Higher rank Clifford indices of curves on a K3 surface

Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in |H|$ 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 $g\geq r^2\geq 4$, 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 $d(\geq 5)$ smooth plane curve is $d-4$, 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 Theorem 1.1. Let (X, H) be a smooth polarized K3 surface with Pic(X) = ZH, 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 .
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 [BPGN95, Mer99,Mer01], which are also included in [LN15a]. Sharp bounds for the case g ≤ 6 and µ < 2 have been determined in [BPGN95,Mer99,Mer01,LN15a,LN15b,LN17]. The upper bound (1) is in general stronger than the bounds in these previous papers unless g ≤ 6 or µ ≤ 2.
This indicates the failure of the Mercat's conjecture in [Mer02] for C which states the higher ranks Clifford indices of the curve C are equal to Cliff(C). 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 in [GL87] 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 inequality. Examples of such surfaces include the projective plane, del Pezzo surfaces and quintic surfaces. We explain more details for smooth plane curves in Section 5. In particular, we show that the first part of the Mercat's conjecture [Mer02] holds for smooth plane curves: Theorem 1.2 (Corollary 5.6). Let C be a degree l(≥ 5) smooth irreducible plane curve, then Cliff r (C) = l − 4, for any positive integer r.
Another concrete example for curves on degree four del Pezzo surfaces is computed in [Li18]. The Clifford type inequality for such curves is the key ingredients in proving the existence of Bridgeland stability conditions on smooth quintic threefolds.
1.1. Approach. The main tool in this paper is the notion of stability condition introduced by Bridgeland [Bri07]. 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 [Fey17], 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.
Generalization. All our results hold slightly more general for a polarized K3 surface (X, H) satisfying the following: Assumption (*): H 2 divides H.D for all curve classes D on X.
To simplify the presentation, we explain our entire argument in the case of Picard rank one and then explain in Section 4.2 how to extend the arguments to this situation.
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 [HRS96,Bri08], 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 : K(X) → C by Note that the function Z β,α , up to the action of GL + (2; R), is the same as the stability function defined in [Bri08, 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, βH, α). Definition 2.1. Let γ : R → R be a 1-periodic function such that for x ∈ [− 1 2 , 1 2 ] is defined as Let Γ(x) := H 2 2 x 2 − γ(x). By abuse of notations, we also denote the graph of Γ by curve Γ (see Figure 1).
We first state Bridgeland's result describing stability conditions on D b (X), and then expand upon the statements.
(a) For a stable object E with respect to any stability condition σ β,α , by [BM14, Theorem 2.15], the point pr(E) is not in The slope ν β,α (E) is just the slope of the line crossing points (β, α) and pr(E).
Proposition 2.5 ([Bri08, Proposition 9.3]). Fix an object F ∈ D b (X). There exists a collection of line segments (walls) W i F in Γ + with the following properties: • the end points of all segments are either on the curve Γ or the on the line segment through (n, H 2 2 n 2 ) to (n, H 2 2 n 2 − 1); • the extension of each line segment passes through pr(F ) if rk(F ) = 0; otherwise it has slope ch 2 (F )H 2 /H. ch 1 (F ); • 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 Figure 2 for a picture and [Fey17] for more details and further references.

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.
Lemma 3.1. Adopt notations as above, we have 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 , we must have ch(ι * E) = k ch(F 2 ) for some real number k = 0 and 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 )) = tH. Since E is of rank r, to make the sequence exact at the term ι * E, we must have Therefore, By Proposition 2.5, the object F 1 is semistable of the same phase as ι * E along the line segment W, in particular it is in the heart Coh β 1 +ǫ X where ǫ → 0 + . Thus by definition of the tilting heart, Therefore inequality (5) implies Similarly, Proposition 2.5 gives Therefore, β 2 − β 1 ≤ 1. By the second property of Proposition 2.5, the slope of W as a line in the projection pr(H * alg (X, R)) is 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 Solving the equation, we get β min We need the following description for the first wall in details.
Proof. Adopt the notations as in the proof of Lemma 3.1.
2. An upper bound on the dimension of global sections. We first recall the result in [Fey17, Section 3]. Define the function Z : K(X) → C as We also define the following non-standard norm on C: The next proposition bounds the dimension of global sections of objects in terms of the length of a polygon.
Proposition 3.3 ([Fey17, Proposition 3.4]). Let F ∈ Coh 0 X be an object which has no subobject (a) There exists ǫ > 0 such that the Harder-Narasimhan filtration of F is a fixed sequence 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 ǫ > 0 such that the Harder-Narasimhan filtration of ι * E with respect to the stability condition σ 0,α for positive α < ǫ 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 /Γ(β 2 ) and the slope of pq is β 1 /Γ(β 1 ), where the real numbers β 1 and β 2 are defined as in Lemma 3.1.
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 that E 1 and ι * E have the same phase. Proposition 2.5 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 assumption, the line L is lower than the wall W for ι * E, see Figure 4. SinceẼ 1 is σ 0,α -semistable where α → 0 + , the point pr(Ẽ 1 ) is not in Γ + . 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 vector bundle E. . Lemma 3.1 implies that the triangle opq is inside the triangle

FIGURE 4. Comparing slopes
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 One can easily show that δ < 2r g and the claim follows.
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 , there exists a stable vector bundle F on X with Chern characters: The restriction F ⊕t | C is semistable (by [Fey16, Theorem 1.1]) 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 polarized smooth K3 surface with Pic(X) = ZH. Let the smooth curve C ∈ |H| be with genus g, E be a semistable vector bundle with rank r and degree d (0 ≤ d ≤ r(g − 1)). Then Cliff(E) > d r − d 2 g 2r 2 (g−1) 2 − 2 g and 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 d > 2r(g−1) 3 2 g . When d ∈ 2r(g−1) 3 2 g , r(g − 1) and g ≥ 3, the function f (d) = d r − d 2 g 2r 2 (g−1) 2 − 2 g reaches its minimum at the left boundary. Therefore, 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 main result in [Fey16], which constructs semistable vector bundles on C by restricting vector bundles on X with low discriminant. By [BM14, Theorem 2.15], there exists a slope stable sheafẼ r on X with Chern character (r, H, g r − r). Define E r :=Ẽ r | C . Theorem 4.1 ([Fey16, Theorem 1.1]). Assume g ≥ max{r 2 , 6} and r ≥ 2, then the sheaf E r is a semistable vector bundle on C with h 0 (C, E r ) ≥ 2r and Proof. We first checkẼ r satisfies conditions in [Fey16, Theorem 1.1]. IfẼ r is not locally free, then the double dual F =Ẽ ∨∨ r is slope stable with Chern characters (r, H, g r − s) for some integer s ≤ r − 1. Yet −χ(F, F ) = H 2 − 2r g r − s − 2r 2 < −2. This contradicts [BM14, Theorem 2.15]. By the assumption on r and g, we have Therefore [Fey16, Theorem 1.1] implies that E r is slope semistable on C.
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 r (g − 1) − 2 r g r .
We first treat with the case that ch 1 (Ẽ 1 ) = H. In this case, as H ch 1 (Ẽ 1 ) H 2 is a positive integer, 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 Γ(−1+1/r) and the vertical coordinate of the point p ′ is equal to 2, see Figure 5.
FIGURE 5. The polygon p ι * E is inside the polygon op 1 p ′ q.
In this case, the point p ′ is on the left hand side of p 1 , so the length of p 1 p ′ is maximum when d is minimum and the horizontal coordinate of p 1 is maximum, i.e. d − 2(g − 1) + g r − r, thus Inequalities (14) and (16) for s = 0 imply that Therefore, l(E) ≤ ⌊ op 1 ⌋ + ⌊ p 1 p ′ + p ′ q ⌋ ≤ (r − 2)(g − 1) + 2 g r + r + 2, so again inequality (12) holds.
Case II. when 1 ≤ s ≤ g 2r , then The point p ′ is still on the left hand side of the point p 1 , so the length of p 1 p ′ is maximum when d is minimum. Therefore, p 1 p ′ ≤ f (s). Combining inequalities (14) and (16) implies that as we required.
Case III. when g 2r ≤ s, the summation of lengths op 1 + p 1 p ′ + p ′ q is maximum when d is minimum and ch 2 (Ẽ 1 ) is maximum, i.e. s = g 2r . In this case, Together with inequality (17) for s = g 2r , we have Since l(E) ≤ ⌊ op 1 + p 1 p ′ + p ′ q ⌋ inequality (12) is satisfied.
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 (14) and some direct computations. Thus we may assume ch 1 (Ẽ 1 ) = H. 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: Moreover, p 1 p ′ = f (t) defined in (15). Thus combining inequality (16) for t ∈ [0, g 2r ] and inequality (14) give Thus the claim follows by Proposition 3.3.
Thus for −g + 4 < x ≤ 0, we have Cliff(E) ≥ g − 1 − g 2 . If x ≤ −g + 4, then again Proposition 3.3 gives Therefore the second part of Theorem 1.1 for r = 2 follows by the fact that Cliff 2 (C) ≤ Cliff 1 (C) = g − 1 − g 2 . 4.2. Higher Picard number case. Theorem 1.1 still holds when the ample divisor H satisfies Assumption (*). Assumption (*): H 2 divides H.D for all curve classes D on X. We explain how to adapt all our arguments from Picard rank one to this more general case. Let Consider stability conditions for which the central charge factors via v H , and denote the space of such stability conditions by Stab H (X). The pair σ β,α := Coh β X, Z β,α defines a stability condition on D b (X) and there is a continuous map from Γ + → Stab H (X). The slope function ν β,α is defined in the same way. All the propositions in Section 2 hold for the higher Picard rank case. The Chern characters in part (a) in Lemma 3.2 should be modified to H. ch(F 1 ) = H 2 . All the other statements do not rely on the Picard rank.

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. 5.1. 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 Γ is replaced by the Le Potier curve (see [DLP85,CHW14,Li17,LZ16]). Since the definition of Le Potier curve is rather involved, we will only use a simpler versionΓ which is enough for our purpose.
For β ∈ R and α >Γ(β), we define the central charge Z β,α : K(P 2 ) → C as By [Li17, Proposition 1.10], we get a slice of stability conditions σ β,α = (Coh β P 2 , Z β,α ) parametrized byΓ + . Results of stability condition and wall-crossings (Theorem 2.2, Remark 2.4 and Proposition 2.5) all hold without any change. One should be cautious that the end points of the first wall may not be on the curveΓ.

5.2.
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 [Fey17, Lemma 3.2] to objects in D b (P 2 ).
Proof. The second claim follows clearly by definition. To prove the first claim, we consider four different cases.
Theorem 5.5. Let C be a degree l(≥ 5) smooth irreducible curve on the projective plane. Let E be a semistable vector bundle with rank r and degree d such that 0 ≤ d ≤ rl(l − 3)/2. Then dim H 0 (C, E) ≤ r + 3 2l + d 2rl 2 d if d ≥ rl max{3r + d − rl, r + rl+r rl 2 −d d} if r(l − 1) ≤ d < rl Proof. When d ≥ rl, Lemma 5.4 implies that the polygon P ι * E is inside the triangle opq wherẽ p = ( d 2 2rl 2 , d l ) and q = (− rl 2 2 + d, rl). Then Lemma 5.3 and convexity of the polygon P ι * E imply that When d < rl, if the range of the slopes in Lemma 5.4 is given by d 2rl − l 2 , − 1 2 , then we may let p be at Also if the range of the slopes in Lemma 5.4 is given by − l−1 2 , d r − l + 1 2 , then we may letp be at d − rl + r 2 , r . Therefore, which completes the proof.
As an interesting consequence, part i) of Mercat conjecture ( [Mer02]) holds for smooth plane curves.