundles E1 ; E2 on Xk equivalent if E1 ' g (E2 ),
where g is an automorphism of Xk , which is identity on the component
X (g could move points on R).
(iii) We set
(
equivalence classes of stable vector bundles on Xk
G(n; d)k =
of rank n and degree (total degree) d
a
G(n; d) = G(n; d)k (disjoint sum):
0kn

Note that G(n; d)0 is the set of isomorphism classes of stable vector bundles
of rank n and degree d on X0 . We shall see that if (n; d) = 1, G(n; d) has
a natural structure of a projective variety with a birational morphism onto
the projective variety U(n; d) = UX (n; d) (the moduli space of stable torsion
free sheaves of rank n and degree d on X0 ) and that it has all the good
properties like specialization (stated in x2).
Let L be a line bundle on X0 . If E is a vector bundle on Xk , then
since  (E
 (L)) '  (E)
L, we note that  (E) is torsion free ()
(E
 (L)) is torsion free. Since stable torsion free sheaves of rank n and
degree d form a bounded family, we see that for l  0 and E 2 G(n; d),
E
 (L) has all the good properties of (d) of Remark 3.2. Thus without
loss of generality we may assume that if E 2 G(n; d), H 0 (E) generates E,
the canonical morphism
E : Xk ! Gr(H 0 (E); n)
224

is a closed immersion and the properties (d) of Remark 3.2 are satis ed.
We can identify Gr(H 0(E); n) with the standard Grassmannian Gr(m; n)
(this identi cation is upto an automorphism of Gr(m; n) i.e. upto an element
of PGL(m)) and E with a morphism (denoted again by E ).
E : Xk ! Gr(m; n) (m = dim H 0 (E)):
Now PGL(m) operates canonically on Gr(m; n) and also on X0  Gr(m; n)
by taking the identity action on X0 . Now E gives rise to a closed immersion:
: Xk ,! X0  Gr(m; n); = (; E ):
E E
Let E1 ; E2 2 G(n; d)k and E1 ; E2 the imbeddings into X0  Gr(m; n).
Then the important remark is the observation:
(
E1  E2 (equivalence relation, see (ii) of Def. 4.1)
() g (Im E1 ) = Im E2 ; g 2 PGL(m).
We observe also that the Hilbert polynomial P1 of Im E is the same for
all E 2 G(n; d). Thus Im E 2 Hilb P1 (X0  Gr(m; n)) (we choose some
polarisation on X0  Gr(m; n)). Note that the action of PGL(m) on X0 
Gr(m; n) induces a canonical action of PGL(m) on Hilb P1 (X0  Gr(m; n)).
The foregoing discussion shows that G(n; d) can be identi ed (set theoret-
ically) as the set of PGL(m) orbits of a certain PGL(m) stable subset of
Hilb P1 (X0 G(m; n)). We observe that given E , E is expressed canonically
as a quotient of the trivial rank m vector bundle
OXk ! E; H 0(OXk )  H 0(E); H 1 (E) = 0:
m m!

Then by (d) of Remark 3.2,  (E) is a quotient of the trivial vector bundle
of rank m on X0
E = OX0 !  (E) and H 0 (OX0 )  H 0( (E))
m m!
()
Let P2 be the Hilbert polynomial of the stable torsion free sheaves on X0
of rank n and degree d of X0 . Let Q = Q(E=P2 ) be the Quot scheme of
quotients of the trivial vector bundle E of rank m on X0 and R; Rs the
PGL(m) stable open subsets of Q, which are now standard (R is de ned
by the condition that the corresponding point of the Quot scheme de nes a
quotient which is torsion free, as well as the second condition in (). The
225
Degenerations of moduli spaces

subset Rs of R corresponds to these quotients which are moreover stable).
Recall that Rs mod PGL(m) ' U(n; d)s and that Rs is a principal bundle
over U(n; d)s .
The following are the main steps in giving a canonical structure of a
quasi-projective variety on G(n; d):
(I) The subset Y s = Y (n; d)s  Hilb P1 (X0  Gr(m; n)), Y s = fIm E g
(E 2 G(n; d)) is PGL(m) stable and has a natural structure of an
(irreducible) variety whose singularities are normal crossings.
(II) The map  : Y s ! Rs de ned by y (represented by E or Im E ) 7 !
the point of Rs represented by () above, is a PGL(m) equivariant
morphism.
(III) The morphism  is proper.
We shall now indicate how admitting I, II and III, we get a nice structure
of a variety on G(n; d).
Let Rs;0 denote the PGL(m) stable open subset of Rs represented by
vector bundles on X0 . Then a point of  1 (Rs;0) is represented by E such
that the equivalence class is in G(n; d)0 i.e. a closed immersion E : X0 ,!
X0  Gr(m; n). Then it is easy to see that the morphism
 :  1 (Rs;0) ! Rs;0
is an isomorphism. Hence it follows that  is a birational morphism. Since
 : Y s ! Rs in PGL(m) equivariant and Rs ! U(n; d)s is a principal
bundle, it is easily seen that the quotient Y s mod PGL(n) exists and in fact
that Y s ! Y s and PGL(m) is a principal PGL(m) bundle. Thus we get
a canonical structure of a variety on G(n; d). Further, since Y s has normal
crossing singularities and Y s ! Y s and PGL(m) is a principal bration,
we see that G(n; d) has normal crossing singularities.
To prove that G(n; d) is quasi-projective, we use GIT, which can also be
used to give simultaneously a variety structure on G(n; d). We can suppose
that Qs = Rs, Qss = Rss (Qs -stable points of a polarisation on Q;   ,
for example, as has been done recently by Simpson). Then since  is a
projective morphism, it is easily seen that we can nd a PGL(m) equivariant
factorisation (instead of taking the Quot scheme we can take the closure of
226

!
Rss in Q, but we use the same notation):
Ys , Z
? ?
!
? ?
y y


Rs , Q
where Z is a projective variety with an action of PGL(m) lifting to an ample
line bundle OZ (1). Consider now the polarisation L =  (OQ (a))
OZ (1)
on Z. Then with the usual notations, one knows that for `a' suciently
large, we have:
(i)  1 (Rs) (=  1 (Qs ))  Z(L)s .
(ii)  maps Z(L)ss onto Rss.
It follows then that Y s mod PGL(m) exists as a quasi-projective variety
and thus G(n; d) acquires a canonical structure of a quasi-projective variety.
Further  induces a canonical (birational) projective morphism G(n; d) !
U(n; d)s . If (n; d) = 1, U(n; d)s = U(n; d) and it follows that G(n; d) is
projective.
For proving I, II, III we require more formal considerations.
De nition 4.2. Let Y be the functor de ned as follows:
Y : (K schemes) ! Sets:
Y(T) = set of closed subschemes  ,! X0  T  Gr(m; n) such that:
(i) the induced projection map p23 :  ! T  Gr(m; n) is a closed
immersion. We denote by E the rank n vector bundle on , obtained
as the pull-back of the rank n tautological quotient bundle on Gr(m; n)
(ii) the projection p2 :  ! T is a proper
at family of curves ft g,
t 2 T, such that t is a curve of the form Xk . Besides, the map (Xk '
)t ! X0 (induced by  ! Xi T) is the canonical  : Xk ! X0
that we have been considering
(iii) the vector bundle Et on t (Et = Ejt ) is of degree d (and rank n)
with d = m + n(g 1)
227
Degenerations of moduli spaces

(iv) by the de nition of E, we get a quotient representation Ot ! Et and
m
we assume that this induces an isomorphism H 0 (Ot )  H 0 (Et ). In
m!
particular, dim H 0 (Et ) = m and it follows that H 1 (Et ) = 0.
Proposition 4.1. The functor Y is represented by a PGL(m) stable sub-
scheme Y of Hilb P1 (X0  Gr(m; n)) (P1 being the Hilbert polynomial of
the closed subscheme t of X0  Gr(m; n), choosing of course a polarisa-
tion). Further Y is an (irreducible) variety with (analytic) normal crossing
singularities.
This is essentially due to Gieseker and some indication of proof will be
given later.
Proposition 4.2. Let  be the universal object representing the functor
Y above. Consider the \universal" closed immersion
 ,! X0  Y  Gr(m; n)
de ned by Y. This de nes a
at family of curves  ! Y . We have also a
vector bundle E on  obtained as the pull-back of the tautological quotient
bundle of rank n of Gr(m; n). Then E de nes a family fEy g of vector
bundles on fy g, y 2 Y . We denote by y : y ! X0 the morphism
induced by the rst projection p1 , to be consistent with our earlier notation.
We observe that (y ) (Ey ) comes with a quotient representation
OX0 ! (y ) (Ey ) with H 0 (OX0 )  H 0 ((y ) (Ey )):
n m!
()
Hence () de nes a point of the open subscheme R of Q(E=P2 ) (Quot scheme
mentioned above). Then we claim that the map  : Y ! R, de ned by
y 7 ! the point of R de ned by (), is a morphism.
Proof. We give a sketch. We have a commutative diagram

X0  Y


p q

Y
228

where  is the projection p12 , p = projection p2 and q = canonical projection
onto Y . Using the fact that the bres of  are connected and the nice nature
of the singularities of Y , it follows that
 (O ) = OX0 Y :
(a)
We have a quotient representation
m
O ! E on :
Using (a) and applying  , we get a homomorphism
m
OX0 Y ! (E):
(b)
The crucial point is to show that
8
> (E) behaves well for restriction to bres over Y ; i:e:
<
> (E)jq y ' (y ) (Ey )
() 1
: q 1(y) ' X0  y ' X0
for one sees easily from () that (b) is surjective and that the Hilbert poly-
nomial of  (E)jq 1 y is P2 . Since Y is reduced,  (E) is
at over Y . Thus
(b) de nes a morphism of Y into Q(EjP2 ) which factors enough R. Thus ()
and hence the above proposition is a consequence of the following:

Lemma 4.1. Suppose that we have a commutative diagram

Z W

p q

T
such that p; q and  are projective morphisms,  (OZ ) = OW and p is
at.
Let E be a vector bundle on Z such that
R1(t )(EjZt ) = 0 i  1; t 2 T:
229
Degenerations of moduli spaces

Then  (E) behaves well for restriction to bres over T i.e.
 (E)jWt ' (t ) (EjZt ); t 2 T:
Further
H 0 (Zt ; EjZt ) ' H 0 (Wt ; (t ) (EjZt )):

Proof. We refer to [NS].
Since  : Y ! R is a morphism and Rs is open in R, it follows that
Y s is open in Y . If we denote by Rtf the open subset of R corresponding
to quotients which are torsion free, we see that Rtf is open in R and hence
Y tf =  1 (Rtf ) is also open in Y . We see that we have a canonical morphism
 : (Y smod PGL(m)) = G(n; d) ! U(n; d)s = (Rsmod PGL(m):)

For properness and good specialization properties, we require to work
with a more general base scheme.
Thus admitting I, II and III we have the following:
Theorem 1. We have a natural structure of a quasi-projective variety on
G(n; d) and the canonical map  : G(n; d) ! U(n; d) is a proper birational
morphism. If (n; d) = 1, G(n; d) is projective. Further the singularities of
G(n; d) are normal crossings.

5 Properness and specialization
De nition 5.1. Let S = Spec A, where A is a d.v.r. (in fact K-algebra)
with residue eld K. Let X ! S be a proper,
at family of curves such
that the closed bre Xs0 ' X0 and the generic bre X is smooth (s0 -closed
point of S;  generic point of S). We suppose also that XS is regular over K
(we may also have to x a section of X over S passing through the smooth
points of the bres).

De nition 5.2. Same as in Def. 4.2 over the base S. The functor is
denoted by YS : (S-schemes) ! Sets. The one point to remember is that,
230

for example, the morphism t ! (X S T)t is an isomorphism if t maps
to the generic point of S [YS (T ) = set of closed subschemes
 ,! X S T S Gr(m; n)(Gr(m; n) = Gr(m; n)S )
 ! X S T
at family of curves etc.].
Now Prop. 4.1 generalizes as:
Proposition 5.1. The factor YS is represented by a PGL(m) stable sub-
scheme YS of the S-scheme HilbP1 (X S Gr(m; n) and YS ! S has the
following properties:
(i) the closed bre (YS )s0 is a variety with (analytic) normal crossing
singularities
(ii) the generic bre (YS ) is smooth
(iii) YS is regular over K (of course YS ! S is
at).
This proposition is essentially to be found in Gieseker [G]. Some indica-
tion of proof will be given later.
Proposition 5.2. Proposition 4.2 generalizes to give a canonical morphism
 : YS ! RS (RS - the obvious open subset of the Quot scheme Q(EjP2 )
associated to X ! S). Then  induces morphisms (denoted by the same
letter) (
YSs ! RS ; YStf ! RS ; (RS  RS )
tf tf
s s
YSs =  1 (RS ); YStf =  1(RS ):
tf
s
Note that the S-morphisms  are isomorphisms over Snfs0 g i.e.  induces
an isomorphism of their generic bres over S. (e.g.  : (YSs ) ! (RS ) ).
s
v
In fact  is an isomorphism over the bigger open subset RS corresponding
to vector bundles i.e. de ned by q 2 QS (EjP2 ) such that the corresponding
quotient sheaf is locally free.
Proposition 5.3. The morphism
 : YSs ! RS (resp. YStf ! RS )
tf
s

is proper.
231
Degenerations of moduli spaces

Proof. We shall now outline the proof giving the main points.
We see that it suces to prove that  : YStf ! RS is proper. We
tf
apply the valuation criterion. Let T = Spec B, B d.v.r with residue eld
K. Given a rational map  : T ! YStf such that (  ) : T ! RS is a tf
morphism, then to show that  is a morphism. We see that (  ) de nes a
family of torsion free sheaves F of rank n on the base change XT = X S T
parametrized by T. We can suppose that F is a quotient of the trivial bundle
of rank m:
m
OXT ! F ! 0:
We can suppose that the morphism T ! S does not map to the closed
point of S so that for the family of curves XT ! T, the bre is smooth
over the generic point of T (i.e. the generic bre of XT ! T is the base
change of the generic bre of X ! S). The question is whether we can lift
this to a T-valued point of YStf (satisfying the required properties). We will
see that there is a canonical candidate for this.
We see also that the closed bre of XT ! T identi es with the closed
bre of X ! S and denote by the same `p' the singular point of the closed
bre of XT ! T, which is ' X0 . Now F is locally free outside `p'. The
quotient representation gives a T-morphism
g : XT nfpg ! Gr(m; n) (Gr(m; n)T ):
We can also suppose that g is an immersion. Let g be the graph of g so
that we have a closed immersion of T-schemes
g ,! XT T Gr(m; n):
()
Let g : g ! XT be the canonical projection which is a T-morphism.
Obviously g induces an isomorphism over the generic bres, in fact an
isomorphism over XT nfpg. Let E be the vector bundle of rank n on g
induced by the quotient bundle on Gr (m; n) (through ()). Then for the
required lifting and properness, it suces to prove the assertion:
(A)8
>
> (i) The closed bres of g ! T is a curve of the form Xk (this
>
>
< implies that the morphism induced by g on the closed bres
is the canonical morphism Xk ! X0 ).
>
>
>
> (ii) (g ) (E) = F.
:
232

Note that g is an isomorphism over XT nfpg and E coincides with F outside
fpg i.e. E and F coincide on XT fpg.
We see that the assertions (A) are really local with respect to XT i.e. it
suces to check them over a neighbourhood of p in XT . More precisely let
C be the local ring of XT at p. Then F is represented by a torsion free C-
module which is B-
at (T = Spec B) and F is the quotient of a free module
of rank m over C (F is of rank n over B) and g should be viewed as the
graph of a morphism
g : Spec Cnfpg ! Gr(m; n)
(restriction of g to Spec Cnfpg) and denoted by the same letter. Thus we
have a commmutative diagram
g
g Spec C

T = Spec B
Let m0 denote the minimal number of generators of the C-module F, we
have a factorisation
Spec C fpg Gr(m; n)

Gr(m0; n)
We can assume without loss of generality that m = the number of mini-
mal generators of F as a C-module.
The point is that we have a concrete description of the C-module F
which facilitates the checking of the local version of (A).
We see that we can suppose that C (resp. B and A) are complete local
rings. It is also not dicult to see that F is equal to its bidual (see [NS]) i.e.
F  = F (F  bidual of F as a C-module):
233
Degenerations of moduli spaces

We suppose that char K is zero, say K = C . Now A = K[[t]] and we see
b
that because of our hypothesis, the completion OX;p of the local ring at `p'
b
is given by K[[x; y]] and the canonical morphism Spec Op ! Spec A(= S),
is given by
A = K[[t]] ! K[[x; y]]; t 7 ! xy:
The canonical homomorphism A ! B (B ' K[[t]]) is given by t 7 ! tr .
From these considerations it follows easily that
C ' K[[x; y; y]]=(xy tr )
and the canonical homomorphism B ! C is de ned by t 7 ! image of t in
C.
Let D = K[[u; v]]. Consider the action of the cyclic group r of order r
operating on D by:
(
(u; v) 7 ! (u; v);  rth root of unit representing our element of r
 complex conjugate of .
(
We see that
C = D r ( r -invariants in D);
taking x = ur , y = vr , t = uv.
Let f be the canonical morphism f : Spec D ! Spec C and f0 : Spec Dnf0g !
Spec Cnfpg, the restriction of f to Spec Dnf0g. Now C is normal with an
isolated (rational) singularity at `p' and of type A. Also f0 is an unrami ed
covering. Then f0(F ) can be extended to a locally free sheaf on Spec D
represented by a free D-modules F. We see that F is canonically a (D r )
module (i.e. D-module with an action of r ). By the re
exivity of F, we
deduce easily that
F ' (F) r :
One knows that a (D r ) free modules is given by a representation of r ,
which is a direct sum of 1-dimensional characters. Thus F is a direct sum
of r -line bundle L of the form
(
L = Spec D  C , action of r is given by
 = f(u; v); C g = f(u; v;  sg;  2 C
(we could view Spec D as a 2-dimensional disc). A -invariant section of
this bundle L is easily identi ed with a function  on the line satisfying:
(u; v) =  s(u; v):
234

Then we check that the r -invariant sections of L are generated by us and
vr s as a C-module. We have
ur s(us ; vr s) = (ur ; (uv)r s ) = (x; tr s ):
Thus we see easily that the C-module F is of the following form:
M
n
(tai ; x); 0  a1  a2      an :
F'
i=1
We see that (tai ; x) is a principal ideal if and only if ai = 0 so that
b M(tai ; x); 0 < a1  a2      a :
nb
' OC
F nb
i=1
As we saw above, we have taken m = minimal number of generators of F
which is equal to b + 2(n b). We see that the morphism g factorises as:

g
Spec C fpg Gr(m; n)

Gr(2(n b); (n b))

Thus we can suppose without loss of generality that b = 0 i.e. m = 2n.
Then we see that g factorises as:

Spec C fpg      P1 (n times)
P1

g

Gr(2n; n)
235
Degenerations of moduli spaces

Let 1 ;    ; l be the distinct ones occurring among the fai g with multi-
plicity i , 1  i  l. Then we see that we have a factorisation

     P1
Spec Cnfpg (l times)
P1

1 l
g

(P1 ) 1      (P1 ) l

Gr(2n; n)
where l are the diagonal morphisms P1 ! (P1 ) i , 1  i  l. We observe
that the pull-back of the tautological quotient bundle on Gr(2n; n) on (P1 )l
identi es with:
( 1
O(1)  O(1) 2      O(1) l (external direct sum)
(a)
i.e. i O(1) i , O(1) i coming from the i-th factor.
We see also easily that the bre of g ! Spec C over the closed point of
Spec C identi es with the chain R of P1 's in (P1 )l (of length l) of the form
(b)
(1
fP  (0; 1)      (0; 1)g [ f(1; 0)  P1  (0; 1)      (0; 1)g
[f(1; 0)  (1; 0)  P1  (0; 1)      (0; 1)g [    [ f(1; 0)  (1; 0)      P1 g:
From these considerations one sees that EjR is strictly standard (EjRi =
O(1) i  Trivial) that the good conditions of Remark 3.2 are satis ed and
in fact that (g ) (E) = F so that (A) follows (see [NS] for more details).

Remark 5.1. Local Theory (outline of proof of Prop. 5.1)
The main steps could be formulated as follows:
236

I Let YS be the functor obtained from YS by forgetting the imbeddings into
0
0
Grassmannians (see Def. 5.2) i.e. YS (T ) is a (
at) family of curves  ! T
together with a T-morphism  ! X S T which induces the canonical
morphisms of bres over t 2 T.
0
We claim that the functor YS ! YS is formally smooth. The proof
of this is quite standard and comes to extending vector bundles and sec-
tions over in nitesimal neighbourhoods, which are possible because of the
vanishing of H 1 in Def. 4.2 (and Def. 5.2) and that we are in the case of
curves.
II We take S = Spec A, A = K[[t]] and W = Spec K[[t0; : : : ; tr ]] endowed
with an S-scheme structure by t 7 ! t0    tr . A crucial point is the con-
0
struction by Gieseker of an element  2 YS (W ) de ned by a family of curves
S ! W and : Z ! X S W, having the following properties:
(a) the closed bre of Z ! W is a curve Xr
(b) the closed subscheme of W corresponding to the singular bres of
Z ! W is the union of ti = 0 so that it has, in particular, nor-
mal crossing singularities and is the inverse image of the closed point
of S (by the morphism W ! S)
(c) Z ! W provides a miniversal (e ective) deformation of the singu-
larities of the closed bre of Z ! W. More precisely, let V be an
ane open subset of the closed bre of Z ! W, containing its singu-
lar points (or we can take the semi-local ring at the singular points).
Then Z ! W de nes a deformation ZV ! W of V . The property
is that ZV ! W is an (e ective) miniversal deformation of V ,
(d)  (M) = L, where L (resp. M) is the dualizing sheaf of Z (resp.
X S W) relative to W.
Note that for r = 0, Z ' X and the property (C) holds, which is a
consequence of the fact that X is regular.
Roughly speaking Z is obtained by taking the base change of X by W !
S and performing certain blow ups.
III We restrict the functor YS to Spec of Artin local rings such that YS
0 0
0
(closed point) ' Xr i.e. we take only YS (T ) where T = Spec B, B is an
0
Artin local algebra with residue eld K and YS (Spec K) ' Xr . In this
237
Degenerations of moduli spaces

way we obtain a functor Yr0 ! Spec (Artin local S-schemes). We view Yr0
0
as studying YS in an in nitesimal neighbourhood of a point which repre-
0
sents Xr . Through the element  2 YS (W ) in II above, we get a canonical
morphism
 : W ! Yr0 :
(i)
Also by (c) of II above, we get a functor
Yr0 Def V (Deformationsof V ):
!
(ii)
The important claim is that and  are isomorphisms
W  r0 
!Y !Def V:
(iii) 
By (c) of II above (  ) is an isomorphism and to prove (iii) it suces to
show that  is formally smooth.
To prove that  is formally smooth we have to do the following. Given
an element  2 Yr0 (T ) de ned by Z 0 ! T and 0 : Z 0 ! X S T, we
suppose further that there is a morphism 0 : T0 ! W such that the
pull backs by 0 of Z ! W and : Z ! X S W coincide with the
restriction 0 2 Yr0 (T0 ) of  to T0 (T0 closed subscheme of T with length one
less than that of T). Then it is required to show that 0 can be extended
to a morphism  : T ! W such that the pull-back of Z ! W and
: Z ! X S W are isomorphic to Z 0 ! T and 0 .
The proof can be sketched as follows: Given Z 0 ; 0 and 0 , we can nd
a morphism  : T ! W such that the pull-back (Z1 ; 1 ) of (Z; ) by
is isomorphic to (Z 0 ; 0 ) over T0 (i.e. the restrictions to T0 are isomorphic);
besides (Z1 ; 1 ) and (Z 0 ; 0 ) are locally isomorphic over T. This is a conse-
quence of (c) of II above. Given these local isomorphisms (whose restrictions
to T0 de ne the given isomorphism of (Z; 1 ) with (Z 0 ; 0 ) over T0 ), we nd
that the obstruction to extending these local isomorphisms to a global one
over T is an element ,
 2 H 1 (Xr ; Hom (
0Xr ; OXr ))
where
0Xr denotes the sheaf of di erentials and Hom denotes the \sheaf
0 0
Hom". Similarly, the obstruction to extending 0 : Z0 ! X S T0 to a
morphism Z 0 ! X S T is an element 0
0 2 H 1 (Xr ; Hom (
0X0 ; OXr ))
238

and we see that  maps to 0 under the canonical homomorphism
H 1(Xr ; Hom (
0Xr ; OXr )) ! H 1 (Xr ; Hom (
0X0 ; OXr ))
(iv)
where  is the canonical morphism Xr ! X0 . But since 0 extends 0 , we 0
see that 0 = 0. One shows that (iv) is injective (see [G]) so that  = 0.
IV From the preceding discussions, we see that if we show that the functor
YS is represented by an open subscheme of H = Hilb p1 (X S Gr(m; n)),
Prop. 5.1 would be proved (of course the closed bre of YS ! S should
also be shown to be irreducible. But this follows easily). Let h 2 H be a
closed point represented by an element YS (Spec K), given by a curve Xr .
We denote by
(
pH : H ! H - the universal object over H and H the
(i)
canonical morphism H : H ! X S H = XH .
Let O be the local ring of H at h. We set C = Spec O and write
pC : C ! C; C : C ! XC
(ii)
for the base changes of (i) by C ! H. For proving the openness, the crucial
point to show is that (ii) represents an element of YS (C). For this one has
to use the property of dualizing sheaves given by II(d). One sees that the
bres of pC have only ordinary double point singularities. Let L (resp. M)
denote the dualizing sheaf of  (resp. XC ) relative to C. The point is the
claim:
 (M) = L:
(iii) C
By II(d) and the arguments in III, it follows that

Cn (Mn ) = Ln
where the subscript `n' denote base changes by Cn ! C where Cn =
Spec O=mn (m-maximal ideal of O). Then by an application of Grothendieck's
comparison theorem, the assertion (iii) follows (see [NS] for more details).
Once one has (iii), that (ii) de nes an element of YS (C) is a consequence of
the following lemma which is easily seen.
Lemma 5.1. Let Y be a connected projective curve with arithmetic genus
g and only ordinary double point singularities. Let f : Y ! D be a
239
Degenerations of moduli spaces

morphism, where D is a smooth projective curve of genus g or D ' X0 .
Suppose that the pull-back of the dualizing sheaf D is isomorphic to the
dualizing sheaf of Y . Then f is an isomorphism if D is smooth; otherwise
Y ' Xr and f identi es with the canonical morphism Xr ! X0 .

Thus we have the following:
Theorem 2. Let X ! S be a proper
at family of curves as in Def.
5.1. Then we have a scheme G(n; d)S which is quasi-projective and
at over
S (and projective if (n; d) = 1). Its generic bre is the moduli space of
stable vector bundles of rank n and degree d of the generic bre of X ! S.
Its closed bre is the variety G(n; d) (see Def. 4.1 and Theorem 1) whose
singularities are normal crossings. Besides, G(n; d)S is regular as a scheme
over k (recall that we have assumed that X is regular as a scheme over k).

Remark 5.2. Let (n; d) = 1. Consider the canonical morphism  :
G(n; d) ! U(n; d). Let F 2 U(n; d) such that its type is r. Then we
claim the following:
()
8The bre of  over F can be canonically identi ed with the \won-9
> >
< =

>derful compacti cation" of PGL(r) (in the sense of De Concini and>
:Processi [D-P]). ;

We shall now outline a proof of () when F is of type n (for type  n,
the proof is similar). Let E 0 2 G(n; d), such that  (E 0 ) = F. Then we see
that  (E 0 jX ) = F (see (c) in the proof of Prop. 3.4). It is not dicult to
see that E 0 jX is a stable vector bundle (intuitively, because of(b) of Remark
3.2). In particular, Aut (E 0 jX ) ' G m . Fix such an E 0 . If E 2 G(n; d) is
an extension to Xk of E 0 jX (1  k  n), we observe that deg EjR = n, so
that  (E) is of type n (by (B) of Prop. 3.4). Then by (c) of Remark 3.2,
the bre of  over F identi es with the set of all E 2 G(n; d) which are
extensions to Xk (1  k  n) of E 0 jX . We shall now describe this more
0 0
concretely. Let I1 = Ep1 and I2 = Ep2 , so that they are considered as xed
240

vector spaces of dimension n. Consider triples (V; 1 ; 2 ) such that:
8
> (a) V is a strictly standard vector bundle on R of rank n and
>
>
> degree n (it follows that length of R  n)
>
>
< (b) i : VP  Ii are isomorphisms (1  i  2).
i!
(i) >
>
> (c) if s is a section of V such that s(p1 ) = s(p2) = 0, then s
>
>
> vanishes identically.
:
We have the obvious notion of isomorphisms between triples. We see that if
: V ! V 0 is an isomorphism and (V; ; 2 ) is a triple, then  extends to
an isomorphism
 : (V; 1 ; 2 ) ! (V 0 ; 01 ; 02 )
of triples; besides,  and 01 ; 02 are uniquely determined. If ( 1 ; 2 ) 2
Aut I1  Aut I2 , then we get a map of triples:
(V; 1 ; 2 ) ! (V; 1  1 ; 2  2 )
and this \action" of Aut I1 AutI2 preserves isomorphism classes of triples.
Let Z1 denote the set of isomorphism classes of triples. Then the above
map de nes an action of Aut I1  Aut I2 on Z1 . Let Z2 denote the set of
equivalence classes of triples (as in Def. 4.1). Let C = G m  G m be the
centre of Aut I1  Aut I2 . The crucial point is that
Z2 ' bre of  over F:
We have of course a canonical action of Aut I1  Aut I2 on Z2 , which is
e ectively an action of (Aut I1  Aut I2 )=C ' PGL(n)  PGL(n).
We know that for V in (i), dim H 0 (V ) = 2n and H 0 (V ) generates V . Let
W be a xed vector space of dimension 2n and W the trivial vector bundle
on R of rank 2n. Consider the set of triples T = (W ! V; 1 ; 2 ) such that
(
(a) (V; 1 ; 2 ) is a triple as in (i) and
(ii)
(b) W ! V induces an isomorphism H 0 (W ) ' W  H 0 (V ).
!
We have canonical commuting actions of Aut W and Aut I1  Aut I2 on T.
We see that:
241
Degenerations of moduli spaces

the orbit space T=Aut W identi es with Z1 .
We x two linear subspaces K1 and K2 of W such that dim Ki = n and
W = K1  K2 . We x also identi cations W=Ki ' Ii , i = 1; 2. Let S1 be
the set of all fW ! V g such that
(iii)
8
> (a) V is as in (i)
<
> (b) the canonical map H (W ) = W ! H (V ) is an isomorphism
0 0
: (c) Ker (W ! Vpi ) = Ki, so that Vpi ' Ii (1  i  2):
Set
(iv) S = set of triples (W ! V; 1 ; 2 ) such that W ! V is as in S1 .
We see that i 2 Aut Ii . We see that given t 2 T, there exists g 2 Aut W
such that g  t is in S. The subgroup H 0 of Aut W which leaves S (also S1 )
stable, is precisely the subgroup which leaves K1 as well as K2 stable and
H 0 ' Aut K1  Aut K2 ' Aut I1  Aut I2 . Let H be the group (H 0 mod
centre) ' PGL(n)  PGL(n). It is easily seen that for s in S there exists
h 2 H 0 such that h  s is the triple
(W ! V; id; id); W ! V in S1
and, in fact, that the orbit space T mod Aut W can be identi ed with the
above set of triples. Thus we see that we have a canonical identi cation
S1 ' Z1 .
We see that S1 is the set of all embeddings : R ! Gr(W; n) such that
(v)8
>
>
> (a) the pull-back of the tautological bundle on Gr(W; n) by is a
>
> bundle V as in (1), and
<
> (b) the canonical map H 0 (W ) = W ! H 0 (V ) is an isomorphism
>
>
>
> (c) (pi ) = ki , ki the points of Gr(W; n) de ned by Ki (i = 1; 2).
:
Through the canonical identi cation S1 ! Z1 we get an action of Aut I1 
Aut I2 and we see that this identi es with the canonical action of H 0 of S1
(H 0 is the subgroup of Aut W which leaves invariant the points ki , i = 1; 2).
242

It is now clear that Z2 can be identi ed with the set of closed subschemes
(R) of Gr(W; n), as in (v). Thus we see that
(vi)
 the bre of  over F can be identi ed with the set of closed sub-
schemes (R) of Gr(W; n), as in (v)
Roughly speaking the above set is the set of all nicely imbedded curves of
type R (with length  n) in Gr(W; n), passing through the two xed points
k1 , k2 in Gr(W; n). We see that all these subschemes of Gr(W; n) have
the same degree. Thus Z2 can be identi ed with a closed subscheme of a
Hilbert scheme  of subschemes of Gr(W; n). We see that H = (Aut K1 
Aut K2 )=C ' PGL(n)PGL(n) acts on Z2 and (Aut K1 Aut K2 ) identi es
with the subgroup of Aut W xing ki , i = 1; 2.
Let R be of length one so that R ' P1 . Then we see that for all the
imbeddings of P1 as in (v), the inverse image by of the tautological
L
bundle on Gr(W; n) is V = n O(1) on P1 . Then the points of Z2 which
correspond to these imbeddings contribute one orbit under H = PGL(n) 
PGL(n) and it is not dicult to show that Z2 is the closure of this orbit.
We can take an imbedding of P1 as follows. Let D be a 2-dimensional
vector space so that P1 = P(D). We x a basis f1 ; f2 of D and we denote by
p1; p2 the points of P1 represented by f1 ; f2 . Set W = D
K, where K is
an n-dimensional vector space and Ki = fi
K so that W = K1  K2 . Let
1 ; : : : ; n be a basis of K. Then we see that
e1 = f1
1 ; e3 = f1
2 ; : : : ; e2n 1 = f1
n
is a basis of K1 and that
e2 = f2
1 ; e4 = f2
2 ; : : : ; e2n = f2
n
is a basis of K2 . Then the embedding
^
n
! Gr(n; W)(Grassmannian of n-dim subspaces of W) ,! P( W)
: P1
is de ned by
(x; y) = xf1 + yf2 7 ! n-dimensional linear subspace of W
spanned by (xe1 + ye2 ); (xe3 + ye4 ); : : : ; (xe2n 1 + ye2n )
243
Degenerations of moduli spaces

i.e.
^
n
(x; y) 7 ! (xe1 + ye2 ) ^ (xe3 + ye4 ) ^    ^ (xe2n 1 + ye2n ) 2 W:
We denote by z0 the point of Z2 de ned by this imbedding. We see that
pi 7 ! ki point of Gr(n; W) represented by Ki . If we identify Aut K as
the subgroup (id on D) Aut K of Aut W, we see that (Aut K) xes (P1 )
(pointwise). Then the isotropy subgroup of H at z0 is (Aut K)= centre, which
can be considered as the diagonal subgroup of H ' PGL(n)PGL(n). Thus
Z2 is a PGL(n)  PGL(n) equivariant closure of PGL(n) and it is natural
to expect that Z2 is the wonderful compacti cation of PGL(n).
We now see that if L is the line bundle OGrn;W (1) on Gr(n; W), then
L restricts to the line bundle OP1(n) on P1 and that
H 0 (Gr(n; W); L) ! H 0 (P1 ; OP1(n)) ! 0
is exact. Let us suppose that this phenomenon is more generally true for
all the imbeddings of R (which seems to be the case). From this we see
V
that if M is the linear subspace of P( n W) generated by (R) ( (R) ,!
V
Gr(n; W) ,! P( n W)), then dim M = n + 1. If z denotes the imbedding
of R for z 2 Z2 , let us suppose that z 7 ! M z de nes a closed immersion
^ ^
n+1 n
 : Z2 ! Gr(n + 1; Q) ,! P( Q) (Q = W):
(vii)
It is H equivariant.
We shall now determine (z0 ), where z0 corresponds to the embedding
of P1 given above. As we saw above, this is given by
(x; y) 7 ! (xe1 + ye2 ) ^    ^ (xe2n 1 + ye2n )
= xn 0 + xn 1 y1 +    + yn n
Say for n = 3, the above expression is given by
(x; y) 7 ! x3 e1 ^ e3 ^ e5 + x2 y(e1 ^ e3 ^ e6 + e1 ^ e4 ^ e5 + e2 ^ e3 ^ e5 )
+xy2 (e1 ^ e4 ^ e6 + e2 ^ e3 ^ e6 + e2 ^ e4 ^ e5 ) + y3 (e2 ^ e4 ^ e6 ):
If we now identify K1 ' K2 ' K (f1 7 ! f2 ), we have
^ ^ M^ ^
n n n i ni
W = (K  K) = ( K

Q= K)
i=0
M ^^
n i i
' Qi; Qi = Hom ( K; K)
i=0
244

and we see that i identi es with the identity homomorphism of
VV
Hom ( i K; i K) = Qi which is Aut K invariant. We see that Q0 and Qn
are 1-dimensional modules and that the H-irreducible module Hom (W ; W )
( = half sum of positive rootsN SL(n) ' SL(K)) is a direct summand of
Nn Q . We have Vn+1 Q = ( for Q )  other irreducible components for
n
i=0 i i=0 i
Aut K  Aut K. Now (z0 ) is de ned by
O! ^
n n+1
(0
  
n ) 2 Qi ,! Q:
i=0
We see that the projection of (0
  
n ) in Hom (W ; W ) is non-zero and
Aut K (in fact Aut K= centre) invariant. Now Hom (W ; W ) is of regular
special weight in the sense of De Concini-Processi and then V their work
by
[D-P] it follows that the closure of the H orbit H (z0 ) in P( n+1 Q) is the
wonderful compacti cation of PGL(n). Hence Z2 ' (Z2 ) is the wonderful
compacti cation of PGL(n) and the principal claim follows.
We have supposed that  is a closed immersion. However, we need not
use this. By the general theory of Hilbert schemes, Z2 gets imbedded in the
Grassmannian of (kn+1) dimensional quotients (kn+1 = dim H 0 (P1 ; O(kn))
of H 0 (P(Q); OPQ(k)), for k  0. This latter vector space is S k (Q ). Hence
we see that we get an imbedding:
 << ñòð. 8(âñåãî 12)ÑÎÄÅÐÆÀÍÈÅ >>