Higher rank Clifford indices of curves on a K3 surface

Let (X, H) be a polarized K3 surface with Pic(X)=ZH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Pic}(X) = \mathbb {Z}H$$\end{document}, and let C∈|H|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\in |H|$$\end{document} be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when g≥r2≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\ge r^2\ge 4$$\end{document}, the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree d(≥5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d(\ge 5)$$\end{document} smooth plane curve is d-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d-4$$\end{document}, which is the same as the Clifford index of the curve.


Introduction
Let U C (r , d) be the set of semistable rank r -vector bundles of degree d on a smooth curve C. For E ∈ U C (r , d), its Clifford index is defined as By the higher rank Clifford Theorem [3, Theorem 2.1], when 0 ≤ d ≤ r (g − 1), the index Cliff(E) is non-negative. The rank r Clifford index of C, first introduce in [15] where it was denoted γ r , is defined as: Our main result is as follows.

Theorem 1.1 Let (X , H ) be a smooth polarized K3 surface satisfying Assumption (*)
, and let C be a smooth curve of genus g in the linear system |H |. Let E be a slope semistable rank r -vector bundle of degree d on the curve C such that d ≤ r (g − 1).

Then we have the bound for the dimension of the global sections of E:
h 0 (C, E) < r + g 4r (g − 1) 2 d 2 + r g . (1) When r ≥ 2 and g ≥ r 2 , the rank r Clifford index of C Assumption (*) H 2 divides H · D for all curve classes D on X . 1 The upper bound for h 0 (C, E) in Theorem 1.1 is much stronger than the higher rank Clifford Theorem, which says h 0 (C, E) ≤ r + d 2 . The bound is not far from the sharp bound, see Remark 3.5. For a smooth curve C of genus g, several upper bounds for the dimension of global sections of vector bundles of low slope μ = d/r have been introduced in [3,22,23], which are also included in [17]. Sharp bounds for the case g ≤ 6 and μ < 2 have been determined in [3,[17][18][19]22,23]. The upper bound (1) is in general stronger than the bounds in these previous papers unless g ≤ 6 or μ ≤ 2.
For r = 2, the second statement of Theorem 1.1 gives so we re-obtain the result [1,Theorem 1.3]. Also for r ≥ 3 and g = 10, we have This indicates the failure of the Mercat's conjecture in [24] for C which states the higher ranks Clifford indices of the curve C are equal to Cliff(C). Meanwhile, when r = 3 and g = 10, we have Cliff 3 (C) = 4 = Cliff(C) for a general curve. When r = 3 and g = 9, the fact that a general curve has Cliff 3 (C) = 10 3 was known according to the results in [16]. When r = 3 and g = 11, our result implies that a general curve has Cliff 3 (C) = 14 3 , which improves the result 11 3 ≤ Cliff 3 (C) ≤ 14 3 in [14,Theorem 3.6].
Let A be a globally generated line bundle on the curve C ⊂ X , the Lazarsfeld-Mukai bundle E C,A on X is defined via the exact sequence In all cases in the second part of Theorem 1.1, there exists a line bundle A on the curve C such that the rank r -Clifford index is computed by the restriction of the corresponding Lazarsfeld-Mukai bundle on the K3 surface X . We expect this result holds without the assumption on the Picard group of X . This can be viewed as a generalization for the result of Green and Lazarsfeld [10] which says that for a curve C on a smooth K3 surface with Cliff(C) < g−1 2 , the Clifford index can be computed by the restriction of a line bundle on the K3 surface.
Our argument can be generalized to curves on other surfaces, especially when the surface admits a stronger Bogomolov-Gieseker type inequality. Examples of such surfaces include the projective plane, del Pezzo surfaces and quintic surfaces. We explain more details for smooth plane curves in Sect. 5. In particular, we show that the first part of the Mercat's conjecture [24] holds for smooth plane curves: The result Cliff 2 (C) = l − 4 for plane curves first appeared in [15,Proposition 8.1]. Further discussions for the rank 3 case appeared in [16]. In particular, the result Cliff 3 (C) = l − 4 was known for l ≤ 6. As Professor Peter Newstead pointed out, it seems to us that all other Clifford indices for smooth plane curves have not been known. In particular, Theorem 1.2 excludes the possibility that Cliff 3 (C) < l − 4 in the assumption in [16,Theorem 5.6].
Another concrete example for curves on degree four del Pezzo surfaces is computed in [13]. The Clifford type inequality for such curves is the key ingredient in proving the existence of Bridgeland stability conditions on smooth quintic threefolds.

Approach
The main tool in this paper is the notion of stability condition introduced by Bridgeland [4]. In general, such a stability condition σ = (A, Z ) is defined on a C-linear triangulated category T , and is consisting of a heart structure A and a central charge Z : K (T ) → C, which is a group homomorphism from the Grothendieck group to complex numbers. The space of stability conditions on T forms a complex manifold which admits a wall and chamber decomposition for any fixed object E ∈ T . In this paper, the triangulated category T will always be the bounded derived category D b (X ) of coherent sheaves on a surface X . We will only make use of a real two-dimensional subspace of stability conditions on D b (X ).
Let ι : C → X be the embedding of a smooth curve C into the surface X , and let E be a semistable vector bundle on the curve C. In [8], a new upper bound for the dimension of global sections of objects in D b (X ) has been introduced. This states the dimension of global sections of ι * E can be bounded by the length of the Harder-Narasimhan polygon at a limit point σ 0 where Z (O X ) → 0. The Harder-Narasimhan polygon geometrically represents the slopes and degrees of the Harder-Narasimhan factors of ι * E with respect to σ 0 . One of the key parts of the paper is to describe the position of the wall for ι * E that bounds the large volume limit chamber at where ι * E is stable. Describing the wall that bounds the large volume limit will enable us to control the length of this Harder-Narasimhan polygon at σ 0 effectively and get the bound for the dimension of global sections of the vector bundle E.
Let (X , H ) be a smooth polarized K3 surface over C with Pic(X ) = ZH . In this section, we review the description of a slice of the space of stability conditions Stab(X ) on D b (X ) given in [5,.
Given an object E ∈ D b (X ), we write ch(E) = (rk(E), ch 1 (E), ch 2 (E)) ∈ H * (X , Z) for its Chern characters. We write H * alg (X , Z) for its algebraic part, in other words, the image of ch(−). The slope of a coherent sheaf E ∈ Coh X is defined by This leads to the usual notion of μ H -stability. For any β ∈ R, we have the following torsion pair in Coh X where − denotes the extension-closure. Following [5,11], this lets us define a new heart of a bounded t-structure in D b (X ) as follows: For any pair (β, α) ∈ R 2 , we define the central charge Z β,α : Note that the function Z β,α , up to the action of GL + (2; R), is the same as the stability function defined in [5, section 6]. The function Z β,α factors via the Chern character The kernel of Z β,α in H * alg (X , R) under the basis {rk, ch 1 , ch 2 } is spanned by (1, β H , α).
We first state Bridgeland's result describing stability conditions on D b (X ), and then expand upon the statements.

Theorem 2.2 [5, Section 1]
For any pair (β, α) ∈ R 2 such that α > (β), the pair σ β,α := Coh β X , Z β,α defines a stability condition on D b (X ). Moreover, the map from We first explain the notion of σ β,α -stability and the associated Harder-Narasimhan filtration. Consider the slope function The Gamma curve E ∈ Coh β X admits a Harder-Narasimhan filtration which is a finite sequence of objects in Coh β X , . The second part of Theorem 2.2 implies that the two-dimensional family of stability conditions σ β,α satisfies wall-crossing as α and β vary. Consider the projection By abuse of notations, we use the same plane for the image of the projection pr and the (β, α)-plane. Note that the point (β, α) is equal to the projection pr (ker Z β,α ) of the kernel of the central charge Z β,α in H * alg (X , Z). We will also write pr (E) instead of pr (ch(E)).

Remark 2.3 (a)
For a stable object E with respect to any stability condition σ β,α , the point pr (E) is not in To see this, note that Thus by Hodge index Theorem, we have By the definition of γ and Assumption (*), we have Together with (4), we have . (n, H 2 2 n 2 ) to (n, H 2 2 n 2 −1) for some n ∈ Z (see the remark below for more details); • the σ β,α -(semi)stability or instability of F does not change when σ β,α changes between two consecutive walls. • the object F is strictly σ β,α -semistable if (β, α) is contained in one of the walls.
• if F is σ β,α -semistable in one of the adjacent chambers to a wall, then it is unstable in the other adjacent chamber.
See Fig. 2 for a picture and [8] for more details.

Remark 2.5
In this paper, we will only apply Proposition 2.4 to an object F = ι * E where E is a slope semistable vector bundle on a curve C ∈ |H |. More precise descriptions for the walls of ι * E are as follows.
• All walls of ι * E are parallel segments with the same slope ch 2 (ι * E) H ch 1 (ι * E) .
• ι * E is σ 0,a -semistable for a 0. • There is at most one wall W intersecting the line {(0, y)|y > 0}. Indeed, if the wall contains passes through (0, α 0 ) for some α 0 > 0, then the destabilizing subobject in Coh 0 (S) will destabilize ι * E for every α < α 0 . So there is at most one α 0 > 0 such that ι * E is strictly semistable with respect to σ 0,α 0 . • Suppose there is a wall W of ι * E intersecting the line {(0, y)|y > 0}. We will see in Lemma 3.1 below that the x-coordinates β 1 and β 2 of the endpoints of W satisfies 0 < β 2 − β 1 < 1. In particular, both endpoints are on the curve . • There are also several walls irrelevant to our study. For each negative integer n < 0 small enough, there is a 'tiny wall' with its 'right endpoint' at (n, H 2 2 n 2 ) and 'left endpoint' on curve . These walls will never intersect the line {(0, y)|y > 0}. So they are irrelevant to the HN factors of ι * E at all. They are the only reason why we give several extra descriptions for the possible endpoints of walls.

Bounds for the dimension of global sections
In this section, we prove the first part of Theorem 1.1 which introduces a new upper bound for the dimension of global sections of vector bundles on a curve over a K3 surface. We always assume X is a K3 surface with Pic(X ) = ZH and C ∈ |H | is a smooth curve of genus g. We denote by ι: C → X the embedding of the curve C into X .

The destabilizing wall for a stable vector bundle on the curve C
Let E be a slope semistable vector bundle on the curve C of rank r ≥ 2 and degree d ∈ [0, r (g − 1)]. By [21,Theorem 3.11], the push-forward ι * E is σ β,α -semistable for any β ∈ R and α sufficiently large. By Proposition 2.4, the walls for ι * E are line segments of slope d r + 1 − g. By Remark 2.5, there is at most one α > 0 such that ι * E is 'destabilized' at σ 0,α , in other words, ι * E is strictly σ 0,α -semistable and not σ 0,α -semistable for every 0 < α < α. Suppose this is the case, in other words ι * E becomes strictly semistable at the wall W which passes through σ 0,α for some α > 0. Denote the x-coordinates of the endpoints of the wall W as β 1 and β 2 for some β 1 < 0 < β 2 (Fig. 3).

Lemma 3.1 Adopt notations as above, we have
Proof Let 0 → F 2 → ι * E → F 1 → 0 in Coh 0 X be the destabilizing sequence at the wall W, then there is an exact sequence in Coh X : If s = 0, then since F 2 and ι * E have the same phase with respect to σ 0,α , it follows that ch(ι * E) = r d 2 ch(F 2 ), so that F 2 cannot make a wall for ι * E. Thus, we may assume s > 0. Let T (F 2 ) be the maximal torsion subsheaf of F 2 and ch 1 (T (F 2 )) = t H. Since E is of rank r , to make the sequence exact at the term ι * E, we must have Therefore, By Proposition 2.4, the object F 1 is semistable of the same phase as ι * E along the line segment W, in particular if −1 < β 1 , it is in the heart Coh β 1 + X where → 0 + . Thus by definition of the tilting heart, By similar reasoning for F 2 /T (F 2 ), it follows from the definition of the tilting heart that Therefore inequality (5) and definition of imply that In particular, β 1 > −1, β 2 < 1, and β 2 − β 1 ≤ 1.
By the second property of Proposition 2.4, the slope of W as a line in the projection It is not hard to see that β 2 (respectively β 1 ) reaches its maximum β max 2 (respectively minimum β min 1 ) when β 2 − β 1 = 1. Substitute this to (9), we get Since 0 ≤ d ≤ r (g − 1), slope of W is not positive, thus 0 < β max 2 ≤ 1 2 and by Definition (2.1), Substituting back into the Eq. (10) gives β max 2 = d r H 2 and β min 1 = d r H 2 − 1. We need the following description for the first wall in details.

Lemma 3.2 Adopt notations from Lemma
Proof Adopt the notations as in the proof of Lemma 3.1. (a) By inequality (7), we know Recall that H ch 1 comparing the first and the last sentences implies the claim. • If r ≤ s, inequality (6) gives Taking ch 1 from the destabilizing sequence gives Since H 2 ≥ −r + 1 and by (12) −r + 1 This finishes the proof of part (a).
r and the claim follows. Thus we may assume otherwise, so part (a) gives We may assume r ≥ 3. Suppose for a contradiction that β 2 > 1 r , then by definition , the slope of the line connecting 1 r , ( 1 r ) and 2−r r −1 , ( 2−r r −1 ) is less than slope of the line connecting (β 2 , (β 2 )) and (β 1 (β 1 )), in other words, Since Substitute them into the left hand side of (14): Since g ≥ r 2 , we get g > r (r − 1) with the last inequality is given by the assumption d ≤ H 2 + r . So this contradicts (14).

An upper bound on the dimension of global sections
We first recall the result in [8,Section 3]. Define the function Z : K (X ) → C as We also define the following non-standard norm on C: x + iy = x 2 + (2H 2 + 4)y 2 .
The next proposition bounds the dimension of global sections of objects in terms of the length of a polygon.
(a) There exists > 0 such that the Harder-Narasimhan filtration of F is a fixed sequence where the last inequality following from the following two cases: (a) If both χ(Ẽ i /Ẽ i−1 ) and p i p i−1 are even or odd, the claim is trivial because Finally by summing up over all stable factors one gets We denote by P F the polygon with the extremal points { p 0 , p 1 , . . . , p n } which is a convex polygon.
Let E be a slope semistable rank r -vector bundle on the curve C of degree d. Proposition 3.3 implies that there exists > 0 such that the Harder-Narasimhan filtration of ι * E with respect to the stability condition σ 0,α for positive α < is a fixed sequence Consider the triangle opq where o is the origin, q = Z (ι * E), the slope of op is equal to β 2 / (β 2 ) and the slope of pq is β 1 / (β 1 ), where the real numbers β 1 and β 2 are defined as in Lemma 3.1 (Fig. 4).

Lemma 3.4 The polygon P ι * E is contained in the triangle opq.
Proof If ι * E is σ 0,α -semistable where α → 0 + , then the polygon P ι * E is just the line segment oq and the claim follows. Thus, we may assume ι * E is not σ 0,α -semistable where α → 0 + . Since the polygon P ι * E is convex, it suffices to show that The phase of the subobjectẼ 1 in the Harder-Narasimhan filtration is bigger than phase of ι * E at the stability condition σ 0,α where α → 0 + . Therefore there are stability condition between large volume limit (σ β,α where α → ∞) and the stability conditions σ 0,α where α → 0 + such thatẼ 1 and ι * E have the same phase. Proposition 2.4 implies that these stability conditions are on a line segment L whose extension passes through the point pr (Ẽ 1 ). Note that rk(Ẽ 1 ) = 0 by the same argument as that in the beginning of the proof of Lemma 3.1. The line L is lower than the wall W for ι * E since otherwise ι * E will already become strictly semistable on L, see Fig. 5.

Fig. 5 Comparing slopes
SinceẼ 1 is σ 0,α -semistable for some α > 0, the point pr (Ẽ 1 ) is not in + by Remark 2.3. Therefore, pr (Ẽ 1 ) is on the dashed part of the line L and the first claim follows. By a similar argument one can show the second claim for ι * E/Ẽ n−1 .
We are now ready to prove the bound for the dimension of global sections of the semistable vector bundle E.

Proof for the first part of Theorem 1.1 Consider the triangle op q where the slope of op is
Lemma 3.1 implies that the triangle opq is inside the triangle op q, so by Lemma 3.4 the polygon P ι * E is also inside the triangle op q. By a direct computation, one can show that the point

Now Proposition 3.3, part (b)
gives where the last equality holds for the non-negative solution δ to the following equation.
This is equivalent to Now we will show that δ < 2r Hence (17) shows f (δ) < f 2r g which gives δ < 2r g . Applying this back into (16) implies

Remark 3.5
The bound for h 0 (C, E) in Theorem 1.1 is not far from the sharp bound.
Let k be an integer in [1, r ], denote t = gcd(r , k). When d = 2k(g − 1) such that g ≥ r t 2 + 2, there exists a stable vector bundle F on X with Chern characters: When k = r , F is a line bundle, so the restriction F ⊕t | C is semistable. When k < r , the rank of F is greater than 1. Since Pic(X ) = Z.H , [9, Proposition 4.6] implies F| C is semistable if where We have Thus (18) clearly holds and the restriction F ⊕t | C is semistable with rank r , degree 2k(g − 1) and dimension of global sections If the · function can be dropped for free, the formula can be simplified as Corollary 3.6 Let (X , H ) be a smooth polarized K3 surface satisfying Assumption (*), and let C be a smooth curve of genus g in the linear system |H |. Let E be a slope semistable rank r -vector bundle of degree d on the curve C such that d ≤ r (g − 1).
Proof The bound for Cliff(E) is by substituting the bounds of h 0 (C, E) into the formula of Clifford index. By the first part of Theorem 1.1, if h 0 (C, E) ≥ 2r , then When g ≥ 7, the value r (g−1) 2 g is in the range of d ∈ 2r (g−1) . To know at which boundary f (d) reaches its minimum, we compare the distances from the two boundaries to r (g−1) 2 Therefore, the function f (d) reaches its minimum at the left boundary. In particular, for any r .

Higher rank Clifford indices
In this section, we compute higher rank Clifford indices of curves over K3 surfaces and prove the second part of Theorem 1.1.

Picard number one case
We assume X is a K3 surface with Pic(X ) = ZH and C ∈ |H | is a smooth curve of genus g. Denote by ι: C → X the embedding of the curve C into X . We first briefly recall the result in [9], which constructs semistable vector bundles on C by restricting vector bundles on X with low discriminant. By [ Proof The stable sheafẼ r is locally-free, otherwise, the double dual F =Ẽ ∨∨ r is slope stable with Chern characters (r , H , g r − s) for some integer s ≤ r − 1.
We now prove the Clifford index of E r is indeed the minimum of Clifford index of any semistable vector bundle E with rank r , degree d and h 0 (E) ≥ 2r . This will involve several different cases. Proof of the second part of Theorem 1.1 for r ≥ 4. Let E be a semistable rank r -vector bundle of degree d ≤ r (g − 1) on the curve C. By Theorem 4.1, it suffices to show that either h 0 (E) < 2r or Cliff(E) ≥ 2 . Denote t:= d − 2(g − 1). The first part of Theorem 1.1 implies that Then Q(t) is a quadratic function with respect to t with negative leading coefficient.
Step 2 We show Cliff(E) ≥ 2 The polygon p ι * E is inside the polygon op 1 p q Applying Proposition 3.3 for the push-forward ι * E implies that there exists > 0 such that its Harder-Narasimhan filtration with respect to σ 0,α for positive α < is a fixed sequence where l(E):= n i=1 p i p i−1 and p i = Z (Ẽ i ). Thus it is suffices to show that SinceẼ 1 is a sheaf supported in dimension ≥ 1 andẼ 1 ∈ T 0 , we get H ch 1 (Ẽ 1 ) H 2 is a positive integer. We first treat with the case that H ch 1 (Ẽ 1 ) H 2 ≥ 2. By Lemma 3.2, β 1 ≥ −1 + 1/r . Applying the same argument as in Lemma 3.4 implies that the polygon P ι * E is contained in the triangle op q where the slope of qp is −1+1/r (−1+1/r ) and the vertical coordinate of the point p is equal to 2, see Fig. 6.
Denote byp the point along the line p q with the vertical coordinate equal to 1. The coordinates of two points p andp are Note that the length qp does not depend on d, The horizontal coordinate of p is negative and is bigger than −g + r + 2. Thus if r ≥ 4, we have This implies l(E) ≤ op + p q ≤ g(r − 2) + 2 g r + r + 2, so inequality (21) holds. Now assume ch 1 (Ẽ 1 ) = H . By Lemma 3.2(b), we have β 2 ≤ 1 r . Therefore We consider three different cases: We first assume r ≥ 5, then the pointλ:= (λ, 1) lies on the right hand side of the line segment op . In Case I, we have which implies op 1 ≤ g r + r . For Case II, write s:= − ch 2 (Ẽ 1 ) − g r + r , then 1 ≤ s ≤ g 2r and Thus op 1 ≤ g r + s + r − 1. For s = g 2r , we indeed have To provide an upper bound for the length p 1 p , we define the function If 0 ≤ x ≤ g 2r , one can easily show that where δ = 1 if x ∈ 0, g r 2 and δ = 2 if x ∈ g r 2 , g 2r .
In Case I, we know the point p 1 lies on the right hand side of op , so the length of p 1 p is maximum when the horizontal coordinate of p 1 is maximum. But the horizontal coordinate of p 1 is less than or equal to d − 2(g − 1) + g r − r because p 1 lies on the left hand side ofp, see Fig. 6, thus In Case II, the length of p 1 p is maximum when the horizontal coordinate of p is minimum, i.e. d is minimum, hence Here s = − ch 2 (Ẽ 1 ) − g r + r as before. Similarly, the length ofλ p is maximum when d is minimum, so Now we apply the above upper bounds to prove inequality (21). In Case I, inequalities (23), (28) and (29) imply that Thus inequality (24) implies so inequality (21) holds. Similarly, in Case II, inequalities (23), (28) and (30) imply that Therefore inequality (25) implies that thus again inequality (21) holds. Finally in Case III, we have Summing up inequalities (23), (26), (28) and (31) show that inequality (21) is satisfied. Finally, we consider the case r = 4. If | ch 2 (Ẽ 1 )| ≤ g 4 − 4, then p 1 lies to the right of op and the same argument as in the Case I above implies the claim. Otherwise, (E) ≤ oλ + λ p + p q whereλ = (λ, 1) for λ = − g 4 + 3. Note thatλ lies to the right of op . We know that the length ofλ p will be maximum when d is minimum so The second inequality follows from (28) for x = 1. Summing up the above inequality with (23) proves our claim (21).
Step 3 We show h 0 (C, E) < 2r if d < 2(g − 1) − 2 g r − r . By using the same notations as in Step 2, we first consider the case ch 1 (Ẽ 1 ) = H . By Proposition 3.3, it suffices to show that One can easily check that the function The last inequality comes from inequality (23) and some direct computations. Thus we may assume ch 1 (Ẽ 1 ) = H . If p 1 is to the left of the line segment op , then the total sum of p i p i−1 is also bounded by op + p q . So we may always assume the polygon op 1 p q is convex.
Define t:=p (x) − ch 2 (Ẽ 1 ) − g r + g r , wherep (x) = d − 2(g − 1) + g/r − r is the horizontal coordinate of the pointp. We consider two different cases: Case I when 0 ≤ t < g 2r , In particular, if t ∈ g r 2 , g 2r , we can improve the bound by 1; Note that p 1 p = f (t) as that defined in (27). Thus combining inequality (28) for t ∈ [0, g 2r ] and inequality (23), we get Hence the claim follows by Proposition 3.3. Case II Suppose g 2r ≤ t. If p 1 lies on the left hand side of op , the polygon P ι * E is inside the triangle op q and the claim follows from (32). Otherwise, the polygon op 1 p q is convex and the summation of the length op 1 + p 1 p + p q is maximum when t = g 2r . Substituting t = g 2r into the formulas of op 1 and p 1 p , we have: Together with (23) for p q , it follows that so the claim follows.

Smooth plane curves
Our method to control the dimension of global sections of semistable vector bundles (first part of Theorem 1.1) can be generalized to curves on more general surfaces, especially for Fano surfaces. As a case study, we follow the argument for curves on K3 surfaces to set up a bound for smooth projective plane curves and finally compute their Clifford indices. We first review Bridgeland stability conditions on the projective plane.

Review: space of geometric stability conditions on D b (P 2 )
The space of geometric stability conditions on the projective plane P 2 is similar but slightly different with that of a K3 surface with Picard number one. In the projective plane case, the curve is replaced by the Le Potier curve (see [6,7,12,20]). Since the definition of Le Potier curve is rather involved, we will only use a simpler version˜ which is enough for our purpose. Let˜ (x) := 1 2 x 2 −γ (x). By abuse of notations, we also denote the graph of˜ by the curve˜ .

Upper bound on the dimension of global sections
Let C be a degree l smooth irreducible curve in the projective plane P 2 . Denote ι: C → P 2 the embedding morphism and H := O P 2 (1). We recollect lemmas from the case of K3 surfaces. The next lemma generalizes [8, Lemma 3.2] to objects in D b (P 2 ). (d) If a 1 /b 1 ∈ I , a 2 /b 2 ∈ J and (a 1 + a 2 )/(b 1 + b 2 ) ∈ J , then there is a non-negative real number k < 1 such that (a 1 + ka 2 )/(b 1 + kb 2 ) = −1, then case (c) implies that Therefore, case (b) gives Now assume the wall W intersects the parabola with the equation y = x 2 /2 at two points (β 2 , β 2 2 /2) and (β 1 , β 2 1 /2) where β 1 < 0 < β 2 . By applying the same argument as in Lemma 3.1, the inequality (41) gives β 2 − β 1 ≤ l. Proposition 2.4 implies that the slope of the wall W is Therefore the wall W is below the line L which has the same slope as W and passes through the point (1 − l, (l−1) 2 2 ). The line L intersects the line x = 1 at the point