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(ii) Let n ∈ N, and let E be stable with d > (n − 1)r or, equivalently, µ>n − 1. Then [n] E is slope stable with respect to Hn. (iii) Let E be semi-stable with d ≤ −r or, equivalently, µ ≤ −1. Then E[n] is slope semi-stable with respect to Hn for every n ∈ N. (iv) Let E be stable with d< −r or, equivalently, µ< −1. Then E[n] is slope stable with respect to Hn for every n ∈ N. [n] d−(n−1)r µ−n+1 The slope of a tautological bundle is given by the formula µ(E )= rn = n ; see Subsection 1.7. Hence, we can reformulate our result as follows: If the slope µ(E[n]) of a tautological bundle lies outside of the interval [−1, 0], the tautolog- ical bundle inherits the properties stability and semi-stability from E. If µ(E[n]) lies on the boundary of this interval, E[n] still inherits semi-stability from E. The key to our proof is a short exact sequence relating the tautological bundles E[n−1] and E[n]; see Proposition 1.5. This exact sequence allows us to prove Theorem 0.1, by a direct argument if deg E > 0, and by induction if deg E < 0. The paper is organised as follows. In Subsection 1.1 and Subsection 1.2, we recall the def- initions of slope stability and of tautological bundles on the symmetric product of a curve. In Subsection 1.3, we introduce some important divisors on C(n) and Cn, and compute their intersection numbers. In Subsection 1.4 we show that stability of E[n] can be tested by com- S ∗ [n] n (n) S puting the slopes of n-equivariant subsheaves of πnE , where πn : C → C is the n- quotient morphism. Then, in Subsection 1.5, we explain how slopes of Sn-equivariant sheaves on Cn can be computed by restriction to appropriate subvarieties. In the next Subsection 1.6, ∗ [n] [n−1] we discuss the key short exact sequence relating πnE and πn−1E . In Subsection 1.7, we compute the slope of tautological bundles and their pull-backs along the quotient mor- phisms, and remark that (semi-)stability of E[n] implies (semi-)stability of E. In Section 2, we carry out the proof of Theorem 0.1. Halfway through the proof, we have to separate the cases of negative and positive degree d. These two cases are treated in Subsection 2.2 and Subsection 2.3, respectively. In the final Section 3, we observe that Theorem 0.1 is already optimal in the sense that the numerical conditions on the slopes cannot be weakened. Conventions. All our varieties are defined over the complex numbers. We denote the set of positive integers by N. Given two varieties X and Y , we write the projections from their X×Y X×Y product to the factors as prX = prX : X × Y → X and prY = prY : X × Y → X. We write VB(X) for the category of vector bundles and Coh(X) for the category of coherent sheaves on X. Acknowledgements. The author thanks Ben Anthes and S¨onke Rollenske for helpful dis- cussions.
1. Preliminaries 1.1. The notion of slope stability. Let X be a smooth projective variety. Let us fix an ample class H ∈ N1(X) in the group of divisors modulo numerical equivalence. For a coherent sheaf A ∈ Coh(X) with rank(A) ≥ 1, we define its degree and its slope with respect to H by
n−1 n−1 degH (A) degH (A) := c1(A) · H := c1(A) · H , µH (A) := . ZX rank(A) A vector bundle E ∈ VB(X) is called slope semi-stable with respect to H if, for every subsheaf A ⊂ E with rank(A) < rank E, we have µH (A) ≤ µH (E). It is called slope stable with respect STABILITY OF TAUTOLOGICAL BUNDLES ON SYMMETRIC PRODUCTS OF CURVES 3 to H if, for every subsheaf A ⊂ E with rank(A) < rank E, we have the strict inequality µH (A) < µH (E). Sometimes, we omit the word ‘slope’ and just speak of semi-stable and stable vector bundles. Note that, if X = C is a curve, the notion of stability and semi- stability is independent of the chosen ample class H ∈ N1(X). 1.2. Symmetric product of a curve and tautological bundles. From now on, let C always be a smooth projective curve, and let n ∈ N. There is a natural action by the symmetric n group Sn on the cartesian product C by permutation of the factors. The corresponding (n) n quotient variety C := C /Sn is called the n-th symmetric product of C. By the Chevalley– (n) n Shephard–Todd theorem, the variety C is smooth, and the quotient morphism πn : C → C(n) is flat. The points of C(n) can be identified with the effective degree n divisors on C. Accordingly, we write them as formal sums: x1 + · · · + xn = πn(x1,...,xn) for x1,...,xn ∈ C. In fact, the symmetric product is the fine moduli space of effective degree n divisors (or, equivalently, (n) zero-dimensional subschemes of length n) on C, with the universal divisor Ξn ⊂ C × C given by the image of the closed embedding (n−1) (n) . (C × C ֒→ C × C , (x1 + · · · + xn−1,x) 7→ (x1 + · · · + xn−1 + x,x Now, the Fourier–Mukai transform along this universal divisor allows us to construct tauto- logical vector bundles on C(n) from vector bundles on C. Concretely, for E ∈ VB(X), the associated tautological bundle on C(n) is given by (n) (n) E[n] := prC ×C (O ⊗ prC ×C∗ E) ∼ a b∗E , C(n)∗ Ξn C = ∗ (n) (n) prC ×C where a:Ξn → C and b:Ξn → C are the restrictions of the projections C(n) and C(n)×C [n] prC , respectively. Since a is flat and finite of degree n, the coherent sheaf E is a vector bundle with rank(E[n])= n rank(E). 1.3. Intersection theory on symmetric and cartesian products of curves. For i = n n n−1 1,...,n, we write pri : C → C for the projection to the i-th factor, and pri : C → C for n n −1 1 n the projection to the other n − 1 factors. We set H = i=1[pri (x)] ∈ N (C ) for any point P −1 x ∈ C. Indeed, modulo numerical equivalence, thee divisor pri (x) is independent from the point x ∈ C. Using the Segre embedding, we see that Hn is ample. 1 (n) ∗ We define Hn ∈ N (C ) as the unique class with π Hn = Hn. One can check easily that (n−1) e (n−1) (n) Hn is represented by C + x, the image of the closed embedding C ֒→ C with n e(n) α 7→ α + x, for any x ∈ C. Since Hn is ample and πn : C → C is finite, Hn is ample too. We always consider stability of bundles on C(n) with respect to this ample class. en Another important divisor on C is the big diagonal δn = 1≤i 1.4. Stability under pull-back along quotient morphism. Let G be a finite group acting on a smooth projective variety X. A G-equivariant sheaf on X is a coherent sheaf B together ∼ ∗ with a G-linearisation, that means a family of isomorphisms {λg : B −→ g B}g∈G such that for every pair g, h ∈ G the following diagram commutes: ∗ λg g λ ∼ / ∗ h/ ∗ ∗ = / ∗ B g B g h B 4 (hg) B . λhg 4 Let π : X → Y := X/G be the quotient morphism. Then, for every g ∈ G, we have π ◦ g = π, ∗ ∼ ∗ ∗ which yields a canonical isomorphism of functors µg : π −→ g π . This gives, for every ∗ ∗ g ∗ ∗ F ∈ Coh(Y ), a G-linearisation {µg : π F −→ π F }g∈G of π F . We call this the canonical G- ∗ ∗ linearisation of the pull-back π F . By a Sn-equivariant subsheaf of π F , we mean a subsheaf ∗ ∗ A ⊂ π F which is preserved by the canonical G-linearisation of the pull-back: µg(A) = g A as subsheaves of g∗π∗F . Lemma 1.1. Let a finite group G act on a smooth projective variety X such that Y = X/G is again smooth and π : X → Y is flat. Let H ∈ N1(Y ) be an ample class and F ∈ VB(Y ). ∗ ∗ ∗ S ∗ (i) If µπ H (A) ≤ µπ H (πnF ) holds for all n-equivariant subsheaves A of π F with rank A< rank F , then F is slope semi-stable with respect to H. ∗ ∗ ∗ S ∗ (ii) If µπ H (A) < µπ H (πnF ) holds for all n-equivariant subsheaves A of π F with rank A< rank F , then F is slope stable with respect to H. ∗ Proof. For every B ∈ Coh(Y ) with rank B > 0, we have µπ∗H (π B)= |G| · µH (B); see [HL10, ∗ ∗ Lem. 3.2.1]. Hence, for F to be semi-stable, it is sufficient to have µπ∗H (π B) ≤ µπ∗H (π F ) for every subsheaf B ⊂ F with rank B < rank F . The assertion of part (i) follows from the fact that π∗B is a G-equivariant subsheaf of π∗F whenever B is a subsheaf of F . The proof of part (ii) is completely analogous. See [Mis06, Sect. 4.2] for a similar criterion for slope stability of sheaves on quotients. 1.5. Some technical lemmas concerning restriction of sheaves. Lemma 1.2. Let X be a smooth variety, D ⊂ X an effective divisor, and F ∈ Coh(X). Then: Tor1(OD, F ) = 0 ⇐⇒ D does not contain an associated point of F . 0 Proof. Let s ∈ H (OX (D)) be a global section defining D. Since s 0 → OX (−D) −→OX → OD → 0 is a locally free resolution of OD, we have Tor1(OD, F )= ker(s⊗idF ). This kernel is non-trivial if and only if D contains an associated point of F . n n Lemma 1.3. Let F1,...,Fk ∈ Coh(C ) be finite collection of coherent sheaves on C , and let m ≤ n. Then there exist points xm+1,...,xn ∈ C such that, if m n (ι: C ֒→ C , (t1,...,tm) 7→ (t1,...,tm,xm+1,...,xn m denotes the closed embedding with image C × xm+1 ×···× xn, the ranks of the Fi do not change under pull-back along ι and all the higher pull-backs of the Fi along ι vanish: ∗ j ∗ ∀ i = 1,...,k : rank(ι Fi)= rank(Fi) , L ι (Fi) = 0 for j 6= 0. STABILITY OF TAUTOLOGICAL BUNDLES ON SYMMETRIC PRODUCTS OF CURVES 5 Proof. We proceed by inverse induction on m. For m = n, we have ι = id, and the assertion is trivial. For general m, by the induction hypothesis, we may assume that we already found xm+2,...,xn ∈ C such that, for m+1 n , (α: C ֒→ C , (t1,...,tm+1) 7→ (t1,...,tm+1,xm+2,...,xn ∗ j ∗ we have rank(α Fi)= rank(Fi) and L α (Fi) = 0 for all i = 1,...,k and all j 6= 0. Now, there m+1 ∗ exists a non-empty U ⊂ C over which all the α Fi are locally free. We choose xm+1 ∈ C m m+1 in such a way that C × xm+1 ⊂ C has a non-empty intersection with U and does not ∗ contain any of the finitely many associated points of the sheaves α Fi. We have ι = β ◦ α m m+1 where β : C → C with β(t1,...,tm) = (t1,...,tm,x) is the closed embedding with image m m C × xm+1. Because of C × xm+1 meeting U, we have ∗ ∗ rank(ι Fi)= rank(α Fi)= rank(Fi) for every i = 1,...,k. ∗ p ∗ ∗ By our choice to avoid the associated points of the α Fi, Lemma 1.2 gives L β (α Fi) = 0 for all p 6= 0. The assertion follows by the spectral sequence p,q p ∗ q ∗ j j ∗ E2 = L β (L α (Fi)) =⇒ E = L α (Fi) . n Lemma 1.4. Let A be an Sn-equivariant sheaf on C . j ∗ n i) Let x2,...,xn ∈ C be points such that L ι (A) = 0 for all j 6= 0 where ι: C ֒→ C is) given by ι(t) = (t,x2,...,xn). Then ∗ deg e (A)= n! deg(ι A) . Hn ii) Let x ∈ C be a point such that Ljι∗(A) = 0 for all j 6= 0 where ι: Cm−1 ֒→ Cn is) given by ι(t1,...,tn−1) = (t1,...,tn−1,x). Then ∗ deg e (A)= n deg e (ι A) . Hn Hn−1 −1 n−1 Proof. In the set-up of part (i), we have ι(C) = pr1 (y) with y = (x2,...,xn) ∈ C . By −1 ∗ ∗ ∗ projection formula, we have [pr1 (y)] · c1(A) = ι c1(A) = c1(ι A) = deg(ι A), where the ∗ ∗ equality ι c1(A) = c1(ι A) is due to the vanishing of the higher derived pull-backs. By S −1 −1 ∗ the n-equivariance of A, we get [pri (y)] · c1(A) = [pr1 (y)] · c1(A) = deg(ι A) for every i = 1,...,n. Combining this with (2) gives n n−1 −1 −1 deg e (A) = (H ) · c (A) = (n − 1)! [pr (y)] · c (A) = (n − 1)!n[pr (y)] · c (A) Hn n 1 i 1 1 1 Xi=1 e = n! deg(ι∗A) . The proof of part (ii) is very similar. 1.6. Pull-back of tautological bundles along the Sn-quotient. For i = 1,...,n, we n consider the divisor δn(i) := j∈{1,...,n}\{i} ∆ij on C ; compare (1). P Proposition 1.5. For every i = 1,...,n, there is a short exact sequence ∗ ∗ [n] ∗ ∗ [n−1] (4) 0 → pri E(−δn(i)) → πnE → pri πn−1E → 0 . ∗ ∗ [n] ∗ [n] The subsheaves Un(E, i) := im pri E(−δn(i)) → πnE of πnE defined by these sequences have pairwise trivial intersections: (5) Un(E, i) ∩ Un(E, i) for i 6= j. 6 A. KRUG Furthermore, these subsheaves get permuted by the natural Sn-linearisation of the pull-back ∗ ∗ ∗ ∗ [n] πnE: If σ(i)= j, we have the equality µσ(Un(E, i)) = σ Un(E, j) of subsheaves of σ πnE . n Proof. The pull-back of the universal divisor Ξn ⊂ C × C along the flat morphism πn × n (n) ∗ n idC : C × C → C × C is given by (πn × idC ) Ξn = Dn := k=1 Γprk . By flat base change along the diagram P π ×id pr Cn × C n C/ C(n) × C C / C pr prCn C(n) π Cn n / C(n) , Cn×C C(n)×C n we get, setting qn := prC = prC ◦(πn ×idC ): C ×C → C, the following isomorphism ∗ [n] ∼ ∗ (6) πnE = prCn∗(ODn ⊗ qnE) . ∗ Now, let us fix some i ∈ [n]. We note that k6=i Γprk = (pri × idC ) Dn−1, which gives ∗ Dn =Γpri + (pr1 × idC ) Dn−1. Hence, we get aP short exact sequence ∗ (7) 0 → O (− [Γ ∩ Γ ]) → O → (pr × id ) O − → 0 Γpri prk pri Dn i C Dn 1 Xk6=i of coherent sheaves on Cn × C. All the sheaves of this sequence are finitely supported over Cn. Hence, combining (6) and (7), gives the short exact sequence (8) ∗ ∗ [n] ∗ ∗ 0 → pr n O (− [Γ ∩ Γ ]) ⊗ q E → π E → pr n (pr × id ) O − ⊗ q E → 0 C ∗ Γpri prk pri n n C ∗ i C Dn 1 n Xk6=i of coherent sheaves on Cn, which will turn out to be isomorphic to the asserted sequence (4). By flat base change along the diagram qn pri×idC qn−1 % Cn × C / Cn−1 × C / C prCn pr n−1 C pr Cn 1 / Cn−1 , ∗ ∗ ∼ ∗ ∗ [n−1] we see that prCn∗ (pri × idC ) ODn−1 ⊗ qnE = pri πn−1E . To bring the first term of (8) into the correct form, we consider the isomorphism n =∼ n t: C −→ Γpr1 ⊂ C × C , (x1,...,xn) 7→ (x1,...,xn; xi) . n Because of Γprk ∩ Γpri = {(x1,...,xn; x) ∈ C × C | xk = xi = x}, we see that ∗ t [Γprk ∩ Γpri ] = δn(i) . Xk6=i ∗ From this, it follows that pr n O (− [Γ ∩ Γ ]) ∼ pr E(−δ (i)), which shows that C ∗ Γpri k6=i pri pri = i n the sheaves in (8) are isomorphic to thoseP in (4). ∗ ∗ ∗ [n] The fact that, for i 6= j, the subsheaves pri E(−δn(i)) and prj E(−δn(j)) of πnE intersect trivially follows from the fact that O (− [Γ ∩ Γ ]) and O (− [Γ ∩ Γ ]) Γpri k6=i prk pri Γprj k6=j prk prj intersect trivially as subsheaves of OD. P P STABILITY OF TAUTOLOGICAL BUNDLES ON SYMMETRIC PRODUCTS OF CURVES 7 The final statement of the proposition follows from the fact that, for σ ∈ Sn with σ(i)= j, we have the equality ν (O (− [Γ ∩ Γ ]) = σ∗O (− [Γ ∩ Γ ]) σ Γpri prk pri Γprj prk pri Xk6=i Xk6=i ∗ S of subsheaves of σ ODn , where ν is the natural n-linearisation of the pull-back ODn = ∗ ∗ (πn × idC ) OΞn . 1.7. Degree and slope of tautological bundles. There are well-known formulae for the Chern classes of tautological bundles; see [Mat65, Sect. 3]. In particular, we have ∗ [n] (9) c1(πnE )= dHn − rδn . e Alternatively, this formula can easily be deduced inductively using the short exact sequence ∗ ∗ ∗ of Proposition 1.5. For doing this, note that Hn = pri [x]+ pri Hn−1 and δn = δn(i)+ pri δn−1 ∗ [n] for every i ∈ [n]. Combining (9) with (3), we get deg e (π E )= n! d − (n − 1)r and e Hn n ∗ [n] (n − 1)!(d − (n − 1)r) (10) µ e (π E )= = (n − 1)!(µ − n + 1) . Hn n r Since πn is finite of degree n!, we also get ∗ [n] µ e (πnE ) (d − (n − 1)r) µ − n + 1 (11) µ (E[n])= Hn = = Hn n! nr n Remark 1.6. For an arbitrary, not necessarily locally free, coherent sheaf A ∈ Coh(C), (n) [n] ∗ we can still define an associated tautological sheaf on C by A := a∗b (A); compare ∗ (n) Subsection 1.2. Since a is finite and b is flat, the functor a∗b : Coh(C) → Coh(C ) is exact; compare [Kru18, Thm. 1.1]. In particular, if 0 → E1 → E0 → A → 0 is a locally free [n] [n] [n] [n] resolution of A, then 0 → E1 → E0 → A → 0 is a locally free resolution of A . It follows that formula (11) extends to a formula for slopes of tautological sheaves of positive [n] µ(A)−n+1 rank, namely µHn (E )= n . It follows that, if E is a vector bundle on C and A ⊂ E is a destabilising sheaf, then A[n] ⊂ E[n] is again destabilising. In other words, (semi-)stability of E[n] implies (semi-)stability of E. 2. Proof of the main result 2.1. General part of the proof. Let E ∈ VB(C) satisfy the assumptions of one of the four parts (i), (ii), (iii), (iv) of Theorem 0.1. By Lemma 1.1, in order to proof stability or semi- [n] ∗ [n] ∗ [n] S stability of E , we need to compare the slopes of A and πnE for A ⊂ πnE a n-invariant subsheaf with s := rank A < nr = rank E[n]. ′ ∗ [n] For i = 1,...,n, we set A (i) := A ∩ Un(E, 1) as an intersection of subsheaves of πnE ; compare Proposition 1.5. We write the corresponding quotient as A′′(i) = A/A′(i). We also 8 A. KRUG set A′ = A′(1) and A′′ = A′′(1), and get a commutative diagram with exact columns and rows (12) 0 0 0 0 / A′ / A / A′′ / 0 / ∗ / ∗ [n] / ∗ ∗ [n−1] / 0 pr1 E(−δn(1)) πnE pr1πn−1E 0 where the bottom row is the short exact sequence form Proposition 1.5. We set s′ = rank A′ and s′′ = rank A′′ which gives s = s′ + s′′. By the last statement of Proposition 1.5 together S ∗ [n] ′ ∼ ∗ ′ S with the n-equivariance of the subsheaf A ⊂ πnE , we have A (i) = σ A for any σ ∈ n with σ(i) = 1. In particular, rank A′(j) = s′ for every j = 1,...,n. By (5), we have n ′ j=1 A (j) ⊂ A. Hence, we get the following inequalities of the ranks L (13) s ≥ ns′ , s′′ ≥ (n − 1)s′ , ns′′ ≥ (n − 1)s . n Now, we divide the proof that µ e (A) ≤ µ e (π∗ E[ ]) (or, for the proof of parts (ii) and Hn Hn n (iv) of Theorem 0.1, that we have have a strict inequality) into the two cases of positive and negative d = deg E, treated in the following two subsections. 2.2. Proof of the main theorem for bundles of positive degree. In this subsection, we proof parts (i) and (ii) of Theorem 0.1. Let E ∈ VB(C) be a semi-stable bundle with d ≥ (n − 1)r, equivalently µ ≥ (n − 1). By Lemma 1.3 and Lemma 1.4, there are points n x2,...,xn ∈ C such that, for ι: C ֒→ C with ι(t) = (t,x2,...,xn) the closed embedding with image C × x2 ×···× xn, we have ∗ (14) deg e (A)= n! deg(ι A) , Hn the rank of objects of diagram (12) remain unchanged after pull-back by ι, and the rows and columns of the diagram (12) remain exact after pull-back by ι. Since C × x2 ×···× xn is a n n n−1 section of pr1 : C → C and a fibre of pr1 : C → C , the restricted diagram takes the form (15) 0 0 0 0 / ι∗A′ / ι∗A / ι∗A′′ / 0 / / ∗ ∗ [n] / ⊕r(n−1) / 0 E(−x2 − x3 −···− xn) ι πnE OC 0 ⊕r(n−1) By the semi-stability of E(−x2 − x3 −···− xn) and OC , we get ∗ ′ (16) µ(ι A ) ≤ µ(E(−x2 − x3 −···− xn)) = µ − n + 1 and µ(ι∗A′′) ≤ 0. Hence, (17) deg(ι∗A)= deg(ι∗A′)+ deg(ι∗A′′)= s′µ(ι∗A′)+ s′′µ(ι∗A′′) ≤ s′(µ − n + 1) . STABILITY OF TAUTOLOGICAL BUNDLES ON SYMMETRIC PRODUCTS OF CURVES 9 By (14), and by the inequality ns′ ≤ s of (13) combined with the assumption µ ≥ n − 1, ∗ ′ deg(ι A) s ∗ [n] (18) µ e (A)= n! ≤ n! (µ − n + 1) ≤ (n − 1)!(µ − n +1) = µ(π E ) . Hn s s n By Lemma 1.1, this shows that E[n] is semi-stable. Let now E be stable and µ>n − 1. Then, by the stability of E(−x2 − x3 −···− xn), the inequality (16) is strict. Accordingly, the inequality in (17) and the first inequality in (18) are strict, except for if s′ = 0. However, for s′ = 0 the second inequality of (18) is strict, due ∗ [n] to the assumption µ>n − 1. Hence, in any case, we have µ e (A) < µ e (π E ) so that Hn Hn n E[n] is stable by Lemma 1.1. 2.3. Proof of the main theorem for bundles of negative degree. In this subsection, we prove part (iii) and (iv) of Theorem 0.1. So, let E ∈ VB(C) be a semi-stable bundle with ∗ µ ≤ −1. We argue by induction on n that µ e (A) ≤ µ e (π E) for every S -equivariant Hn Hn n n [n] ∗ [1] subsheaf with s := rank A < nr = rank E . For n = 1, the assertion is trivial as π1E = E. Let now n ≥ 2. By Lemma 1.3 and Lemma 1.4, there is an x ∈ C such that, for ι: Cn−1 ֒→ Cn n−1 with ι(t2,...,tn) = (x,t2,...,tn) the closed embedding with image x × C , we have ∗ (19) deg e (A)= n deg e (ι A) , Hn Hn−1 the rank of objects of diagram (12) remain unchanged after pull-back by ι, and the rows and ∗ columns of the diagram (12) remain exact after pull-back by ι. Noting that ι (δn(1)) = Hn−1 and pr ◦ι = id n−1 , the restricted diagram takes the form 1 C e (20) 0 0 0 0 / ι∗A′ / ι∗A / ι∗A′′ / 0 / ⊕r / ∗ ∗ [n] / ∗ [n−1] / 0 O(−Hn−1) ι πnE πn−1E 0 By the induction hypothesis,e together with (10), we get ∗ ′′ ∗ [n−1] (21) µ e (ι A ) ≤ µ e (π E ) = (n − 2)!(µ − n + 2) . Hn−1 Hn−1 n−1 ∗ ′ ⊕r Furthermore, the inclusion ι A ֒→ O(−Hn−1) combined with (3) gives ∗ ′ ⊕r (22) µ e (ι A ) ≤ µ e e (O(−Hn ) )= −(n − 1)! . Hn−1 Hn−1 −1 Combining (21) and (22), we get e ∗ ′′ ∗ ′′ ′ ∗ ′ deg e (ι A)= s µ e (ι A )+ s µ e (ι A ) Hn−1 Hn−1 Hn−1 (23) ≤ (n − 2)! s′′(µ − n + 2) − s′(n − 1) . By the assumption µ ≤ −1, we have µ − n + 2 ≤ −(n − 1). Hence, (23) is maximised if the inequality s′′ ≥ s′(n − 1) from (13) is an equality. This gives ∗ ′′ ′′ ′′ deg e (ι A) ≤ (n − 2)! s (µ − n + 2) − s = s (n − 2)!(µ − n + 1) Hn−1 10 A. KRUG We get the following chain of inequalities deg e (A) ′′ Hn ns (n − 2)!(µ − n + 1) ∗ [n] (24) µ e (A)= ≤ ≤ (n − 1)!(µ − n +1) = µ e (π E ) , Hn s s Hn n where the first inequality is due to (19), the second is due to the inequality ns′′ ≥ (n − 1)s of (13) together with the fact that µ − n + 1 is non-positive, and the last equality is (10). This proves that E[n] is semi-stable. Let now E be stable and µ < −1. Proceeding as before by induction, we see that, if s′′ < (n − 1)r, the inequality (21) is strict. Hence, the first inequality of (24) is strict too. If s′′ = (n − 1)r, we have ns′′ > (n − 1)s since s < nr. It follows that, in this case, the second inequality of (24) is strict. 3. The numerical conditions are sharp In this section, we observe that the numerical conditions in Theorem 0.1 on the slope cannot be weakened. For this, we consider examples of (semi)-stable bundles on C with various values µ(E) ∈ [−1,n − 1] such that E[n] is unstable. Let ℓ ≥ 0, x ∈ C, and L = OC (ℓ·x). Any non-zero section of L induces a non-zero section of [n] [n] L ; see [Mat65, Corollary of Prop. 1]. Hence, OX(n) is a subsheaf of L . For 0 ≤ ℓ [AO94] Vincenzo Ancona and Giorgio Ottaviani. Stability of special instanton bundles on P2n+1. Trans. Amer. Math. Soc., 341(2):677–693, 1994. STABILITY OF TAUTOLOGICAL BUNDLES ON SYMMETRIC PRODUCTS OF CURVES 11 [BD18] Suratno Basu and Krishanu Dan. 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