Let $M$ be the moduli space of generalized parabolic bundles (GPBs) of rank
$r$ and degree $d$ on a smooth curve $X$. Let $M_{\bar L}$ be the closure of
its subset consisting of GPBs with fixed determinant ${\bar L}$. We define a
moduli functor for which $M_{\bar L}$ is the coarse moduli scheme. Using the
correspondence between GPBs on $X$ and torsion-free sheaves on a nodal curve
$Y$ of which $X$ is a desingularization, we show that $M_{\bar L}$ can be
regarded as the compactified moduli scheme of vector bundles on $Y$ with fixed
determinant. We get a natural scheme structure on the closure of the subset
consisting of torsion-free sheaves with a fixed determinant in the moduli space
of torsion-free sheaves on $Y$. The relation to Seshadri--Nagaraj conjecture is
studied.Comment: 7 page

Bhosle, Usha N

Proceedings of the Indian Academy of Sciences. Mathematical sciences

English

arXiv

Let M be the moduli space of generalized parabolic bundles (GPBs) of rankr and degree dona smooth curve X. Let ML&#x0305; be the closure of its subset consisting of GPBs with fixed determinant L&#x0305;. We define a moduli functor for which ML&#x0305; is the coarse moduli scheme. Using the correspondence between GPBs on X and torsion-free sheaves on a nodal curve Y of which X is a desingularization, we show that ML&#x0305; can be regarded as the compactified moduli scheme of vector bundles on Y with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves on Y. The relation to Seshadri-Nagaraj conjecture is studied

Bhosle, Usha N.

Publications of the IAS Fellows

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 115, No. 4, November 2005, pp. 445–451.
© Printed in India
Vector bundles with a fixed determinant on an irreducible
nodal curve
USHA N BHOSLE
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
E-mail: usha@math.tifr.res.in
MS received 7 July 2005; revised 31 August 2005
Abstract. Let M be the moduli space of generalized parabolic bundles (GPBs) of rank
r and degree d on a smooth curve X. Let ML¯ be the closure of its subset consisting of
GPBs with fixed determinant L¯. We define a moduli functor for which ML¯ is the coarse
moduli scheme. Using the correspondence between GPBs on X and torsion-free sheaves
on a nodal curve Y of which X is a desingularization, we show that ML¯ can be regarded
as the compactified moduli scheme of vector bundles on Y with fixed determinant. We
get a natural scheme structure on the closure of the subset consisting of torsion-free
sheaves with a fixed determinant in the moduli space of torsion-free sheaves on Y . The
relation to Seshadri–Nagaraj conjecture is studied.
Keywords. Nodal curves; torsion-free sheaves; fixed determinant.
1. Introduction
Generalized parabolic vector bundles (GPBs) on a smooth curve X are vector bundles on
X together with parabolic structures on finitely many disjoint divisors Dj, j = 1, . . . , m
[1, 2]. There is an open subscheme M ′′ of the moduli space M of GPBs on which one can
define a determinant morphism into the moduli space of generalized parabolic line bundles
L¯, the map does not extend to M . Let M ′′
L¯
be its locally closed subset consisting of GPBs
with a fixed determinant L¯. In this note, we define a moduli functor and construct a coarse
moduli scheme ML¯ for it. The moduli scheme contains M ′′L¯ as an open dense subscheme.
Let Y be an irreducible projective nodal curve with nodes yj , j = 1, . . . , m and p: X →
Y its desingularization with Dj the inverse image of yj . Denote by U the moduli variety
of torsion-free sheaves of rank r , degree d on Y . Let U ′ be the open subvariety of U
corresponding to vector bundles on Y . There is a surjective morphism f from M onto U
[1, 2]. The restriction of the morphism f to M ′ = f−1U ′ is an isomorphism onto the
open subvariety U ′ of U . A GPB L¯ gives a torsion-free sheaf L on Y . If L is locally free,
let U ′L be the closed subset of U
′ consisting of vector bundles with fixed determinant L.
Using f, U ′L may be identified with M
′
L¯
and ML¯ can be regarded as compactified moduli
variety of vector bundles on Y with determinant L.
We show that f (ML¯) = U ′L, the closure ofU ′L inU , thus givingU ′L the scheme structure
of an image subscheme of U . Let Ij denote the ideal sheaf at the node yj . For a torsion-
free sheaf F of rank r on Y , let N = rF/(torsion) where (torsion) denotes the torsion
subsheaf. Then we show that for any L¯, the image f (ML¯) can be described (as a set) by
f (ML¯) = {F ∈ U : I rjL ⊂ N ⊂ L, ∀j}.
445
446 Usha N Bhosle
This gives a proof of a conjecture by Seshadri and Nagaraj (Conjecture (a), p. 136 of [3]).
Proving Seshadri–Nagaraj conjecture was not the aim of this note. The conjecture was
proved by Sun [6] by degeneration methods. However he does not get a scheme structure
on U ′L or a moduli functor (except in some low rank cases). Our aim is to give a moduli
functor and an explicit construction of a projective moduli space for it which contains
an open subvariety isomorphic to U ′L if L is a line bundle. We also deal with the case
when L is torsion-free but not locally free. The construction is much simpler than that
of Schmitt [4] and hence the moduli space is easier to study. For example, properties
like reduced, irreducible, Cohen–Macaulay follow immediately for our moduli space.
Normality is true in rank 2 and is expected to be true in general. These properties have
been used in computation of Picard groups in the rank two case.
2. The moduli scheme of GPBs with fixed determinant
2.1.
Let X be a nonsingular projective curve over an algebraically closed base field k. Let
Dj, j = 1, . . . , m be disjoint divisors on X with Dj = xj + x′j , where xj , x′j are distinct
closed points. We recall here some basics on generalized parabolic bundles (GPBs), details
may be found in [1, 2].
DEFINITION 2.1
A generalized parabolic bundle (GPB, in short) of rank r and degree d on X is a vector
bundle E of rank r and degree d on X together with r-dimensional vector subspaces
Fj (E) of Exj ⊕Ex′j . For a subbundle N of E, define Fj (N) = Fj (E)∩ (Nxj ⊕Nx′j ) and
fj (N) = dim Fj (N).
DEFINITION 2.2
Fix a rational number α ∈ (0, 1]. A GPB (E, Fj (E)) is α-stable (resp. α-semistable) if
for every proper subbundle N of E, one has (d(N) + αjfj (N))/r(N) < (resp. ≤)
(d(E) + αrm)/r.
DEFINITION 2.3
Let pj : Fj (E) → Exj , p′j : Fj (E) → Ex′j be the projections. Assume that for each j , at
least one of pj , p′j is an isomorphism. The subspace Fj (E) determines an element Fj (E)
of Gr(r, Exj ⊕Ex′j ) ⊂ P(r(Exj ⊕Ex′j )). One has a (rational) morphism δ: P(r(Exj ⊕
Ex′j )) → P(rExj ⊕ rEx′j ). Let det Fj (E) denote the one-dimensional subspace of
rExj ⊕rEx′j determined by δ(Fj (E)). Define the determinant of (E, Fj (E)) to be the
generalized parabolic line bundle (det E, det Fj (E)).
DEFINITION 2.4
A family of GPBs of rank r , degree d parametrized by a scheme T is a tuple (E, Fj (E)j )
where E → T × X is a family of vector bundles of rank r , degree d on X which is flat
over T and Fj (E) is a rank r subbundle of E |T×xj ⊕ E |T×x′j . The notion of equivalence
of families is the obvious one.
Vector bundles with a fixed determinant 447
We fix a generalized parabolic line bundle L¯ := (L, Fj (L)). Fix isomorphisms
hj : Lxj → k, h′j : Lx′j → k. Then Fj (L) can be identified to a point Fj (L) of P1 of the
form (1 : 0), (0 : 1) or (1 : λj ), λj ∈ k∗.
2.2 The moduli functor
For simplicity, let us assume that there is only one divisor D = x1 + x2. Let (E, F (E)) →
T × X be a family of GPBs of rank r , degree d on X with Et , t ∈ T , of fixed determinant
L. For i = 1, 2 we have vector bundles
Exi = E |T×xi→ T .
Let Gr → T denote the Grassmannian bundle of rank r subbundles of Ex1 ⊕ Ex2 . It is
embedded as a closed subvariety in P(r(Ex1 ⊕ Ex2)) by Plu¨cker embedding. Note that
F(E) defines a section of Gr . Since det E |t×X= L, it follows that det E = pT ∗N ⊗ p∗XL
for some line bundle N on T . Hence for i = 1, 2 one has det Exi = N ⊗ Lxi = N , using
the isomorphism hi . Let qi : r(Ex1 ⊕ Ex2) → det Exi = N be the projections, i = 1, 2.
Define a hyperplane subbundle H of P(r(Ex1 ⊕ Ex2)) by q2 = 0 if F(L) = (1 : 0), by
q1 = 0 if F(L) = (0 : 1) and by q2 − λjq1 = 0 if F(L) = (1 : λ). Let HT := Gr ∩H.
It is a closed reduced subscheme of Gr ⊂ P(r(Ex1 ⊕ Ex2 )). Note that HT is independent
of the choice of h1, h2.
More generally, if we consider parabolic structures over finitely many disjoint divisors
Dj = xj + x′j , for each j one constructs the hyperplane bundle Hj and Grassmannian
bundle Grj over T . Let Gr be the fibre product of Grj over T and HT the fibre product of
Grj ∩Hj over T .
DEFINITION 2.5
Let F ss
L¯
be the functor F ss
L¯
: Schemes → Sets which associates to a scheme T the set of
equivalence classes of families (E, F (E)) → T × X of α-semistable GPBs of rank r and
degree d with det Et ∼= L for all t ∈ T such that the section of Gr defined by (Fj (E)j )
maps into HT .
One similarly defines a full subfunctor F s
L¯
of F ss
L¯
with semistable replaced by stable.
2.3 Construction of the moduli space
Let S denote the set of semistable GPBs (E, F (E)), where E is a vector bundle of rank r ,
degree d with fixed determinant L and F(E) is a subspace of Exj ⊕ Ex′j of dimension r
with fixed weights (0, α), 0 < α < 1. For m  0, all GPBs in S satisfy the condition
(∗) H 1(E(m)) = 0, H 0(E(m)) ∼= Cn,H 0(E(m))→H 0(E(m) ⊗ (⊕jODj ))
is surjective.
Let Q denote the quot scheme of coherent quotients ofOnX⊗OX(−m) with fixed Hilbert
polynomial determined by r, d. Let R be the nonsingular subvariety of Q corresponding to
quotient vector bundles E satisfying condition (∗). Denote by R0 the nonsingular closed
subvariety in R corresponding to E with r(E) = L. Let E → R × X be the universal
quotient bundle. Over R, we have the fibre bundle Gr with each fibre an m-fold product
of Gr(r, 2r) as in §2.2. Over R0, we have the fibre bundle R˜0 := HR0 whose fibres are
448 Usha N Bhosle
m-fold products of hyperplane sections of Gr(r, 2r). Then R˜0 is a closed subvariety of Gr .
Let R˜0
s (resp. R˜0ss) be the open set of stable (resp. semistable) points in R˜0.
The GIT quotient M of Gr by PGL(n) for a suitable polarization (depending on α) is
the coarse moduli space for GPBs [1, 2]. It is a normal projective variety. Since R˜0 is a
PGL(n)-invariant closed subscheme (subvariety) of Gr , the GIT quotient ML¯ of R˜0
ss by
PGL(n) is a closed subvariety of M (with a natural reduced subscheme structure). The
GIT quotient Ms
L¯
= R˜0s//PGL(n) is an open subscheme of ML¯. It is easy to see that ML¯
(resp. Ms
L¯
) is the coarse moduli space for the functor F ss
L¯
(resp. F s
L¯
).
Theorem 1. Let α ∈ (0, 1). Then there is a coarse moduli space ML¯ (resp. MsL¯) for
the functor F ss
L¯
(resp. F s
L¯
). The moduli space ML¯ is a projective (irreducible) variety,
containing Ms
L¯
as an open subvariety.
Let M ′ (resp. M ′
L¯
) be the open subscheme of M (resp. ML¯) consisting of GPBs
(E, Fj (E)) such that the projections pj , p′j are isomorphisms for all j . Then M ′L¯ corre-
sponds to GPBs in M with fixed determinant L¯ with Fj (L) = (1 : λj ), λj ∈ k∗ and ML¯
is the closure of M ′
L¯
.
3. Application to nodal curves
3.1.
Let Y be an irreducible projective nodal curve with nodes yj , j = 1, . . . , m and X its
desingularization with Dj , the inverse image of yj . Then there is a correspondence from
GPBs on X of rank r , degree d to torsion-free sheaves on Y of the same rank and degree
[1, 2]. The correspondence induces a surjective morphism f from M onto U , where U is
the moduli space of torsion-free sheaves of rank r , degree d on Y . The restriction of the
morphism f to M ′ is an isomorphism onto the open subvariety U ′ of U corresponding to
vector bundles on Y . One has f−1U ′ = M ′.
For r = 1, a GPB L¯ corresponds to a torsion-free sheaf L on Y . Then L is a line bundle
if and only if Fj (L) = (1, λj ), λj ∈ k∗, ∀j . Suppose that L is a line bundle and let
U ′L be the closed subset of U
′ corresponding to vector bundles with fixed determinant L.
Then f−1U ′L = M ′L¯ and the morphism f maps M ′L¯ isomorphically onto U ′L. Hence, if L
is a line bundle, then f (ML¯) contains U ′L as an open subset. Since ML¯ is an irreducible,
closed subscheme of M , the image f (ML¯) is a closed, irreducible subscheme of U and
U ′L, being open, is dense in it. It follows that f (ML¯) is the closure of U
′
L.
Remark 3.1. The projective scheme ML¯ can be regarded as the compactified moduli space
of vector bundles of rank r with determinant L on the nodal curve Y .
Remark 3.2. In fact, for any torsion-free sheaf L of rank 1, the image f (ML¯) is a closed,
irreducible subscheme of U containing the subset of U consisting of torsion-free sheaves
with fixed determinant L as an open dense set.
3.2 Relation to Seshadri–Nagaraj conjecture
For a torsion-free sheaf F of rank r on Y , let N := (rF )/(torsion), where (torsion)
denotes the maximum subsheaf with proper support. Denote by Ij the ideal sheaf of the
Vector bundles with a fixed determinant 449
node yj . Define UL as the set
UL = {F ∈ U : I rjL ⊂ N ⊂ L, ∀j}.
Seshadri and Nagaraj had defined this set for Y with one node and conjectured that if L
is a line bundle, then UL is the closure of U ′L (Conjecture (a), page 136 of [3]). We prove
this conjecture.
Let R˜1−ss0 denote the subset consisting of 1-semistable points, then R˜
ss
0 ⊂ R˜1−ss0 . Let P
be the moduli space of 1-semistable GPBs [5]. One has morphisms f : R˜ss0 → U inducing
f : ML¯ → U and f1: R˜1−ss0 → U inducing f1: P → U .
PROPOSITION 3.3
Let L¯ be any GPB of rank 1 on X and L the corresponding torsion-free sheaf of rank 1 on
Y .
(1) If (E, Fj (E)) ∈ R˜1−ss0 , then F = f1((E, Fj (E))) ∈ UL and hence f (ML¯) ⊂ UL.
(2) The morphism R˜1−ss0 → U surjects onto UL.
(3) f (ML¯) = UL for α sufficiently close to 1.
Proof. We may assume that Y has only one node y. It is easy to see (from the proof) that the
general case follows exactly on same lines. Consider a GPB (E, F (E)). Let pi : F(E) →
Exi , i = 1, 2 be the projections and ai = dim ker pi . Let E0 = p∗(F )/(torsion), then
E0 ⊂ E. Since F |Y−y = p∗E |Y−y , one has N |Y−y = (p∗L)|Y−y = L |Y−y . Hence to
check that I rL ⊂ N ⊂ L, we have only to check it locally at the node y.
Let (A,m) be the local ring at y. Its normalization A¯ is a semilocal ring with two
maximal ideals m1,m2. The inclusion
Fy ⊂ (p∗E)y
may be identified with the inclusion
(r − a1 − a2)A ⊕ a1m1 ⊕ a2m2 ⊂ rA¯
(Proposition 4.2 of [2]).
(1) We consider the following cases separately.
Case (i). Suppose that p1, p2 are both isomorphisms. Then det(E, F (E)) = L¯ cor-
responds to a torsion-free sheaf L which is locally free at y. In this case, F is locally
free at y with (det F)y = Ly so that N = L ⊃ I rL.
Case (ii). Assume that p1 is an isomorphism, p2 is not an isomorphism (the opposite
case can be dealt with similarly). Then det(E, F (E)) = (L,Lx1) = L¯ corresponds
to L which is not locally free at y. One always has N ⊂ p∗L and Ny ⊂ Ly if and
only if Ny ⊗ k(y) ⊂ F(L). Locally, a1 = 0 so that Fy = (r − a2)A ⊕ a2m2. Then
Ny = (m2)a2 so that Ny ⊗ k(y) ⊂ Lx1 = F(L). Hence N ⊂ L. Since mr ⊂ ma22 , it
follows that I rL ⊂ N ⊂ L.
Case (iii). If both p1 and p2 are not isomorphisms, one has a1 ≥ 1, a2 ≥ 1. Then
locally, Ny = ma11 ma22 ⊂ m1m2 = m. It follows that Ny maps to zero in p∗L ⊗ k(y)
so that N ⊂ IL ⊂ L. Since mr ⊂ ma11 ma22 , one has I rL ⊂ N ⊂ L.
450 Usha N Bhosle
Note that any (E, F (E)) ∈ ML¯ with F(L) = (1 : λ), λ ∈ k∗ (i.e. L locally free
at y) occurs only in cases (i) or (iii). For (E, F (E)) ∈ ML¯ with F(L) = (1 : 0)
(or F(L) = (0 : 1)) only cases (ii) and (iii) occur. Part (1) now follows. Note that
(E, Fj (E)) is 1-semistable if and only if F is semistable [1, 2].
(2) Let F ∈ UL. Since I rL ⊂ N ⊂ L it follows that L|X−D = p∗N |X−D where
D = ∑j (xj + x′j ). Since det E0 = p∗N outside D, one has L = det E0 outside D.
It follows that L = det E0 ⊗ OX(
∑
j (aj xj + a′j x′j )), aj + a′j ≤ r . Let E be given
by an extension
0 → E0 → E → ⊕j (k(xj )aj ⊕ k(x′j )a
′
j ) → 0.
The composite F ↪→ p∗E0 ↪→ p∗E induces a linear map F ⊗ k(yj ) → p∗(E) ⊗
k(yj ). Let Fj (E) be the image of this linear map. Then (E, Fj (E)) maps to F and it is
1-semistable asF is semistable [2]. By construction, det E = det E0⊗OX(
∑
j (aj xj+
a′j x
′
j )) = L. It follows that (E, Fj (E)) ∈ R˜1−ss0 .
(3) Let PL¯ denote the closure of R˜1−ss0 /PGL(n) in P . For α close to 1, there is a surjective
birational morphism φ: M → P with f = f1 ◦ φ. It maps ML¯ birationally into PL¯.
Since both these spaces are irreducible and of the same dimension, it follows that
φ(ML¯) = PL¯. Since f (R˜1−ss0 ) surjects onto UL, it follows that f1(PL¯) ⊃ UL and
hence f (ML¯) ⊃ UL. From (1), it follows that UL = f (ML¯) = f1(PL¯). 
COROLLARY 3.4
If L is a line bundle, then UL is the closure of U ′L.
Proof. From Proposition 3.3, one has (as sets) f (ML¯) = UL. Since f (ML¯) is the closure
of U ′L if L is a line bundle, the result follows. 
Remark 3.5. The proof of Proposition 3.3(1) easily generalizes to (E, Fj (E)) replaced by
a family (E, Fj (E)) → T × X.
Remark 3.6. Sun [6] had proved the conjecture (a) of Seshadri–Nagaraj by considering a
smooth curve X degenerating to an irreducible nodal curve Y . However he does not get a
moduli functor or a scheme structure on UL except in some cases (e.g. one node, rank 2,
degree 1).
Remark 3.7. Schmitt [4] has constructed a moduli space M of α-semistable descend-
ing singular SL(r)-bundles (A, q, τ ) where (A, q) is a GPB on X and τ : Sym∗(A ⊗
Cr )SL(r,C) → OX a nontrivial homomorphism. It is shown that det A = OX and for
α ∈ (0, 1) ∩ Q, there is a forgetful morphism h: M → M (§5.1 of [4]). For α close
to 1, one has a forgetful morphism M → U = U(r, 0) whose set theoretic image is
UL (Proposition 5.1.1 of [4]). Then, since det A = OX, it follows that h(M) = ML¯ (as
sets).
Acknowledgements
I would like to thank I Biswas and A Schmitt for useful comments and careful reading of
the previous version.
Vector bundles with a fixed determinant 451
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[2] Bhosle Usha N, Generalised parabolic bundles and applications II. Proc. Indian Acad. Sci.
(Math. Sci.) 106(4) (1996) 403–420
[3] Nagaraj D S and Seshadri C S, Degenerations of the moduli spaces of vector bundles on
curves I. Proc. Indian Acad. Sci. (Math. Sci.) 107(2) (1997) 101–137
[4] Schmitt A, Singular principal G-bundles on nodal curves. J. Eur. Math. Soc. 7 (2005)
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Vector bundles with a fixed determinant on an irreducible nodal curve

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Vector bundles with a fixed determinant on an irreducible nodal curve

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