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 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 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 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* 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). LetA be a common specialization of F and G. Use the openness of stability and the fact that the local deformation space of A is smooth because the local to global spectral sequence of the Ext-functor and the fact that dim(X) = 1 give Ext2(^4, ^4) = 0. □ It is important to extend [10, Satz 2.2] to the case of a nodal curve proving the following result. Theorem 2.6. Let X be an integral nodal curve of arithmetic genus g 2. > Fix integers x,y,u with x > u > 1. Let E be a general element ofM(X;x,y). Then for every saturated rank u subsheaf F of E we havefi(F) + g — 1 < fi(G/F) and 1 62 E. BALLICO if 9* := 9 ~ card(Sing(X)) > 2 and we have equality, then F and G/F are locally free and stable. Proof. Fix a rank u subsheaf F of E with maximal degree. Since F is saturated, E/F is torsion free. Let {m(i)}i Remark 2.7. The inequality in Theorem 2.6 is equivalent to the inequality (rank(F) — rank(F)) deg(F) + (rank(F) - rank(F))rank(F)(g - 1) < rank(F)(deg(F)—deg(F)), i.e. to the inequality rank(F/F)(deg(F)—deg(F/F)) + rank(F/F)(rank(F) — rank(F))(g — 1) < (rank(F) — rank(F/F)) deg(F/F). Using Theorem 2.6 the same proof as in the case of smooth curves (see e.g.[4, §2 and §3]) gives the following result. Proposition 2.8. Let X be an integral nodal projective curve with g := pa(X) > 2. Fix integers x ,y with x > 1. Fix P £ X reg. The general positive transformation supported by P of a general E £ M(X;x,y) is stable. The general elementary transformation supported by P of a general £ F M(X;x,y + l) is stable. Making a general positive transformation supported by P we obtain a dominant rational map from M(X;x,y) and M(X;x,y + 1). Making a general elementary transforma tion supported by P we obtain a dominant rational map fromM(X\x, y + 1) and M(X;x,y). We have the following key results proven in[7] and [10, p. 458] for smooth curves. Theorem 2.9. Let X be an integral nodal projective curve with g := pa(X) > 2. Fix integers x and k with x > k > 1. Every torsion free sheaf E on X with rank(E) = x has a rank k subsheaf F such that k(deg(E)) — x(deg(F)) < k(x — k)(g — 1) + x — 1. Proof. We use that E is a limit of stable sheaves (Proposition 1.5), that U (X x , deg(E)) is irreducible, that (up to the finite group Aut(X)) we may see U(X;x, deg(F)) as limit of a moduli space of semistable rankx vector bundles on smooth curves of genus g ([12, Th. 8.2.1]) and that the result is true for smooth genus g curves ([8]). □ EXACT SEQUENCES OF STABLE VECTOR BUNDLES ON NODAL CURVES 163 Theorem 2.10. Fix integers g,x,k and y with g > 3 and x > k > 0. Let X be an integral nodal curve withpa{X ) = x. Let e be the integer with 0 < e < x and k(x — k)(g — 1) -f e = ky mod (x). Set ml := ky — (g — 1) k(r — k) — e and m := m'/k(r — k). Then there is a Zariski open subset U ofM(X;x,y), U ^ 0 , such that for every E G U m is the maximal degree of ranka k subsheaf of U. Proof. By [10, p. 458] the result is true for every smooth genus g curve. By [12, Th. 8.2.1] we know the existence of an integerm" > m and of a Zariski open subset U of M(X;x, y) such that every E G U fits in an exact sequence (1) with rank(A) = k and deg(A) = m". We need to show that m" = m. Assume m" > m. Since E is stable, it is simple and henceH°(X, Hom(i?, A)) = 0. Assume that A (resp. B) has isomorphic type (k — m(i), m(i)) (resp. (x — k — n(i), n(i)) at P(i) G Sing(X). Hence dim (Extx(B,A)) = k(x — k)(m(B) — m(A)+g — l) = k(y —m") — (x — k)m" + k(x-k)(g-l) + Y^i Theorem 2.11. Fix integers g,r,s,a,b with g > 3 , r > 0, s > 0 and a/r — b/s < g — 1. Let X be an integral nodal curve withpa{X) — g. Then for a general (A, B ) G M(X;r,a) x M(X; s, b) we have h° (X , Hom(A, B)) = 0. Proof of Theorem 1.1. The proof is just the justification (see below) of the re mark that the proof for smooth curves given in [4, proof of Th. 3.2] works verbatim writing “saturated subsheaf’ instead of “subbundle” . We fixP G X reg and we make elementary transformations of torsion free sheaves supported by the pointP. The related results proven in [4, §1 and §2] are true with no change in the proofs because every torsion free sheaf onX is locally free at P. The numerical computations used for the proof of [4, Th. 3.1 and Th. 3.2] and given in [4, §3] work with no change. We only need to justify the quotations of[10, p. 458] for a nodal curve. This is the key result 2.11. Now we study the stable bundles, E , fitting in (1) when n(B) — fi(A) < g — 1. Theorem 2.12. Fix integers g,r,s,a,b with g > 3, r > 0, s > 0 and 0 < b/s — a/r < g — 1. Let X be an integral nodal curve withpa{X) = g. The set of all stable vector bundles E G M(X;r + s,a + b) fitting in an exact sequence (1) with A G M(X-,r,a) and B G M(X\s,b) forms an irreducible, non-empty subvariety T ofM(X;r + s, a + b) with dim(T) = (r2 + rs + s2)(g — 1) + (br — as) + 1 birational M(X;r, a) x M(X\ s, b) x PN with N := rs(g — l) + (6r — as) — 1. A general E G T fits in a unique exact sequence (1). Proof. Let EXT be the set of all extensions (1) with A G M(X;r,a) and B G M(X] s,b). Since h°(X, Hom(i?, ^4)) = 0 for all such A and B, by the theory of the relative Ext (see [9] or [3]). EXT is a vector bundle (in the Zariski topology) of rank N + 1 over M(X;r,a) x M(X;s,b) (or, more precisely, over the product of finite coverings of the moduli spaces on which there is a “universal” family of stable vector bundles). Since proportional extensions give isomorphic bundles we 164 E. BALLICO obtain a morphism, u, from a non empty Zariski open subset of the projectiviza- tion P(EXT) of EXT (which is birational to M (X ;r , a) x M(X\s,b) x P N) to M(X\r + s,a + b). Since char(lf) = 0 it is sufficient to show the existence of an open subset U of P(EXT), U ^ 0 , such that u | U is injective, i.e. to prove that a general bundle E G Im(u) has a unique subbundle A G M(X;r,a) such that E/A G M(X-s,b). We will take as U a Zariski open subset of the restriction of P(EXT) to the projective bundle overM' x M ", where M' (resp. M") is the Zariski open subset of all bundles of M (X ;r ,a) x M(X;s,b) satisfying the thesis of Theorem 2.11. The proof of the injectivity for any non-split extension of such B G M" by A G M' given in [2, Sublemmas 1.2 and 1.3] works for a nodal curve, because it uses essentially only Theorem 2.11 and numerical computations. As far as we know the last assertion of Theorem 2.12 was not known for smooth curves, except in the case of rank 2 bundles([11, Prop. 3.3]). Fix integers g,r,s,a,b with g > 3, r > 0 and s > 0 with a/r < b/s. Let X be an integral nodal curve with pa(X) = g. Let T(X;r, s,a,b) be the set of all stable vector bundles E G M(X\r + s, a + b) fitting in an exact sequence (1) with A G M(X;r,a) and B G M (X ;s , b). By Theorem 1.1 T(X;r,s,a,b) ^ 0 and T(X-,r, s, a, b) is an irreducible variety. □ Theorem 2.13. Let X be an integral nodal curve withpa{X ) = g > 3. For all integers r,s,a and b with r > 0, s > 0 and a/r < b/s T(X;r, s,a,b) is in the closure in M(X;r + s,a + b) of T(X;r, s, a— 1,b 4-1). Proof. Fix a general extension (1) with respect to the quadruple (r, s, a,b). Note that the general elementary transformationE' of E fits in an exact sequence 0 -► A' -»• E' -> B -> 0 (4) with A’ general elementary transformation ofA and that the general positive ele mentary transformation E" of E' fits in an exact sequence 0 -> A' -»• E" -» B' -+ 0 (5) with B' general positive elementary transformation ofB'. By Proposition 2.8 we may assume that A' is a general element of M (X ;r ,a — 1) and B' is a general element ofM(X] s,b+ 1). Note that for every elementary transformation,E' , of E there is an “inverse” positive elementary transformation ofE' which gives E. Since the family of all elementary transformations ofE is parametrized by an irreducible variety and the same is true for positive elementary transformations, we obtain that E is in the closure of T(X\r, s, a — 1, b + 1). Since T(X;r, s, a, b) is irreducible, we obtain the result. It seems easy to use the proof of Theorem 2.13 and an extension of [2, Th. 0.1] to the nodal case to prove Theorem 1.1 wheng — 1 > 2(max{r, s}). It seems likely to us that this approach could be justified to obtain Theorem 1.1 in full generality. However, in any case this would not give a proof of 1.1 conceptually different from the one in [4]. □ References 1. E. Ballico, Maximal subsheaves of rank 2 stable torsion free sheaves on integral curves, Preprint. 2. E. Ballico, L. Brambila-Paz and B. Russo, Exact sequences of stable vector bundles on projective curves, Math. Nachr. (to appear). EXACT SEQUENCES OF STABLE VECTOR BUNDLES ON NODAL CURVES 165 3. C. Banica, M. Putinar, and G. Schumaker, Variation der globalen Ext in De- formationen kompakter komplexer Raume, Math. Ann. 250 (1980), 135-155. 4. L. Brambila-Paz and H. Lange A stratification of the moduli space of vector bundles on curves, Preprint. 5. P.R. Cook, Local and Global Aspects of the Module Theory of Singular Curves, Ph.D. Thesis, Liverpool 1993. 6. G.M. Greuel and H. Knorrer, Einfache Kurvensingularitaten und torsionfreie, Moduln. Math. Ann. 270 (1985), 417-425. 7. A. Hirschowitz, Problemes de Brill-Noether en rang superieur, unpublished preprint, partially published as the next reference. 8. A. Hirschowitz, Problemes de Brill-Noether en rang superieur C.R. Acad. Sci. Serie I, Paris 307 (1988), 153-156. 9. H. Lange, Universal families of extensions, J. Alg. 83 (1983), 101-112. 10. H. Lange, Zur Klassification von Regelmannigfaltigkeiten, Math. Ann. 262 (1983), 447-459. 11. H. Lange and M.S. Narasimhan, Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266 (1983), 55-72. 12. R. Pandharipande, A compactification over M~ of the universal moduli space of slope-semistable vector bundles, Preprint. 13. C.J. Rego, Compactification of the space of vector bundles on a singular curve, Comment. Math. Helvetici 57 (1982), 226-236. 14. C.S. Seshadri, Fibres vectoriels sur les courbes algebriques, Asterisque 96 (1982). E. Ballico Department of Mathematics University of Trento 38050 Povo (TN) ITALY [email protected]