ñòð. 8 |

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

7 Comments 262

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 dierent 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 compactications 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 dened 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-dened 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 identied 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 dened 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 ane scheme, dened 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, identies 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-

pactications 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

Denition-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.

Denition-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 ane neighbourhood

V of p and set U = 1 (V ), V 0 = U \ X. Since V 0 is ane, 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 ) identies with the subspace of

H 0(EjV 0) consisting of elements `s' such that s(p1) = s(p2). This shows

that (E) identies with the vector bundle on X0 dened 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 dicult. 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 dicult 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-dened 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-dened 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 dened by Lq1 for EjR2 . In-

ductively, we thus dene 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 dene 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 identied 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 dened 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 dicult 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 dierent 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 dierent

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 ane 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 dened 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 = a1 ; 2 = a2; 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 identication

(

: 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 identies 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.

Denition 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):

0kn

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 satised.

We can identify Gr(H 0(E); n) with the standard Grassmannian Gr(m; n)

(this identication 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 identied (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 dened

by the condition that the corresponding point of the Quot scheme denes 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 dened 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' suciently

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.

Denition 4.2. Let Y be the functor dened 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 ft 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 = Ejt ) is of degree d (and rank n)

with d = m + n(g 1)

227

Degenerations of moduli spaces

(iv) by the denition of E, we get a quotient representation Ot ! Et and

m

we assume that this induces an isomorphism H 0 (Ot ) 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)

dened by Y. This denes 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 denes a family fEy g of vector

bundles on fy 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 () denes a point of the open subscheme R of Q(E=P2 ) (Quot scheme

mentioned above). Then we claim that the map : Y ! R, dened by

y 7 ! the point of R dened 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) denes 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

Denition 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).

Denition 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. dened 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 suces 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 ( ) denes 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 identies 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 suces 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

suces 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 dicult 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 dened 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 unramied

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; sg; 2 C

(we could view Spec D as a 2-dimensional disc). A -invariant section of

this bundle L is easily identied 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

identies 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 identies 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 satised 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 innitesimal 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 ) dened 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 (eective) deformation of the singu-

larities of the closed bre of Z ! W. More precisely, let V be an

ane 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 denes a deformation ZV ! W of V . The property

is that ZV ! W is an (eective) 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 innitesimal 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 suces to

show that is formally smooth.

To prove that is formally smooth we have to do the following. Given

an element 2 Yr0 (T ) dened 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 dene 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 dierentials 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) denes 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 identies 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 identied with the \won-9

> >

< =

>derful compactication" 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 dicult 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 identies 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 denes 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

eectively 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 identies with Z1 .

We x two linear subspaces K1 and K2 of W such that dim Ki = n and

W = K1 K2 . We x also identications 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 identied with the

above set of triples. Thus we see that we have a canonical identication

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) dened by Ki (i = 1; 2).

:

Through the canonical identication S1 ! Z1 we get an action of Aut I1

Aut I2 and we see that this identies 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 identied 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 identied 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 identied 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 ) identies

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 dicult 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 dened 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 dened 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 compactication of PGL(n).

We now see that if L is the line bundle OGrn;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 denes 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 y1 + + 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 identies 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 dened 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 compactication of PGL(n). Hence Z2 ' (Z2 ) is the wonderful

compactication 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); OPQ(k)), for k 0. This latter vector space is S k (Q ). Hence

we see that we get an imbedding:

ñòð. 8 |