# Cohomology of desingularization of moduli space of vector bundles

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## C OMPOSITIO M ATHEMATICA

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### 3 (1989), p. 309-339

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309

### Cohomology of desingularization of moduli space of vector bundles

NITIN NITSURE

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400 005, India

Received 25 September 1987; accepted in revised form 27 June 1988

Compositio 309-339,

Introduction

Let X be a smooth

curve over

and let

### Uo

be the moduli space of rank 2 semistable vector bundles on X with trivial determinant. When X is

a canonical

of

has been con-

structed

### by

Narasimhan and Ramanan

the Hecke

Let

denote this

The main result

### proved

in this paper is the

### following.

THEOREM 1: Let X be a smooth

curve over C

genus

3 which

is

and let

### Uo

be the moduli space

### of rank

2 semistable vector

bundles on X with trivial determinant. The

group

### H3(H0, Z) of the

Narasimhan-Ramanan

is

and

rank

### 2g.

The motivation for

that

### H3

is torsion-free is as follows. The moduli space

is known to be

### unirational,

but it is not known whether it is rational. As the torsion

of

### H3(V, Z)

is a birational invariant of a smooth

### projective variety (see [1]),

it was of interest to determine

for some

model

of

### Uo.

Narasimhan and Ramanan state that

page

the

tion

### Ho

can be blown down

certain

### projective

fibrations to obtain another

model of

A

of this is

in

below

This new

### desingularization

has the same third

group as the

The

### arrangement

of this paper is as follows. Let K be the

locus

of

Let Z c

be the

of the fibres of

over the

of K of the

### form j E9 j-l where j

is an element of order 2 in the

Jacobian. For any x

### E X,

there is a natural conic bundle C over Z which

(3)

into

### pairs

of lines over the divisor Y which is the inverse

### image

of K in Z. Sections 1 to 3 are devoted

to

### proving

that the

total space of this conic bundle is smooth. This allows us to

the result

in

### [6]

which relates the

### topological

Brauer class of the

### P1-bundle

on Z-Y determined

C to the

class in

### H2(Y, Z)

which

is associated to the 2-sheeted cover of the

locus Y deter-

mined

C.

In

we show that

cl

an

map from

to

where

### IVy,z is

the normal bundle to Y in Z.

in

we

the

### proof

of Theorem 1.

Remark on notation: In this paper, we have followed the notation used in

In

for a

on the

### curve X, Lx

denotes the line bundle defined

the divisor x,

### and Ux

denotes the moduli space of all rank 2 stable vector bundles E on X for which det E is

to

### Lx.

The moduli space of all rank 2 stable vector bundles on X whose determinants are

to

### Lx

for some x E X is denoted

The

from

to Hilb

Let

c Hilb

be the

of

and let x E X be

any

### point.

As a subscheme of Hilb

a flat

W c

x

of subschemes of

### UX. By [5]

Lemmas 7.9 and 7.12 for any

E

the Hilbert

of the subscheme

of

with

to the

of Pic

is 2m + 1 where

= Wy

### ~ Ux

where

Wy is the fibre of W over y. As

is an

### integral scheme,

it follows from the

of the Hilbert

that the

x

of subschemes of

is a flat

The above flat

has a

Hilb

It

follows from

is

In this

we prove

that if 1 =

is a limit Hecke

of

for the

over a non-nodal

of K c

### Uo,

then the differential

of

Hilb

is

at the

1 E

### Ho.

LEMMA 1.1: Let S be a

T a

### non-singular

curve, and

P: S ~ T a smooth

### morphism.

Let V ce S be a 2-dimensional closed reduced subscheme such that either

### (i)

V is irreducible and the

V - T is

or

V

is a union

### of exactly

two irreducible components each

and the

T

are smooth

n

### V2

is the scheme theoretic intersection

the reduced

### schemes Vl and V2

which we assume is a

### non-singular

curve.

(4)

Let t E T be a closed

and let

denote the

S

over t. Then

V is a local

### complete

intersection in S.

is a local

intersection in

### (3)

The normal bundle

in

### S,

is the restriction

the normal

bundle

V in S.

Lemma 1.1:

are

### nonsingular

subvarieties of the

S such that the

is a

### non-singular variety,

the reduced scheme V

is a local

intersection

### by [5]

Lemma 8.2. This proves

If V is

the

### morphism

V - T is smooth. Hence

is a

in

a local

### complete

intersection. If V

then as

both

and

are

varieties. As

n

T is also

### smooth,

it follows that the scheme theoretic intersection of

and

is a

### regular

scheme and hence the

is

is a local

### complete

intersection as above. This proves

To prove

### (3)

we prove the dual statement about conormal

that

### N*Vt,St ~ N;s |Vt.

A local section of

is

defined

### by

a local section of

### N:s

which is a differential 1-form on S which kills vectors

### tangent

to V. This defines a 1-form on

### S,

which kills vectors

to

This defines a

### morphism

of vector bundles from

to

From the

### hypothesis

of this lemma it is easy to see that this

is

### injective

on

each fibre hence an

### isomorphism

as these vector bundles are of the same rank.

LEMMA 1.2 : Let P: S - T be as in lemma 1.1. Let W c S x R be

subschemes

S

a

R, such

each

r E

### R,

the subscheme wr c S

the conditions

### imposed

on V c S

in lemma 1.1. Let t E T be a

and let

W n

x

subschemes

R

Let 0: R - Hilb

and

R ~ Hilb

be the

W and

For

Hilb

Hilb

### (St)

be the restriction map

to

where

lemma 1.1. Then

### the following diagram of vector

spaces is commutative.

(5)

312

1.2 :

it is

### enough

to prove a similar statement where the

R is

### replaced by Spec k[03B5]/(03B52).

In this set up, the lemma follows

### immediately by taking

a suitable affine open cover of S and

at the

of the

with

### respect

to this open cover.

Now let

be the

of

and

Hilb

be

the

defined at the

### beginning

of this section.

LEMMA 1.3: Let 1 =

U

be a limit Hecke

over a non-nodal

e

and let 1 be

a

y e

Then the

map

Hilb

is

### Proof.-

Let P: S ~ T be the map det:

X. Then the

of lemma

1.2 is satisfied

### by

the restricted universal

W

and

so

### do,,

is the restriction map from

The

### injectivity

of this map follows from lemma 1.4 below.

LEMMA 1.4:

section

### of Nl,UX

which restricts to zero an

= l ~

is

zero.

For the

of lemma

we need the

### following

LEMMA 1.5 : Let U be a

and let

be closed non-

### singular surfaces

in U which intersect

a

curve

X. Let Y =

U

is a local

### complete intersection).

Then there is a short exact sequence

sheaves on

In

any

section

which vanishes

over

X is the

a

section

U .

1.4 : Let N 1 =

and N2 =

Then we have the

### following

exact sequence on 1.

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Let 0 E JIÜ

### (1, Nl,UX)

which restricts to zero

= 1 n

### Ux .

Then 0 defines sections

of N’ and N2

such

0

n

0 on

and

on X

n

### l2 . By

lemma 1.5 we have

an exact sequence.

Now

=

and

=

### P(E2)

where the vector bundles

### El and E2

occur

in exact sequences

In 1 =

u

the bundles

and

are identified

### along

the sections defined

and

Hence

we

### get

the short exact sequence

E

vanishes over

n

Hence the

in

### H° (X, OX)

must vanish at x E X and hence be

zero. Hence

### by

exactness of the sequence,

E

is the

of an

E

also be zero on

Hence to

the

of lemma

it is

to show that any

section

which is

zero on

=

is

zero.

The bundle

occurs in the

### following

exact sequence.

Hence we have to prove that any

section

and any

section

### of NP(Dj),UX lj

which vanish over

are

zero.

is

to a direct sum

2

### copies

of i where i is the

bundle on

=

As

### El

occurs in the exact sequence

it follows that dim H0

### (l1, 03C4) = 1,

and the non-zero sections vanish

any section

vanishes over

is identi-

### cally

zero. We now consider

We have the

exact

(7)

sequence on

lemma

### 6.22)

where "trivial" denotes a trivial vector

03C0:

is the

and

### F(j)

is a vector bundle on X.

attention to

c

where

~ X.

### Now,

it is obvious that any

section of r-’ Q

is

zero, while any

### global

section of the trivial bundle is constant.

any

section of

which is zero on

is

zero. This

the

of lemma 1.4.

Lemma 1.5:

Lemma

### 8.2,

there is an exact sequence

We

the functor Hom

### ( -, OY1)

to the above sequence to get the short

exact sequence

where L is the cokernel sheaf of the first

### morphism.

Note that a non-zero

section of L is defined

a

a:

### Niu Y1 ~ (Dy,

which does not extend to a

from

to

### OY1.

Hence it follows that the

### support

of L lies inside X.

let P E X and

a

function on

defined in a

of P which

### locally

defines the divisor X. Then there exists a local basis

... ,

for

### N*Y1,U

and a local basis

... ,

for

in a

### neighbourhood

of P such that the

defined

el

- e; for 2 i n -

### 2,

where n = dim U. Note that when restricted to

### X,

el defines a local frame

of

the exactness of

Let a be a local

from

to

such that

(8)

where

are

### regular

functions defined

### locally.

To a we associate a local

section ex of

Q

### X,Y2

defined as follows where

### Lx

is the line bundle

defined

### by

the divisor X. Note

### that f-1

is a local frame for the line bundle

### LX.

Then we define - =

~

### él . Now,

a extends to a local homo-

from

to

if and

vanishes

X. Hence we

a well-defined

between and

Q

### Ñx, Y2.

It is easy

to see that these local

to

a

iso-

between and

Q

Now

the

and hence

Q

@

which

the

of the lemma.

The

Hecke

### cycle

A. Preliminurie.s

Let x E X be fixed. Then for a

E

we can

a

of

E

which

a limit Hecke

some

### special

property. In this

we define such a

Hecke

1

definition

and then

### explicitly

determine the normal bundle

of its

restriction

= 1

in

lemma

A crucial

of the

### argument

in this section is the

### following

tech-

nical lemma about the normal bundle of the union of two smooth curves

which intersect

inside a smooth

### variety.

We shall use this

lemma for

LEMMA 2.1: Let U be a smooth

and

and

### L2

smooth closed curves in U

at a

### single point

,sr. Let the tangent space

U have a

direct sum

=

### TsL2.

Let the normal bundles

and

in U

have direct sum

as

(9)

Under the

map

let the

...,

be contained

in the

### subspaces (F2)s,..., (Fm)s of (NL1,U)s,

and let the induced

i =1 1 be

let the

...,

under the

map ~2:

U -

be contained

in the

...,

let the

i =1 2 be

Under the above

### hypothesis,

the normal bundle

in U has the

direct sum

i

V

### Gi

denote the bundle

on

obtained

and

their

with

Let

and

### F2(s)

denote the line bundles

Q

and

=

1 0

Then

where

v

### G1 (s)

is the line bundle on

obtained

sonie

tion between the

and

at s.

### Proof of Lemma

2.1: We shall prove the dual statement about the conormal bundle of

### L, u L2, namely,

We have an exact sequence

~ ··· C

and the map

maps

to zero

3. Hence

is

to F ~

~ ··· ~

### £,/*

where F is the kernel sheaf of the map

which

### is just F2* ( - s),

the sheaf of all section

### of F*

which vanish at s. We thus get

an

(10)

Note that at the

the

of

for 1 a 3 in

is

the

of

to

c

U.

we have an

such that

3 the

of

under gs in

is

the

to

c

U.

Hence under the

### isomorphism gs 1 · fs

between the fibres over s of

~

~ ···

and

C

C - ’ ’ Ef)

the

of

is

### precisely (G,*),.

Now it is easy to see that the induced maps from

to

### (Gi*)s

are all zero for i 3. Hence under

maps

onto

Hence

is isomor-

### phic

to the direct sum

This proves the lemma.

B. The

Hecke

### cycle

In this subsection we define the

Hecke

Let

H’

@

and let

be imbedded in

as in the

of

a

of

and hence the

has a lift

where

is the dual

### projective

Poincaré bundle. Let

n

and

n

### P(Dj-1,x).

Let J’ be the subset of J where

1. When x is

we

f.1: J’ ~

and

J’ ~

where

=

=

and

=

=

Note

is the lift

to

### Hx.

Remark 2.2: It can be checked that the differential of f.1 is

on J’.

### As fi

is the lift of f.1, it follows that

is also

### everywhere injective

on J’.

LEMMA 2.3: Consider the

J ~

which is

R E9

### R-’ of vector bundles on X parame terized by J where R ~ X

x J is the

Poincaré line bundle. Then the open subset

on which the

this

### map is injective

is a non-empty set.

Let K c

be the

### singular

set with reduced subscheme structure.

Then the

J -

factors

K ~

c K be

(11)

the smooth open

of K and let

### J*

c J be its inverse

As J - K

is

is

dim

dim

g, and both

and

### K*

are smooth. Hence there exists a

in

at which the map

### J* ~ K*

is of maximal rank. This proves the lemma.

DEFINITION 2.4: Let x E X be fixed.

there exists a

### non-empty

sub-set of J such that

the differential of the map J1: J -

H

is

### injective (in particular

the differential of

J -

is

### injective)

and the differential of the map J -

is

for

### any j

in this open subset of J.

We will call such

a

### general point

of J.

LEMMA 2.5: Let x E X

and

be a

### general point of J (see 2.4).

Then

the tangent space to

at the

has

direct sun1

### Proof-

We have a commutative

where

=

Now

E J is a

dv is

at

Hence

the

of the above

dh is

on the

of

inside

On the other

### hand,

dh is zero on both

and

### TsP(Dj-1,x),

which in turn are

as shown

in

Hence the three

are

in

As

### 3g -

2 which is the dimension of

the

lemma is

~

be the

the

~

at the

s =

### s(j, x).

Then there exists a

E J such

that the

on oach

### of

the three direct summands

### by

lemma 2.5 is non-zero.

(12)

Remark 2.7:

### Though

we do not need it

### here,

with some extra calculations we can in fact prove that the conclusion of lemma 2.6 is true for all

Lemma 2.6:

and

intersect

inside

This shows

### that Lj

is not contained in

E9

Hence the

of

on

is

non-zero for a

as

is

remark

### 2.2, Lj

is not contained in

Hence the

tion

on

EB

is

non-zero for a

### Now,

the condition that the

of

on

### TsP(Dj,x)

is zero is a closed

condition in the Zariski

the

### irreducibility

of the open subset of J of

### general points j,

we see that if the lemma is

then either im

c

for all

or

c

for all

### general j.

Note that the open subset c J of all

is stable

Hence we

### get

a section of the canonical map of J to the Kummer

over a

open set

the

of

Now choose a

lemma

and let

E

E

and

E

be the

### projections

of a non-zero vector in

Note that

+

E

are the

of

### Mi, û2, Ù3 respectively,

then each ui is non-zero, but ul + U2 + U3 = 0. Let

and

be lines in

and

each

s =

n

the

vectors ul E

and

U 2 E

of

there exists a

limit Hecke

1 such

that

=

where

denotes 1 n

### Ux.

DEFINITION 2.8: The limit Hecke

### cycle

1 above will be called a

Hecke

and

=

### L, u L2

will be called a restricted

Hecke

### cycle.

Remark 2.9: Note that

and

are defined

the

### tangent

vectors ul , u2 E

### T, U,

above. The vector U3

### above,

which satisfies u1 + u2 + u3 =

is

to the

of f.1: J’ ~

### Cf,.

Remark 2.10: The lines

c

and

c

have the

direct

we do not

is

### given by

the 2-dimensional kernel of the linear map

Q

~

~

and a similar

holds for

The vector u3 E

lies in the

under

### df.1

of the kernel of the map

~

where

=

C. Normal bundle

the restricted

Hecke

### cycle

LEMMA 2.11: Let x E X

and

E J be a

this x. Let

c

the restricted

Hecke

to

with

(13)

320

=

~

where each

line in the

space

Then the

normal bundle

has

direct sum

into

subbundles.

where

and D ~

### Qg-’ Olx

is the trivial bundle

rank g - 1.

Choose

of the

spaces of

n

and

n

at the

s =

n

as follows. Let

~

and let

while

are

Let U3 E

be as in remark

and

choose an

Then

lemma

we

a direct sum

of

as follows.

Note that

dim

2 and

### dim V5 =

g - 1. Now the normal bundle

=

in

### t/,

occurs in the exact

sequence

### (see [5]

lemma 6.22 and the

of lemma

### 7.10)

such that under the canonical map

the

c

maps

on

c

Hence the

as

induces a

### splitting

of the trivial bundle

for i =

### 1, 2,

and we denote these

H1

0

and H1

### (X, OX)P2 ~ 1 ~ 5

where the bundles

are all trivial.

(14)

as

= 0 for any

### projective

space, we can choose a

### splitting

of the exact sequence

for i =

to

direct sum

as follows.

and

### Now,

the normal bundles

for i =

### 1,

2 fit in the exact sequence

As

### NLt,Pt ~ ~g-2 (DL, (1),

the above exact sequence

### splits,

and so we can

choose a direct sum

Let

### F2, F5

be the restricted bundles

and

while let

and

Note that

### F2, F5, G1, GS

are trivial bundles.

Now define

on

and

Then we

the direct sum

### decompositions

while

Now from the definitions of the

and

### Gi’s

above it is immediate that the

### hypothesis

of lemma 2.1 is satisfied. Hence

lemma

the normal

(15)

322

bundle of

=

in

### Ux

has the direct sum

### decomposition

while it is obvious that

and

This

the

of the lemma.

### §3.

The conic bundle over the

### desingularization

We first recall the definition of a conic bundle as

in

of

### [5].

DEFINITION 3.1: A scheme C

with a very

### ample

line bundle h is said

to be a conic if its Hilbert

### polynomial

is 2m + 1. A scheme C over T is said

to be a conic bundle over T if with

### respect

to a line bundle on C its fibres are

conics and the

### morphism

C ~ T is proper and

flat.

For

let Hilb

### ( Ux )’

be the open subset of Hilb

where the

Hilbert

### polynomial

is 2m + 1. Then the

c

x Hilb

### (Ux)’

is a conic bundle over Hilb

### (Ux)’.

Remark 3.2: Let E be a rank 3 vector bundle on T

with a

of the

bundle

where

is the second

### symmetric

power of E *. This

### section q

defines a subscheme C c

### P(E )

such

that C ~ T is a conic bundle as defined above.

Remark 4.4

### (iv),

every conic bundle C ~ T arises in the above manner, where a

### possible

canonical

choice for the vector bundle E is the first direct

where 7r’:

C ~ T is the

### projection

and cv is the relative

### dualizing

sheaf on C over T.

In the

### subsequent

discussion we assume that every conic bundle C is defined

as a subscheme

where E =

it can be shown

(16)

323 that

we do not need

### it)

the type of the conic bundle

is

### independent

of the choice of the vector bundle E.

Note that a

section of

### P(S2E*)

is the same as a line subbundle

L c

### S2E*

which defines a

on E with values in L*. The

discriminant

### det q of q

is a section of the line bundle

0

The

### degeneration

locus Y c T of the conic bundle C ~ T is defined

the

of det q. At a

E

the fibre

### C,

is a double line or a

of

distinct lines

on whether

### rank q

is 1 or 2.

DEFINITION 3.3: Let t be a smooth

### variety

and C ~ T be a conic bundle for which the

### degeneration

locus Y is an irreducible divisor. Let C

into

### pairs

of distinct lines at all

### points

of Y. Then the type

of the conic

bundle is

definition the

of

### vanishing

of the discriminant det q over the divisor Y.

Consider the

Hilb

defined in

where Hilb

### (Ux)’

is the open subset of Hilb

### ( Ux )

for which the Hilbert

is 2m + 1.

The

over Hilb

### ( Ux )’

is a conic bundle 16 and hence its

to

### Ho

is a conic bundle on

The

locus of

### 0

*L is the closed subset 03C0-1

which is the

of the

locus K c

of

### Uo

under the map 03C0:

Let

### Ko

c K be the set of

nodal

of K. Let Z c

### Ho

be the open subset Z =

### Ho - 03C0-1 (Ko).

Let

the conic bundle C ~ Z be the restriction

Z c

Then the

### degeneration

locus for C is the irreducible divisor Y =

Z.

LEMMA 3.5: The

locus Y is a

### Pro of.

Let R - X x J’ be the Poincaré line

where J’ c J

### {j|j2 ~ 1}.

Then the first direct

and

are

free sheaves on

each of

1. Let

J’ be the

x

### Pg-2

bundle which is the fibred

of the

### corresponding projective

bundles. As described in

each

of

### Y

determines a limit Hecke

1 =

U

ce

### U,

in a canonical manner, and so

an

of subschemes of

### U,

each of which is a limit Hecke

### cycle

and hence has the same Hilbert

As

is an

the

above

### family

is flat and so we

a

Hilb

### (UX).

It is easy to see that Y is the

of this

As

is

to prove

the

of Y it is

to show that the

level map

is

Let g:

Hilb

be the

map

Hilb

### ( Ux )

is as defined earlier. Then it is

to prove that

is

since the one

union

imbeds in

it is

(17)

324

obvious that

is

the fibres of

J’. On the other

let b e be

e

and let v e

### Tb project

to a non-zero vector

w e

Let the

of b in Hilb

### (Ux)

be the restricted Hecke

denoted

Note that

~

is the

defined

in

Then v e

### Tb Y corresponds

to an infinitésimal deformation

such that the

infinitesimal

### displacement

of the intersection

in

is the

vector

### d03BC(w),

which is non-zero

### by §2.

Hence

the infinitesimal deformation of

### LI U L2

is non-zero. Hence

is

### injective,

which proves the lemma.

Over

### Y,

the conic bundle C ~ Z

into

### pairs

of distinct lines.

This defines a 2-sheeted

of Y. The

### following

lemma is obvious and

we omit its

### proof.

LEMMA 3.6: The 2-sheeted cover

the

locus Y

### defined by

the

conic bundle C - Z is not

has the same

structure

and K’ = K -

### K0)

under the map 7r: Y - K’.

Next consider the restriction

### Cz-

y of C ~ Z to the open set Z -

### Y, over

which it defines a

Under 7r:

### Ho - Uo

the open subset Z - Y maps

### isomorphically

onto the open set

K. Let

be the dual

Poincaré bundle

where

and consider the

Hecke map h :

as defined in

### [5] §5.

LEMMA 3.7: There exists an

h-’

which makes

commutative

Let

be the

### correspondence variety

over the Hilbert scheme Hilb

which contains

### Ho

as a closed subscheme

c Hilb

Let

H be the dual

### projective

Poincaré bundle on

### UX,

with the Hecke map h:

H ~

### Uo.

Then it follows from the identification

c Hilb

with an

### appropriate

closed subset of the relative Hilbert scheme Hilb

### §5 onward)

that the restriction of the

over

Hilb

### (UX)

to the subset Z - Y is

### canonically isomorphic

to the fibration h : H -

K under the

n : Z - Y 5

K.

Now

### by

definition of the

Hilb

we

have to

intersect the

H -

1

c

x

with the subset

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