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Exact Sequences of Stable Vector Bundles on Nodal Curves

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Exact Sequences of Stable Vector Bundles on Nodal Curves NEW ZEALAND JOURNAL OF MATHEMATICS Volume 27 (1998), 159-165 EXACT SEQUENCES OF STABLE VECTOR BUNDLES ON NODAL CURVES E . B a l l i c o (Received April 1997) Abstract. Let X be an integral projective curve with g pa(X) > 3 and with only nodes as singularities. Fix integers r, s, a, with b r > 0, s > 0 and a/r < b/s. Here (using the corresponding result proven recently by Brambila- Paz and Lange) we prove the existence of stable vector bundlesA, E, onB X with rank(A) = r, deg(A) = a, rank(B) = s, deg(B) = b, A subbundle of E and E/A — B. Then we study several properties of maximal degree subbundles of stable bundles on X. 1. Introduction Let X be an integral projective curve of arithmetic genusg := pa{X) > 3 with only nodes as singularities. We work over an algebraically closed base fieldK with ch&r(K) = 0. In this paper we want to construct “suitable” stable bundles onX. To prove (as in [10] and [4] for smooth curves) the existence of suitable families of stable bundles on X it is important to solve the following question. Fix integers r, s, a, b with r > 0, s > 0 and a/r < b/s. Is there an exact sequence 0^ A-* E ^ B ^ 0 (1) of stable vector bundles on X with rank(A) = r, rank(B) = s, deg(A) = a and deg(£?) = s? An affirmative answer to this question was found in [4] when X is smooth. The main aim of this paper is to show that this question has an affirmative answer for nodal X and prove the following result. Theorem 1.1. Let X be an integral projective curve of arithmetic genus g pa{X) > 3 with only nodes as singularities defined over an algebraically closed field K with char(.K') = 0. Let r,s,a,b be integers with r > 0, s > 0 and a/r < b/s. Then there is an exact sequence (1) with A ,B and E stable vector bundles, rank(A) = r, rank(£) = s, deg(A) = a and deg(B) — b. To prove Theorem 1.1 we will use strongly the results and proofs given in [4]. We will extend to the case of nodal curves several interesting results about maximal degree subsheaves of stable vector bundles (see Theorems 2.6, 2.9, 2.10 and 2.11). We believe that this general part will be useful also for the Brill - Noether theory of rank r torsion free sheaves on singular curves. In the last part of the paper we will study the general stable bundleE fitting in the exact sequence (1) when a/r < b/s < a/r + g — 1 (see Theorem 2.12) and the closure in the moduli space of the set of all such bundles (see Theorem 2.13). It seems to us that Theorem 2.13 is new even for smooth curves. For stronger results on singular curves in the rank 2 case, see also [1]. 1991 A M S Mathematics Subject Classification: 14H60. 160 E. BALLICO The author was partially supported by MURST and GNSAGA of CNR (Italy). He wishes to thank P. Newstead, B. Russo and the authors of [4]. 2. Statements and Proofs Recall that for any integral projective curve Z and for any rank r algebraic coherent sheaf E on Z , the integer deg(i?) is defined by the Riemann - Roch formula £(E) = r(l — pa(Z)) + deg(E) and is called the degree ofE ; if E is torsion free set /i(E) deg(£')/rank(£') and call n(E) the slope of E. Let X be a reduced and irreducible nodal curve, i.e. assume that every singular point of X is a singularity of type A\. Let f : Y —► X be the normalization. Set T := Sing(X), g pa{X) and g' := g — card(T) = pa(Y). For the elementary theory of torsion free sheaves onX, see [14, parts VII and VIII], or [5] or [6]. In particular ([14, Prop. 2 at page 164], or [5, pages 24 and 25], or see [6] for much more) for every P E T every torsion free Ox p-module of finite rank is the direct sum of a free module and some copies of the maximal idealrrip of P. By [5, Lemma 2.5.9] we have Ext1 (m /p,m /p) = K ®2 and Ext1(mp, p ) = Extx(0^- p , Trip) = 0; the dimension of each if-vector space Ext*(*, *) is invariant under completion. We will assume g > 2. Let U(X]x, y) be the reduction of the moduli scheme of semistable torsion free sheaves onX with rank x and degree y. It is known (see [13] or [14, Prop. 9 at page 173]) that U(X;x,y) is irreducible, i.e. that U(X-,x,y) is the closure of its open subset M (X;x,y) parametrizing the stable vector bundles. Since stability is an open condition Theorem 1.1 is equivalent to the assertion that for general A E M (X ;r , a) and B G M(X;s,b) the general extension (1) has a stable vector bundle as middle term. Every algebraic subset of U(X; r, d) will be considered a scheme with the reduced structure. Fix an order P(l),... , P(z) of the points of Sing(X). Let U(X\x, y, r i ,... , rz) be the subset of U(X;x,y) corresponding to points represented by a torsion free sheaf,E, such that (over the completion or the henselianization) the stalkEx,p(i) <8> p^ of E at P(i) is isomorphic to 0 /^SpX^ri') © m^®p^) at each point P(i); we will say often that E has type {x — ) at P(i). Proposition 2.1. If g' := g — z >2, then U(X;x,y,ri , . ,rz) has codimension > z ( £ i<i<zri) in U(X-x,y). Proof. For every rank x stable vector bundle A o n F there is a stable vector bundle B o n I with A = f*(B). Indeed, the family of all such bundles B has dimension r2. Using a Jordan - Holder filtration we see that the same is true for semistable bundles. Note that the Grassmannian G(ri,x) has dimension r^(x — r*) and that dim (GL(r*)) = r2. Now the result follows from a dimensional calculation using that dim (U(Y-,x,y)) = dim (U(X ; x, y)) — zx2 and the description/construction of all bundles in U(X;x,y,ri, ... ,rz) made in [14, Th. 17, p. 178-179], as,glueing of subspaces of dimension r* of the fibers over the points / -1 P(i( )) of bundles f*{U) with U suitable bundle on X. □ The same proof gives the following results. EXACT SEQUENCES OF STABLE VECTOR BUNDLES ON NODAL CURVES 161 Proposition 2.2. Assume that the normalization Y of X is an elliptic curve. Then U(X-,x,y, r\,... ,rz) has codimension > x(Xa <i<zri) — I in U(X;x,y). We have dim (U(X; x, y, r i ,... , rz)) < dim (U(X;x, y)) unless r* = 0 for every i. Proposition 2.3. Assume that the normalization Y of X is a rational curve. Then U(X;x,y, r i ,... ,rz) has codimension > x {J 2 ia <z ri) ~2 in U(X;x,y). We have dim (U(X;x,y,r\,... ,rz)) < dim (U(X]x,y)) unless r* = 0 for every i. Now we will extend (with the same proof) [7, Prop. 2.1 and Cor. 2.2]. Proposition 2.4. Fix an integer x > 1 and let F and G rank x torsion free sheaves on X such that the rank 1 torsion free sheaves det(F) and det(G) are isomorphic and for every P E Sing(X) the stalks of F and G are isomorphic over the completion O % p ofOx,p ■ Then F and G have a common specialization in a flat family of torsion free sheaves with constant determinant and constant type at each P E Sing(X). Proof. We use induction on x. If x = 1 we have F = G by assumption. Assume x > 2. Fix a very ample L E Pic(X). There is an integer n > 0 such that F* ® L®n and G* ® L®n are generated by global sections. Taking general sections of F* <g> L®n and G* 0 L®n we obtain surjective maps u : F —► M and v : G —> M with M subsheaf of and /M supported by the points P € Sing(X) at which F ® P = (and hence G ® O% p — m ^p)- Hence F and G fit in the exact sequences 0 -> Ker(w) F -> M -* 0 (2) 0 -► Ker(v) -> G -* M 0 (3) Call e (resp. f) the extension class of (2) (resp. (3)). Taking te and t f, t e lf\ {0 } and sending t to zero we see that F has Ker(w) ® M as specialization and G has Ker(u) © M as specialization. By the inductive assumption Ker(w) and Ker(v) have a common specialization in a flat family with constant determinant and constant type at every P E Sing(X). □ Hence we conclude. Corollary 2.5. Every torsion free sheaf is the limit of a flat family of stable torsion free sheaves with constant determinant. Proof. Fix a rank x torsion free sheaf F. It is easy to check the existence of a rank x torsion free sheafG with det(G) = det(F) and such that the stalks of G and F are isomorphic at every point of Sing(X).
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