# Moduli spaces of vector bundles on reducible curves.

Consider a stable curve C without rational components. Write C = [union][C.sub.i] where the [C.sub.i] for i in I are the irreducible components of C. We shall say that a sheaf is torsion-free if every torsion element vanishes identically on some component.Choose rational numbers [([[Lambda].sub.i]).sub.i[element of]I] (called weights) with 0 [less than] [[Lambda].sub.i] [less than] 1 and [summation of][[Lambda].sub.i]i = 1. Seshadri defined the notion of [[Lambda].sub.i]-semistable torsion free sheaf on C in the following way. If E is a sheaf on C and [r.sub.i] is the rank of E restricted to [C.sub.i], define the slope of E as [Mu](E) = [Chi](E)/[summation of][[Lambda].sub.i][r.sub.i]. Then the sheaf is said to be semistable when for every subsheaf F of E [Mu](F) [less than or equal to] [Mu](E) and stable when the inequality is strict for every proper subsheaf F.

The notion of equivalence classes of semistable sheaves is defined in the same way as in the case of an irreducible curve: given a semistable torsion-free sheaf, one can find a chain of subsheaves 0 = [E.sub.0] [subset] ... [subset] [E.sub.k] = E such that [E.sub.i]/[E.sub.i-1] is stable. This chain is not unique but the sheaf [symmetry][E.sub.i]/[E.sub.i-1] is well defined up to the order of its summands and is called the graduate associated to E. Then, two semistable vector bundles are said to be equivalent if they have the same graduate.

Seshadri (cf. [S] Ch VII) proved the existence of a moduli space U(C, n, d, [[Lambda].sub.i]) parametrising [[Lambda].sub.i]-stable sheaves on C of given ranks [r.sub.i] and given Euler-Poincare characteristic. Its natural compactification is the set of equivalence classes of semistable vector bundles on C.

The purpose of this work is to give a description (cf. Th. 3.1 and 3.2) of the components of this moduli space in the case when all the components of the normalisation of C have genus at least one and the rank on each component is the same. This description is similar to the one given by Oda and Seshadri for the case of rank one although our approach is the opposite of theirs. We are not trying to prove the existence of a moduli space but only finding out what it looks like. In fact we associate to every torsion free sheaf a polyhedra which depends only on the degree of the restriction of the sheaf to every component and the rank at every node. If the sheaf is stable, the associated polyhedra is stable too. The main difference with the rank one case is that stability of the polyhedra does not automatically imply stability of the sheaf. We prove however that if the associated polyhedra is stable, the sheaf is a vector bundle and the restriction to each component is semistable, then the vector bundle in the whole curve is stable too. Hence each choice of degrees on the [C.sub.i] allowed by the stability of the polyhedra gives rise to a component of the moduli space provided a semistable sheaf of the prefixed degree on each component exists. This is the reason why we must exclude rational components. A rational curve has semistable bundles only for those degrees which are a multiple of the rank. In fact, it is easy to show that some degrees which are allowable for components of genus at least one are not allowable for components of genus zero.

Assume that C is the special fiber of a family C [approaches] B of curves. Given a vector bundle E[prime] on C - C, assume there is an extension E of the E[prime] to C. The restriction of E to C is not completely determined by E[prime]. In fact, we can modify any E with a line bundle which has support on components of C and this does not modify [Mathematical Expression Omitted]. We show (cf. Th. 3.3) that the components of the moduli space correspond to choices of degrees on components of C modulo those that can be obtained with modifications as above. This interpretation was suggested by Joe Harris.

The next question that presents itself is how different components of U(C(n, d, [[Lambda].sub.i])) fit together. The points of intersection of different components correspond to torsion-free sheaves which are not locally free at some of the nodes in the following way: assume that [P.sub.j] is in the intersection of [C.sub.i(0)] and [C.sub.i(1)]. We want to consider a sheaf which has rank n - 1 at [P.sub.j]. Assume it has degree [d.sub.i] on each [C.sub.i]. Then it can be considered as the limit of vector bundles of degree [d[prime].sub.k] = [d.sub.k] on each [C.sub.k] k [not equal to] i(0), i(1) and [d[prime].sub.i(0)] = [d.sub.i(0)] and [d[prime].sub.i(1)] = [d[prime].sub.i(1)] + 1 or [d[prime].sub.i(0)] = [d.sub.i(0)] + 1 and [d[prime].sub.i(1)] = [d.sub.i(1)]. Hence, these sheaves are in the intersection of the two components corresponding to the above choice of degrees. We can easily generalize this construction to the case of dropping rank simultaneously at several nodes and by any quantity [Alpha], 1 [less than or equal to] [Alpha] [less than or equal to] n. We must warn though that, unlike the vector bundle case, stability of the associated polyhedra does not guarantee the existence of a stable torsion free sheaf with the preassigned degrees and ranks on the components (see Prop. 4.5 and Remark 4.6).

This work generalises our [T] which deals with the tree-like case only.

1. Basic conditions for the stability of a vector bundle. As in the introduction, we write C = [[union].sub.i[element of]I][C.sub.i] and we denote by [{[P.sub.j]}.sub.j[element of]J] set of nodes of C.

Let A be the following subset of the set of parts of J. For [Mathematical Expression Omitted], [Mathematical Expression Omitted] if and only if the curve [Mathematical Expression Omitted] that we define next is connected. The curve [Mathematical Expression Omitted] consists of all components which contain at least one of the nodes of [Mathematical Expression Omitted] glued only at the nodes of [Mathematical Expression Omitted].

We choose a complete ordering of the set A, A = {J(1) = J,...,J(M)} so that J(i) has at least as many elements as J(i + 1).

Given a vector bundle E on C, denote by [E.sub.i] the restriction of E to [C.sub.i] modulo torsion. For each J(t) [element of] A, consider the subsheaf [E.sub.t] of E consisting of those sections of E which vanish identically on the complement of [C.sub.J(t)]. The condition that [E.sub.t] does not contradict semi(stability) may be written as

(1) [[summation over i] [Chi]([E.sub.i]) - [summation over j] [r[prime].sub.j]]/[summation of][[Lambda].sub.i]([less than or equal to]) [less than] [Chi](E)

Here the index i ranges over the components of [C.sub.J(t)] and the index j over the nodes on [C.sub.J(t)] and [r[prime].sub.j] is the dimension of the locally free part in the stalk of [P.sub.j].

PROPOSITION 1.2. Assume that E is a vector bundle. If condition (1.1) is satisfied for every J(t) in A and all [E.sub.i] are semistable, then E is semistable. If the inequailitites are strict, all the [E.sub.i] are semistable and at least one of them is stable, then E is stable.

Proof. Let F be a subsheaf of E. Denote by [r[prime].sub.j] the dimension of the fiber of F at the point [P.sub.j]. For every node [P.sub.j] and every subset J(t) in A, define numbers [a.sub.i] and [r[prime].sub.j,t] by induction on t in the following way.

[r[prime].sub.j,0] = [r[prime].sub.j]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Moreover, for each component i, we define [r.[double prime].sub.i] = [r.sub.i] - [summation of][C.sub.J.(t)][implication][[Ci.sup.a].sub.t]. From these definitions, the [r[prime].sub.j,t] and [a.sub.t] are positive. We want to see that the numbers [r[double prime].sub.i] are positive too. This will follow from the following Lemma.

LEMMA 1.3. For each t, define j(t) by j(t) [element of] [J.sub.t] and [r[prime].sub.j(t)] [less than or equal to] [r[prime].sub.j] for any other j in [J.sub.t]. Then [a.sub.t] = [r[prime].sub.j(t),t-1].

Proof. By induction on t. For t = 1, this follows from the definitions. Assume the result true up to t - 1 and let us show that it holds for t. Define B = {j [where] j [union] J(t) [element of] A, j [not an element of] J(t)}.

Case a) Assume first that there exists a [j.sub.1] [element of] B such that [r[prime].sub.[j.sub.1]] [greater than or equal to] [r[prime].sub.j(t)]. As J(t)[union]{[j.sub.1]} [element of] A and has more elements than J(t), J(t)[union]{[j.sub.1]} = J(t[prime]) with t[prime] [less than] t. Then j(t[prime]) = j(t). By induction hypothesis, [a.sub.t[prime]] = [r[prime].sub.j(t),t[prime]-1]. Hence [Mathematical Expression Omitted] J where D = {k [where] k [less than or equal to] t - 1, j(t) [element of] [J.sub.k]}. Now this expression can be written as [Mathematical Expression Omitted] where D[prime] = {k [where] t[prime] + 1 [less than or equal to] k [less than or equal to] t - 1,j(t) [element of] [J.sub.k]} and this equals zero. Hence [a.sub.t] = [min.sub.j[element of]J(t)] {[r[prime].sub.j]} = min (0, [min.sub.j[element of]J(t)-{j(t)}][r[prime].sub.j]) = 0 = [r[prime].sub.j(t),t-1].

Case b) Assume now that for every j [element of] B [r[prime].sub.j] [less than] [r[prime].sub.j(t)]. Let [j.sub.1] [element of] [J.sub.t], [j.sub.1] [not equal to] j(t). Then, [Mathematical Expression Omitted] where D = {k [where] k [less than or equal to] t - 1, [j.sub.1] [element of] J(k)}. As k [less than or equal to] t - 1, J(k) has at least as many elements as J(t). If j(t) [not equal to] J(k), there exists a [j.sup.2] [element of] B such that [j.sup.2] [element of] J(k). By assumption, [r[prime].sub.[j.sup.2]] [less than] [r[prime].sub.j(t)]. We can choose a few indices j(t), ..., [j.sub.[Alpha]] such that [J.sub.k] [union] {j(t),...,[j.sub.[Alpha]]} = [J.sub.k[prime]] and k[prime] [less than] k. Then by induction hypothesis, [a.sub.k[prime]] = [r[prime].sub.[j.sub.i],k[prime]-1] where [j.sub.i] [not equal to] j(t), ..., [j.sub.[Alpha]] Hence as in a), we obtain [a.sub.k] = 0. Therefore [Mathematical Expression Omitted]. As [r[prime].sub.j] [less than or equal to] [r[prime].sub.[j.sub.1]], it follows that [r[prime].sub.j(t),t-1] [less than or equal to] [r[prime].sub.[j.sub.1],t-1]. Therefore, [a.sub.t] = [r[prime].sub.j,t-1] as we had to prove.

COROLLARY 1.4. Let j [element of] J(t). If [r[prime].sub.j] minimum of the [r[prime].sub.j] for j in J(k)for some k [less than or equal to] t, then [a.sub.t] = 0.

Let now [C.sub.i] be a connected component and [P.sub.i,1], ..., [P.sub.i[[Alpha].sub.i]] be the nodes of [C.sub.i]. Assume [r[prime].sub.1] [less than or equal to] ... [less than or equal to] [r[prime].sub.[[Alpha].sub.i]]. We claim then that [Mathematical Expression Omitted].

We know that [Mathematical Expression Omitted]. We want to see that if [[Alpha].sub.i] [not element of] J(t), i [element of] J(t) for some i, then [a.sub.t] = 0.

Proof of the claim. As J (t) [union] {[[Alpha].sub.i]} = J (t[prime]), t[prime] [less than] t and [r[prime].sub.j] [less than] [r[prime].sub.[[Alpha].sub.i]] for some j [element of] J(t), the Corollary implies [a.sub.t] = 0.

We now turn to the proof of the (semi)stability of E.

Note first that from the exact sequence [Mathematical Expression Omitted] we find [Chi]([F.sub.i]) [less than or equal to] [Chi]([F.sub.i](- [P.sub.j])) + [r[prime].sub.j] [less than or equal to] ... [less than or equal to] [Chi]([F.sub.i](- [summation of] [P.sub.j] where j=1 to [[Alpha].sub.i])) + [summation of] [r[prime].sub.j] where j=1 to [[Alpha].sub.i] [less than or equal to] ([r.sub.i]/n)([Chi]([E.sub.i]) - n[Alpha]) + [summation of] [r[prime].sub.j] where j=1 to [Alpha]. Now

[Mathematical Expression Omitted]

Now this expression is smaller than or equal to [Chi](E)/n because of (1.1). If the inequalities in (1.1) are strict, this last expression is strictly smaller than [Chi](E)/n unless all [r[double prime].sub.i] are zero and all [a.sub.t] except [a.sub.1] are zero. This implies that all [r[prime].sub.i] are equal. In this case the inequality is still strict if one of the [E.sub.i] is stable and [F.sub.i] [not equal to] [E.sub.i]. Now, if [F.sub.i] = [E.sub.i], then the rank of [F.sub.i] is n. As the rank of all [F.sub.j] is the same, F = E.

2. Polyhedral associated to a torsion-free sheaf. Let C be as above. Choose an orientation for the graph of C. We recall from [O,S] the following construction: we associate to C two real vector spaces V and W with basis [{[v.sub.i]}.sub.i[element of]I] and [{[w.sub.j]}.sub.j[element of]J] respectively. Inner products [,] and <,> are given by [[v.sub.i], [v[prime].sub.i]] = [[Delta].sub.i,i[prime]]<[w.sub.j], [w.sub.j[prime]]> = [[Delta].sub.j,j[prime]]. There are also natural maps [Mathematical Expression Omitted] defined by [Delta]([w.sub.j]) = 0 if [P.sub.j] is a loop for the graph, [Delta]([w.sub.j]) = [v.sub.i] - [v.sub.i[prime]], if [P.sub.j] has origin at [C.sub.i] and end point at [C.sub.i[prime]]. Define next [Mathematical Expression Omitted] by [Delta]([v.sub.i]) = [summation of] [[v.sub.i], [Delta]([w.sub.j])][w.sub.j]. If J[prime] [subset] J, [[Delta].sub.J[prime]] denotes the map [Delta] associated to the curve C normalized at J - J[prime].

In all what follows we fix a rank n (the same for all components), an Euler-Poincare characteristic [Chi] = d + n(1 - g) and weights [[Lambda].sub.i]. Let E be a torsion-free sheaf on C of rank n on each component and Euler-Poincare characteristic [Chi]. Denote by [n.sub.E]([P.sub.j]) the dimension of [E.sub.[P.sub.j]]/[M.sub.[P.sub.j]][E.sub.[P.sub.J]].

DEFINITION 2.1. Let E be a torsion-free sheaf on C of rank n on each component and Euler-Poincare characteristic [Chi]. Denote by [n.sub.E]([P.sub.j]) the dimension of [E.sub.[P.sub.j]]/[M.sub.[P.sub.j]][E.sub.[P.sub.j]]. Define [J.sub.[Alpha]] = {j [element of] J [where] [n.sub.E]([P.sub.j]) = [Alpha]}. Then J is the disjoint union of the [J.sub.[Alpha]] for [Alpha] = 1,...,n. Denote by deg (E) the element of V = [summation of][a.sub.i][v.sub.i] where [a.sub.i] is the degree on the component [C.sub.i] of the pullback modulo torsion [E.sub.i](E) of E to the normalisation of [C.sub.i].

As in [O,S], denote by e([w.sub.j]) = origin of [w.sub.j]-end point of [w.sub.j], d([w.sub.j]) = origin of [w.sub.j]+end point of [w.sub.j] and define e and d on any subset of J by linearity. For a subset [Mathematical Expression Omitted].

DEFINITION 2.2. Given a partition of J as the disjoint union of [J.sub.0],...,[J.sub.n], define the following convex polyhedra

[Mathematical Expression Omitted]

DEFINITION 2.3. We assoctate to E the polyhedral D(E) = b(D) + [Delta]V[J.sub.0],...,[J.sub.n] where b(D) = deg E + [summation over [Alpha]] (n - [Alpha])d([J.sub.[Alpha]])/2. We shall call b(D) the baricenter of the polyhedra D.

DEFINITION 2.4. Define a vector [Phi] with rational coefficients by the formula [[Lambda].sub.i] = (1/[Chi]){[[v.sub.i], - nd(J)/2 + [Phi]] + n[Chi]([[Sigma].sub.[C.sub.i]])}.

PROPOSITION 2.5. Let G be the subsheaf of sections of E which vanish identically on the curve [[union].sub.i[element of]I-I[prime]][C.sub.i]. Then G does not contradict (semi)stability of E if and only if [b(D) - [Phi], v(I[prime])] [less than or equal to] [summation over [Alpha]] [Alpha]([[Delta].sub.[J.sub.[Alpha]]](v(I[prime])), [[Delta].sub.[J.sub.[Alpha]]](v(I[prime])))/2 where b(D) denotes the baricenter of the polyhedral D.

Proof. We need to prove that the condition [Chi](G) [less than or equal to] [Chi](E)([summation of] [[Lambda].sub.i] where i[element of]I[prime]) is equivalent to the inequality in the statement of the proposition.

In order to compute [Chi](G), we use the exact sequence

[Mathematical Expression Omitted]

where B is the set of nodes which are intersection of two components in I - I[prime]. Hence [Chi](G) = [Chi](E) - [summation of] [Chi].sub.i](F) where i[element of]I-I[prime] + [summation of][n.sub.E]([P.sub.j]) where j[element of]B. Note now that the number of points which are intersection of two components, both lying in a certain subset K of I may be computed as [v(K),d(K)/2] - ([Delta](v(K)), [Delta](v(K)))/2 (cf. [O,S] Corollary 4.4).

Hence

(1) [Chi](G) = [v(I), [Chi]([E.sub.i](E) - [summation over [Alpha]] [Alpha]d([J.sub.[Alpha]])/2]

+ [summation over [Alpha]] ([Alpha]/2)([[Delta][J.sub.[Alpha]]](v(I)), [[Delta].sub.[J.sub.[Alpha]]](v(I)))

- [v(I - I[prime]), [Chi]([E.sub.i](E)) - [summation over [Alpha]] [Alpha]d([J.sub.[Alpha]])/2]

- [summation of]([Alpha]/2)([[Delta].sub.[J.sub.[Alpha]]](v(I - I[prime])), [[Delta].sub.[J.sub.[Alpha]]](v(I - I[prime])))

On the other hand, from the definition of D,

(2) [v(I[prime]), b(D) - [Phi]] = [v(I[prime]), deg ([E.sub.i](E)) + [summation of] (n - [Alpha])d([J.sub.[Alpha]])/2 - [Lambda][Chi](E)

- n(d(J)/2) + n[Chi]([[Sigma].sub.[X.sub.i]])[v.sub.i]]

= [v(I[prime]), [[Chi].sub.i](E) - [summation over [Alpha]] [Alpha]d([J.sub.[Alpha]])/2 - [Lambda][Chi](E)]

Then the result follows from (1) and (2) using the fact that ([[Delta].sub.[J.sub.[Alpha]]](v(I - I[prime])), [[Delta].sub.[J.sub.[Alpha]]](v(I - I[prime]))) = ([[Delta].sub.[J.sub.[Alpha]]](v)(I[prime]), [[Delta].sub.[J.sub.[Alpha]]](v(I[prime]))).

PROPOSITION 2.6. The set [Delta]([V.sub.[J.sub.0]],...,[J.sub.n]) consists of those x in [Delta]([C.sub.1]([Gamma], R)) satisfying [x, v(I[prime])] [less than or equal to] [summation over [Alpha]] [Alpha]([[Delta].sub.[J.sub.[Alpha]]] (v(I[prime])), [[Delta][J.sub.[Alpha]]](v(I[prime])))/2 for every I[prime] contained in I.

Proof. This proof follows closely the proof of (5.2) in [O,S]. One of the inclusions is straightforward: assume that x [element of] [Delta][V.sub.[J.sub.0]],...,[J.sub.n]]. Then x = [Delta](y) where [Mathematical Expression Omitted] and [Mathematical Expression Omitted].

Hence [Mathematical Expression Omitted].

We now need to see the inverse inclusion. As [V.sub.[J.sub.0]],...,[J.sub.n]. is a convex poly-hedra, its projection on [Delta]([C.sub.0]([Gamma], R)) is convex too. It is enough to identify the hyperplanes limiting the faces in [Delta]([C.sub.0]([Gamma], R)) and write them as {x [element of] [Delta]([C.sub.0]([Gamma], R)) [where] (x, v(I[prime])) = [summation over [Alpha]] ([Alpha]/2)([[Delta].sub.[J.sub.[Alpha]]](v(I[prime])), [[Delta].sub.[J.sub.[Alpha]]] (v(I[prime])))} for some I[prime] [subset] I.

The vertices of [V.sub.[J.sub.0]],...,[J.sub.n] (0) are of the form [summation over [Alpha]] [Alpha](e([J.sub.[Alpha]]) - e([J.sub.[Alpha]] - [J[prime].sub.[Alpha]))/2 for arbitrary choices of [J[prime].sub.[Alpha]] [subset] [J.sub.[Alpha]].

Let a in [Delta]([C.sub.1]([Gamma], R)) be the equation of a hyperplane face of [Delta]([V.sub.[J.sub.0],...,[J.sub.n]]). We mean by this that the face is given by {x [element of] [Delta]([C.sub.0]([Gamma], R)) [where] (a,x) = c}. Then (a, x) = c for some of the vertices that determine the face while either (a, x) [less than] c for all other vertices or (a, x) [greater than] c for all other vertices. Up to a choice for the signs of a and c, we may assume that the inequality [less than] occurs. Denote by [x.sub.1],...,[x.sub.m] the set of vertices that satisfy the equation (a, x) = c. Denote by [J[prime].sub.i,[Alpha]] the subsets of [J.sub.[Alpha]] such that [x.sub.i] = [summation of] [Alpha](e([J.sub.i,[Alpha]]) - e([J.sub.[Alpha]] - [J[prime].sub.i[Alpha]]))/2.

Then (1/2) [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i[Alpha]]) - e([J.sub.[Alpha]] - [J[prime].sub.i,[Alpha]])) = c and for any choice of subsets [J[prime].sub.[Alpha]] of [J.sub.[Alpha]] [summation over [Alpha]] [Alpha](a, e([J[prime].sub.[Alpha]])) [less than or equal to] [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i,[Alpha]])).

Write now

J = {([J[prime].sub.i,1],...,[J[prime].sub.i,n]) i = 1,...,m}

as a subset of the product of the sets of parts of [J.sub.1], ..., [J.sub.n]. We want to see that J is closed by the operations of sum and intersection taken componentwise. In fact the following equation is satisfied

[summation over [Alpha]] [Alpha](a, e([J[prime].sub.i,[Alpha]]) + [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i[prime],[Alpha]]) = [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i,[Alpha]] [intersection] [J[prime].sub.i[prime],[Alpha]]) + [summation over [Alpha]] [Alpha](a, e(J[prime].sub.i,[Alpha] [union] [J[prime].sub.i[prime],[Alpha]])

Now, [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i,[Alpha]]) = c, [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i[prime],[Alpha]]) = c and [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i,[Alpha]] [intersection] [J[prime].sub.i[prime],[Alpha]) [less than or equal to] c, [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i,[Alpha]] [union] [J[prime].sub.i[prime],[Alpha]]) [less than or equal to] c. Hence [summation over [Alpha]] [Alpha](a, e([J[prime].sub.i,[Alpha]] [intersection] [J[prime].sub.i[prime],[Alpha]]) = c and [summation over [Alpha]] [Alpha](a, e[J[prime].sub.i,[Alpha]] [union] [J[prime].sub.i[prime],[Alpha]]) = c.

Let us now write [Mathematical Expression Omitted]

LEMMA 2.7. The following inequalities are satisfied for [Beta] [greater than] 0, [w.sub.j] [element of] [J.sub.[Beta]]: (a, [w.sub.j]) [greater than] 0 iff j [element of] [J.sub.[Beta]+], (a, [w.sub.j]) = 0 iff j [element of] [J.sub.[Beta],0], (a, [w.sub.j]) [less than] 0 iff j [element of] [J.sub.[Beta]-].

Proof. Assume that [Mathematical Expression Omitted]. Choose a k with 1 [less than or equal to] k [less than or equal to] m. Then ([J.sub.k,0],...,[J.sub.k,[Beta]] - {[e.sub.j]},...,[J.sub.k,n]) is not in J. Hence [summation over [Alpha]] [Alpha](a, e([J.sub.k,[Alpha]])) - [Beta](a, [e.sub.j]) [less than] c while [summation over [Alpha]] [Alpha](a, e([J.sub.k,[Alpha]])) = c. Because of the assumption [Beta] [not equal to] 0, this implies (a, [w.sub.j]) [greater than] 0. On the other hand, if [Mathematical Expression Omitted], then [Mathematical Expression Omitted]. Therefore (a, [w.sub.j]) [less than] 0 and the statement for [J.sub.[Alpha]+] is proved.

Assume now [Mathematical Expression Omitted]. Then [Mathematical Expression Omitted] while [Mathematical Expression Omitted]. It follows that (a, [w.sub.j]) = 0.

If [w.sub.j] [element of] [J[prime].sub.i,[Beta]], say [w.sub.j] [element of] [J[prime].sub.[i.sub.0],[Beta]], then c = [summation over [Alpha]] [Alpha](a, e([J.sub.[i.sub.0],[Alpha]])) + [Beta](a, e([J.sub.[i.sub.0]] - {[w.sub.j]})) + [Beta](a,[w.sub.j]) [less than or equal to] c + [Beta](a, [w.sub.j]). Hence (a, [w.sub.j]) [is less than or equal to] 0. This proves the statement for [J.sub.[Beta],0]. The statement for [J.sub.[Beta-]], follows from the other two.

Let us now write a = [summation of][[Lambda].sub.[Alpha]][w.sub.[Alpha]] where [w.sub.[Alpha]] has support on the nodes [J.sub.[Alpha]]. Note that a is determined by the condition (a, [x.sub.i]) = c i = 1,...,n. Then from (2.7) and up to changing the value of c, we may assume that a = [summation of][[Lambda].sub.[Alpha]][w.sub.[Alpha]] where [w.sub.[Alpha]] is a cycle, namely all coefficients of the [e.sub.j] are [+ or -]1.

Then, ([Mathematical Expression Omitted]). The maximum is [Alpha][[Lambda].sub.[Alpha]]([w.sub.[Alpha]], [w.sub.[Alpha]])/2 = [summation of] [Alpha]([[Delta].sub.[J.sub.[Alpha]]] w, [[Delta].sub.[J.sub.[Alpha]]] w)/2 where w = [summation of][[Lambda].sub.[Alpha]][w.sub.[Alpha]] and this maximum is attained when [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. This is equivalent to [Mathematical Expression Omitted]. Therefore, the constant c is [summation of] [Alpha]([[Delta].sub.[J.sub.[Alpha]]] w, [[Delta].sub.[J.sub.[Alpha]]] w)/2. Hence the hyperplane that gives the hce is (x, w) = [summation of] [Alpha]([[Delta].sub.[J.sub.[Alpha]]] w, [[Delta].sub.[J.sub.[Alpha]]] w)/2. Hence the hyperplane that gives the face is (x, w) = [summation of] [Alpha]([Delta].sub.[J.sub.[Alpha]] w, [Delta].sub.[J.sub.[Alpha]] w)/2. As we have an inclusion [Delta]([V.sub.[J.sub.0],...,[J.sub.n]]) [subset] {x [element of] [Delta]([C.sub.0]([Gamma], R)) [where] (x, w) [is less than or equal to] [summation over [Alpha]] ([Alpha]/2) ([[Delta].sub.[J.sub.[Alpha]]](w), [[Delta].sub.[J.sub.[Alpha]]](w)) for every w} and the faces of [Delta]([V.sub.[J.sub.0], ...,[J.sub.n]]) are the sets where the equality holds for certain w, the inclusion is in fact an equality.

PROPOSITION 2.8. The interior of [Delta]([V.sub.[J.sub.0], ...,[J.sub.n]]) in [Delta]W is empty if and only if the curve obtained by desingularizing C at the nodes in [J.sub.0] is not connected. The interior of [Delta]([V.sub.[J.sub.0], ...,[J.sub.n]]) if nonempty consists of those x such that [x, v(I[prime])] [is less than] [summation over [Alpha]] [Alpha]([[Delta].sub.[J.sub.[Alpha]]](v(I[prime])), [[Delta].sub.[J.sub.[Alpha]]] (v(I[prime])))/2 for every I[prime] contained in I.

Proof. Consider the curve C[prime] obtained by desingularizing C at the nodes of [J.sub.0]. Consider the polyhedra [V[prime].sub.[null set],[J.sub.1],...,[J.sub.n]] associated to the curve C[prime]. The vector space V[prime] associated to C[prime] (cf. beginning of [section]2) is identical to the one V associated to C. Also [Delta][prime]([V[prime].sub.[null set],[J.sub.1],...,[J.sub.n]]) = [Delta]([V.sub.[J.sub.0],...,[J.sub.n]]). Now V = [H.sup.0](C, R) [symmetry] [Delta]W (cf. [O,S] p. 19)

If C[prime] is not connected, [Delta]W[prime] has dimension smaller than [Delta]W. As [V[prime].sub.[null set],[J.sub.1],...,[J.sub.n]] is contained in [Delta]W[prime], its interior must be empty.

Conversely, assume that the interior of [Delta]([V[prime].sub.[J.sub.0],...,[J.sub.n]]) is empty. This polyhedral has dimension equal to the dimension of [Delta][prime]([V[prime].sub.[null set],[J.sub.1],...,[J.sub.n]]) and this in turn has dimension equal to that of [Delta]W[prime]. Hence C[prime] is not connected.

The proof of the second statement follows from (2.5).

DEFINITION 2.9. Define the n-dual of the polyhedra D = b + [Delta][V.sub.[J.sub.0],...,[J.sub.n]] as [D.sup.*] = b + [Delta][V.sub.[J.sub.n],...,[J.sub.0]]. Note that this definition makes sense, namely from (2.6) [D.sup.*] depends only on D and not on the particular choice of [J.sub.0],...,[J.sub.n] used in representing D.

DEFINITION 2.10. Given a polyhedra D, we shall say that D is n-[Phi]-semistable when [D.sup.*] [subset] [Phi] + [Delta][V.sub.[null set],...,[null set],J]; n-[Phi]-stable when [Delta]D [subset] [Phi] + Int([Delta]([V.sub.[null set],...[null set],J])).

PROPOSITION 2.11. The following two conditions are equivalent for a polyhedra D = b + [V.sub.[J.sub.0],...,[J.sub.n]](0)

i) [D.sup.*] [subset] [prime] + [Delta]([V.sub.[null set],...[null set],J])

ii) b(D) - [Phi] [subset] [Delta]([V.sub.[J.sub.0],...,[J.sub.N]]).

PROPOSITION 2.12. The following two conditions are equivalent for a polyhedra D = b + [Delta]([V.sub.[J.sub.0]],...,[J.sub.n](0))

i) D* [subset] [Phi] + Int([Delta]([V.sub.[Phi]],...,[Phi],J))

ii) b(D) - [Phi] [element of] Int([Delta]([V.sub.[J.sub.0]],...,[J.sub.n]).

We prove (2.12), the proof of (2.11) being similar.

Let us show first that condition 2 implies condition 1. Let y be an element in D*. Write [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. Because of condition 2, [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. Hence [Mathematical Expression Omitted]. Now, [Mathematical Expression Omitted]. Hence y [element of] [Phi] + [Delta]([V.sub.[Phi]],...,[Phi],J) as we had to prove.

Let us show now that condition 1 implies condition 2. Recall that (from (2.8))

Int([Delta]([V.sub.[J.sub.0]],...,[J.sub.n]) = {x [where] [x, v(I[prime])] [less than] [[Sigma].sub.[Alpha]] [Alpha]([[Delta].sub.[J.sub.[Alpha]]](v(I[prime])), [[Delta].sub.[J.sub.[Alpha]]](v(I[prime])))/2 for every I[prime] contained in I}. We can write [[Delta].sub.J](v(I)) = [[Sigma].sub.k[greater than or equal to][Alpha]] [[Delta].sub.[J.sub.k]](v(I)) + [[Sigma].sub.k[less than][Alpha]] [[Sigma].sub.j [element of] [J.sub.k]] [v(I), [Delta]([w.sub.j])][w.sub.j]. Hence ([[Delta].sub.J](v(I)), [[Delta].sub.J](v(I))) = [[Sigma].sub.k[greater than or equal to][Alpha]] ([[Delta].sub.[J.sub.k]](v(I)), [[Delta].sub.[J.sub.k]](v(I)) + [[Sigma].sub.k[less than][Alpha]] [[Sigma].sub.j [element of] [J.sub.k]] [[v(I), [Delta]([w.sub.j])].sup.2]. Adding these equalities for [Alpha] = 0,...,n and dividing by 2, we get (n/2)([[Delta].sub.J](v(I)), [[Delta].sub.J](v(I))) = [[Sigma].sub.[Alpha]] ([Alpha]/2)([[Delta].sub.[J.sub.[Alpha]](v(I)), [[Delta].sub.[J.sub.[Alpha]]](v(I))) + [[Sigma].sub.j [element of] [J.sub.[Alpha]]] ((n - [Alpha])/2)[[v(I), [Delta]([w.sub.j])].sup.2].

Now [b - [Phi], v(I[prime])] - [Sigma] ([Alpha]/2)([[Delta].sub.[J.sub.[Alpha]]](v(I[prime])), [Mathematical Expression Omitted]. Hence, from condition 1), the difference between the first two terms of this expression is strictly negative. If we choose the [Mathematical Expression Omitted], the difference between the third and fourth term in the expression is 0. This proves the result.

COROLLARY 2.13. The basic conditions in (1.1) are equivalent to the (semi)stability of the polyhedra D(E) associated to the sheaf E.

3. Number of components of the moduli space. From (2.13), we know that if a torsion-free sheaf is (semi)stable, then D(E) is [Phi]-semistable too. We want to prove a kind of converse here in the case of vector bundles.

THEOREM 3.1. Consider a choice of degrees [[Delta].sub.i] for each i [element of] I. Consider the polyhedra D = [Sigma] [[Delta].sub.i][v.sub.i] + [Delta]([V.sub.[Phi]],...,[Phi], J]). Assume this polyhedra is stable. Then, this choice of [[Delta].sub.i] gives rise to exactly one component of the moduli space U(C, n, d, [[Lambda].sub.i]).

Proof. Choose semistable vector bundles of degrees [[Delta].sub.i] on the corresponding components [C.sub.i] and glue them together at the nodes with arbitrary nondegenerate gluings. Then from (1.2) and (2.13) the vector bundle E that we obtain is semistable. If at least one of the [E.sub.i] is stable, then E is stable.

Note that a torsion-free sheaf is determined by giving a vector bundle on each component and a gluing of the quotient of the fibers at each node. Assume that, by taking two different sets of such data, we obtained isomorphic sheaves E and E[prime]. Then the restriction of the isomorphism to each component would give rise to an isomorphism between the restrictions [E.sub.i] and [E[prime].sub.i] to [C.sub.i]. If the [E.sub.i] are stable, any such isomorphism is multiplication with a scalar [[Lambda].sub.i]. Moreover the matrices of the gluings should be compatible with this isomorphism. Hence if [P.sub.j] is the intersection of [C.sub.[i.sub.1]] and [C.sub.[i.sub.2]] the matrix at [P.sub.j] corresponding to E should differ in [[Lambda].sub.[i.sub.1]]/[[Lambda].sub.[i.sub.2]] from the matrix corresponding to E[prime]. We deduce that the fibers of the map [Mathematical Expression Omitted] to U(C, n, d, [[Lambda].sub.i]) have dimension #I - 1. Hence the dimension of the image is [Mathematical Expression Omitted]. If [g.sub.i] is the genus of [C.sub.i], g the genus of C, then g = [Sigma] [g.sub.i] + #J - #I + 1. Hence, [n.sup.2](g - 1) + 1 = [Sigma] [[n.sup.2]([g.sub.i] - 1) + 1] + (#J)[n.sup.2] - #I + 1. Now, the left hand-side member is the dimension of U(C, n, d, [[Lambda].sub.i]). We deduce that a generic point of any component of U(C, n, d, [[Lambda].sub.i]) must come from a generic vector bundle on each [C.sub.i] and a generic isomorphism at each node.

We recall that for a curve of genus g [greater than] 1, a generic vector bundle is stable. For curves [C.sub.i] of genus 1, reasoning exactly as in p. 347 of [T], one deduces that the generic point corresponds to an E such that [E.sub.i] is direct sum of stable indecomposable bundles of equal rank and degree. The fact that each choice of degrees gives rise to exactly one component of the moduli space of bundles on C now follows.

THEOREM 3.2. The number of components of the moduli space of semistable sheaves of rank n on C is k[n.sup.(#I - 1)] where k is the number of generating trees for the graph of C or equivalently, the determinant of the pairing induced on [H.sub.1] (C) by the intersection form. A generic point of a given component of the moduli space is a vector bundle whose associated polyhedra is stable. If the component [C.sub.i] has genus at least 2, the restriction of the generic vector bundle to [C.sub.i] is stable. If [C.sub.i] has genus one, the restriction is a direct sum of indecomposable vector bundles of equal rank and degree.

Proof. Consider a vector bundle E and its associated polyhedra D(E) with baricenter b. The choice of degrees on the components [C.sub.i] corresponds to the choice of b. We only need to compute the possibilities for b for a stable D. Moreover, we can assume that D corresponds to a vector bundle and not just a torsion-free sheaf, as these give the generic points on each component. By definition of stability, this is equivalent to b(D) - [Phi] [element of] [Delta][V.sub.[Phi]],...,[Phi],J]. Therefore, we need to compute the number of points of intersection of [Delta][V.sub.[Phi],...,[Phi],J](0) with the net [H.sup.1] (C, Z). When n = 1, this equals k (cf. [O,S] Th. 7.7). For rank n, [Delta][V.sub.[Phi]],...,[Phi],J] is obtained by applying an homothety of ratio n to [Delta][V.sub.J] for rank one. As [Delta][V.sub.J] is a polyhedra of dimension dim [Delta]W = #I - 1, the result follows.

THEOREM 3.3. Consider a family of curves [Mathematical Expression Omitted] where B is one dimensional. Assume that the generic fiber is nonsingular and the special fiber is C. Given vector bundles E and E[prime] on C, we shall say that they are equivalent if E[prime] = E([[Sigma].sub.i[element of]I] [[Zeta].sub.i][C.sub.i]) where [[Zeta].sub.i] are integers. The set of all possible degrees on C modulo those which correspond to the restriction of equivalent bundles is in one to one correspondence with the components of the moduli space of vector bundles on C.

Proof. From (3.1), the components of the moduli space correspond with the baricenter of stable polyhedra i.e. with the intersection of the interior of [Phi] + [Delta][V.sub.[Phi],...,[Phi],J] with the image by [Delta] of the vectors [W.sub.Z] in W which have integral coefficients. By a straightforward generalization of Corollary 6.3 in [O, S], the number of elements in this intersection equals the index of [Delta][Delta][V.sub.nZ] in [Delta][W.sub.Z]. Let us check that this index corresponds with the degrees modulo the equivalence relation above. In fact, if [v.sub.i] is a basic element of V corresponding to a component [C.sub.i], [Delta][Delta][v.sub.i] = m[v.sub.1] - [v.sub.i(1)] - ... - [v.sub.i(m)] where m is the number of external nodes on [C.sub.i] and [C.sub.i(1)], ..., [C.sub.i(m)] are the (not necessarily different) components that intersect [C.sub.i] at the nodes. On the other hand, tensoring a vector bundle E with O([C.sub.i]) modifies the degree by substracting mn to the degree on [C.sub.i] and n to the degrees on [C.sub.i(i)], ..., [C.sub.i(m)] (if one of these components appears [Alpha] times, one should add [Alpha]n to the degree). Hence, this tensorialization modifies the degree by n(m[v.sub.1] - [v.sub.i(1)] - ... - [v.sub.i(m)]).

The total degree of the vector bundle is fixed. Moreover, because of the connectivity of the curve, [Delta]W coincides with the vectors in V of total degree zero. Hence [Delta]W can be identified, up to a translation, with the set of all possible degrees. This proves the assertion.

4. Incidences among components. We want to study now the incidences among components of the moduli space. Recall (cf. 3.1) that each component of the moduli space corresponds to a choice of degrees of the restriction of E to each [C.sub.i]. Hence the points of intersection correspond to those sheaves that are not locally free. Then, part of the degree is transformed into torsion at a node. The next two propositions show that this can be done for any two adjacent components shifting the degree from one to the other.

PROPOSITION 4.1. Let E be a torsion-free sheaf. Assume that E has a rank [Alpha] [less than] n at the node [P.sub.0]. Let [C.sub.1], [C.sub.2] be the two components of C that intersect at [P.sub.0]. Then, E is the limit of torsion-free sheaves which have rank [Alpha] + 1 at [P.sub.0] such that the degree of the restriction to say [C.sub.[i.sub.1]] is one higher.

Proof. Consider the partial normalisation C[prime] of C at the node [P.sub.0] and let [P.sub.1] and [P.sub.2] be the inverse images of [P.sub.0] on C[prime]. Denote by [Pi] the natural map from C[prime] to C.

Given a sheaf on a curve and a point P on the curve, denote by [E.sub.P] the [O.sub.P] module fiber of E at P and by [E.sub.(P)] the vector space obtained as [E.sub.P]/[M.sub.P][E.sub.P] where [M.sub.P] is the maximal ideal of [O.sub.P].

Let [E.sub.0] be [Pi]*(E)/torsion. Choose [a.sub.1] and [a.sub.2] such that [a.sub.1] + [a.sub.2] = n - [Alpha] and consider an extension

[Mathematical Expression Omitted].

Consider the composition of inclusions [Mathematical Expression Omitted]. By restricting to the fiber at [P.sub.0] we obtain an inclusion of [E.sub.([P.sub.0])] into [([Pi]*E[prime])([P.sub.0])]. Now, [([Pi]* E[prime]).sub.([P.sub.0])] can be identified with [E[prime].sub.([P.sub.1]) [symmetry] [E[prime].sub.([P.sub.2]). Therefore, we obtain a subspace of dimension n of E[prime]([P.sub.1]) [symmetry] [E[prime].sub.([P.sub.2])]. Because of our choice of E[prime], the map from V to [E[prime].sub.([P.sub.1])] has a kernel of dimension [a.sub.1] and the map to [E[prime].sub.([P.sub.2])] has a kernel of dimension [a.sub.2]. With a suitable trivialization of [E[prime].sub.([P.sub.1])] [symmetry] [E[prime].sub.[P.sub.2)] a basis of V could be of the type (0 ... 1 ... 0) with 1 in the n + i place i = 1,...,[a.sub.1], [e.sub.i] = (0 ... 1 ... 0 ... 1 ... 0) with 1 in the i and n + i places for i = [a.sub.1] + 1, ... n - [a.sub.2] and [e.sub.i] = (0 ... 1 ... 0) with 1 in the i place for i = n - [a.sub.2] + 1, ..., n. Define the vector [e[prime].sub.t,[a.sub.1]] = (0 ... t ... 0 ... 1 ... 0) with t in the [a.sub.1] place and 1 in the n + [a.sub.1] place. Consider the family of subspaces [V.sub.t] = ([e.sub.1],...,[e.sub.[a.sub.1]] - 1, [e[prime].sub.t],[a.sub.1]], [e.sub.[a.sub.1] + 1],...,[e.sub.n]) of [E[prime].sub.([P.sub.1])] [symmetry] [E[prime].sub.([P.sub.2])]. Then the kernel of the projection of [V.sub.t] onto [E[prime].sub.([P.sub.1])] has dimension [a.sub.1] - 1 for t [not equal to] 0 and dimension [a.sub.1] for t = 0. Hence the kernel of the map [Mathematical Expression Omitted] gives the required defformation of E with degre of [C.sub.1] one higher (compare with our propositions 4.5 and 4.9 in [B]).

PROPOSITION 4.2. Let E be a torsion-free sheaf on the curve C. Denote by [[Delta].sub.i] the degree of E on [C.sub.i] and by [J.sub.[Alpha]](E) the set of nodes where E has rank [Alpha]. Let [j.sub.0] [element of] [J.sub.[Beta]] for some [Beta] [less than] n. Let [C.sub.1], [C.sub.2] be the two components that meet at [j.sub.0] (eventually [C.sub.1] = [C.sub.2]). Consider a torsion-free sheaf E[prime] whose restriction to [C.sub.i] for i [not equal to] 1 has degree [[Delta].sub.i] and whose restriction to [C.sub.1] has degree [[Delta].sub.1] + 1. Assume [J.sub.[Alpha]](E[prime]) = [J.sub.[Alpha]](E) for [Alpha] not equal to] [Beta],[Beta]+ 1, [J.sub.[Beta]](E[prime]) = [J.sub.[Beta]](E) - {[j.sub.0]} [J.sub.[Beta]+1](E[prime]) = [J.sub.[Beta]+1](E) [union] {[j.sub.0]}. If the polyhedra associated to E was stable, so is the polyhedra associated to E[prime].

Note. The same result is valid interchanging [C.sub.1] and [C.sub.2].

Proof. Let b be the baricenter of the polyhedra associated to E, b[prime] the baricenter of the polyhedra associated to E[prime]. From the definition, [Delta]b[prime] = [Delta]b + ([v.sub.1] - [v.sub.2])/2.

The stability condition for D(E) is [Delta]b - [Phi] [element of] [Delta]([V.sub.[J.sub.0]],...,[J.sub.n]]). Hence, there exist real numbers [Mathematical Expression Omitted] with [Mathematical Expression Omitted] such that [Mathematical Expression Omitted]. Then [Mathematical Expression Omitted]. As [Mathematical Expression Omitted], the result is proved.

LEMMA 4.3. Assume that [Mathematical Expression Omitted]. Define [Mathematical Expression Omitted] by [Mathematical Expression Omitted] for k = 1, 2. Define [Mathematical Expression Omitted]. Define [Mathematical Expression Omitted]. Then, [Mathematical Expression Omitted] if and only if [Mathematical Expression Omitted].

Proof. The if part is easy and left to the reader. Let us prove the only if part.

From (2.8), we need to show that [Mathematical Expression Omitted].

Write [Mathematical Expression Omitted]. Also, [Mathematical Expression Omitted].

Hence [Mathematical Expression Omitted], [Mathematical Expression Omitted].

Write now [Mathematical Expression Omitted], [Mathematical Expression Omitted]. Now [Mathematical Expression Omitted].

Hence our assumption implies that * [greater than or equal to] 0. This concludes the proof.

THEOREM 4.4. Let [E.sub.1], [E.sub.2] be torsion free sheaves on C. Denote by [Mathematical Expression Omitted] the [J.sub.[Alpha]] corresponding to [E.sub.i] for i = 1 and 2. Then [E.sub.1] is in the closure of the set of torsion-free sheaves with same degrees on the components and same ranks at the nodes as [E.sub.2] if and only if D([E.sub.1]) [subset] D([E.sub.2]) and [Mathematical Expression Omitted].

Proof. Assume that [E.sub.1] is in the closure of the set of vector bundles of the type of [E.sub.2]. Then the rank of [E.sub.2] at each node must be greater than or equal to the rank of [E.sub.1]. Hence [Mathematical Expression Omitted]. Moreover the missing rank at the nodes must be accounted for by missing degree in adjacent components. Hence, we can write [Mathematical Expression Omitted] where [Mathematical Expression Omitted], [Mathematical Expression Omitted] are positive and for each component [v.sub.i] if [Mathematical Expression Omitted], [Mathematical Expression Omitted] are the degrees of [E.sub.1] and [E.sub.2] on [C.sub.i], then [Mathematical Expression Omitted] where A(i) is the set of nodes for which [C.sub.i] is the origin and B(i) is the set for which [C.sub.i] is the end point.

Write [Mathematical Expression Omitted].

Note that [Mathematical Expression Omitted]. Hence, if [Mathematical Expression Omitted] then [Mathematical Expression Omitted] as we had to prove.

Let us now prove the converse. From the inclusion [Mathematical Expression Omitted] for every j. Now from D([E.sub.1]) [subset] D([E.sub.2]) and (4.3) (with the notations there) [Mathematical Expression Omitted]. Hence [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. Hence [Mathematical Expression Omitted]. Note now that [Mathematical Expression Omitted], [Mathematical Expression Omitted] and the sum of these two number is [Mathematical Expression Omitted]. Hence this shows that the difference of degrees on each components comes from the differences of ranks at the nodes lying on them. Then (4.2) shows that [E.sub.1] is in the closure of the set of torsion free sheaves of the same type as [E.sub.2].

PROPOSITION 4.5. If the components of genus one of C have at least two nodes, the moduli space U(C, n, d, [[Lambda].sub.i]) is connected.

Proof. We need to prove that there are stable torsion free sheaves of rank n - 1 at one given node, rank n at all remaining nodes for every choice of a stable polyhedra corresponding to a torsion-free sheaf with these ranks. If the node does not disconnect the curve, we can degenerate the torsion-free sheaf to a torsion-free sheaf of rank zero at the node. The latter can be identified to a vector bundle on the normalization of the curve at the node. We know (cf. Th. 3.1 or [T]) that such a stable vector bundle exists.

If the node does disconnect, let us assume for simplicity that the curve has only two components [C.sub.1] and [C.sub.2] that intersect at P.

Assume given a torsion-free sheaf E on C of rank k at P. Denote by El and [E.sub.2] the restrictions of E to [C.sub.1] and [C.sub.2] modulo torsion. Let F be a subsheaf of E which has rank [r.sub.1] and [r.sub.2] on [C.sub.1] and [C.sub.2] respectively and rank r[prime] at P. As in (1.2), one can check that if

(*) [Chi]([F.sub.i])/[r.sub.i] [less than] [[Chi]([E.sub.i]) - (n - 1)]/n + (r[prime]/2[r.sub.i])((n - 1)/n +1)

for every subsheaf F of E, then E is stable.

From [H] Th. 4.4, for a generic [E.sub.i],

(**) [r.sub.i][Chi]([E.sub.i]) - n[Chi]([F.sub.i]) [greater than or equal to] [r.sub.i](n - [r.sub.i])/([g.sub.i] - 1)

where [g.sub.i] is the genus of [C.sub.i].

In order to check existence of a stable bundle E, we need to find [E.sub.1] and [E.sub.2] such that (*) is satisfied for every subsheaf [F.sub.i] on [E.sub.i].

From the exact sequence [Mathematical Expression Omitted], [Chi]([F.sub.i]) [less than or equal to] [Chi]([F.sub.i](- P)) + rk[F.sub.iP]. Then, unless r[prime] = rk[F.sub.iP] or r[prime] = rk[F.sub.iP] - 1, we can replace [F.sub.i] and [E.sub.i] by [F.sub.i](- P) and [E.sub.i](- P) and obtain a better inequality.

Let us assume first that r[prime] = [r.sub.i]. Then, (*) reads [Chi]([F.sub.i])/[r.sub.i] [less than] [[Chi]([E.sub.i]) - (n - 1)]/n + (1/2)((n - 1)/n + 1) = [[Chi]([E.sub.i]) + 1]/n - 1/2n this is equivalent to [r.sub.i][Chi]([E.sub.i]) - n[Chi]([F.sub.i]) + [r.sub.i]/2 [greater than or equal to] 0 and it follows easily from (**).

Assume now r[prime] = [r.sub.i] - 1. Then, (*) is [Chi]([F.sub.i])/[r.sub.i] [less than] [[Chi]([E.sub.i]) - (n - 1)]/n + ([r.sub.i] - 1/2[r.sub.i])((n - 1)/n + 1) = [[Chi]([E.sub.i]) + 1]/n - 1/2n - 1/2[r.sub.i] + 1/2n[r.sub.i]. This is equivalent to [r.sub.i][Chi]([E.sub.i]) - n[Chi]([F.sub.i]) [greater than or equal to] n - [r.sub.i]/2 - 1/2. From (**), if [g.sub.i] [greater than or equal to] 2, [r.sub.i][Chi]([E.sub.i]) - n[Chi]([F.sub.i]) [greater than or equal to] [r.sub.i](n - [r.sub.i]) [greater than or equal to] n - [r.sub.i] [greater than or equal to] n - [r.sub.i]/2 - 1/2. This proves the assertion.

Remark 4.6. Proposition (4.5) may fail if C has an elliptic component attached to the rest by just one point.

Proof. Assume for simplicity that C consists of just two components [C.sub.1] and [C.sub.2] that intersect at P. Assume [C.sub.1] is elliptic. The condition for the stability of the polyhedra associated to a torsion-free E which has rank n - 1 at P is [[Lambda].sub.1][Chi](E) [less than] [Chi]([E.sub.1]) [less than] [[Lambda].sub.1][Chi](E)+n - 1. Notice that E is obtained from an identification of a quotient of the fibers of [E.sub.1] and [E.sub.2] at P [E.sub.1P]/[V.sub.1] [approximately equal to] [E.sub.2P]/[V.sub.2].

Let [F.sub.1] be a sheaf on [C.sub.1]. From Riemann-Roch, [h.sup.0]([E.sub.1] [symmetry] [F*.sub.1]) = [r.sub.1][Chi]([E.sub.i]) - n[Chi]([F.sub.1]). Hence, the set of all irreducible subsheaves of [E.sub.1] of given Euler-Poincare characteristic [Alpha] and given rank r has dimension 1 + r[Chi]([E.sub.1])-n[Alpha]. The Grassmannian of subspaces of dimension r of [E.sub.1P], has dimension r(n - r). The flag variety of those r-dimensional subspaces containing V has dimension (r - 1)(n - r), hence codimension n - r in the Grassmannian. This implies that, when 1 + r[Chi]([E.sub.i])-n[Alpha] [greater than] n - r, there is a subsheaf of E of rank r and Euler-Poincare characteristic [Alpha] such that its fiber at P contains V. Then, stability of E implies [[Alpha] - (r - 1)]/[[Lambda].sub.1]r [less than] [Chi]/n. This condition will fail for certain values of [[Lambda].sub.1] unless [[Alpha] - (r - 1)]/r [less than] [Chi]([E.sub.1]) - (n - 1)/n. This latter condition is r[Chi]([E.sub.1]) - n[Alpha] + r - n [greater than] 0 and does not follow from (*) above.

DEPARTMENT OF MATHEMATICS, TUFTS UNIVERSITY, MEDFORD, MA 02155

REFERENCES

[B] U. Bhosle, Generalized parabolic vector bundles and applications to torsion free sheaves, Arkiv for Mathematik 30 (1992), 187-215.

[H] A. Hirschowitz, Problemes de Brill-Noether en rang superieur, preprint, 1986.

[O,S] T. Oda and C. S. Seshadri, Compactifications of the generalized Jacobian, Trans. Amer. Math Soc. 253 (1979), 1-90.

[S] C.S. Seshadri, Fibres vectoriels sur les courbes algebriques, Asterisque 96 (1982).

[T] M. Teixidor, Moduli spaces of (semi)stable vector bundles on tree-like curves, Math. Ann 290 (1991), 341-348.

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Author: | Bigas, Montserrat Teixidor I. |
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Publication: | American Journal of Mathematics |

Date: | Feb 1, 1995 |

Words: | 9312 |

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