Slope inequalities and a Miyaoka-Yau type inequality

We prove several slope inequalities for a relative minimal surface fibration in positive characteristic. As an application, we prove a Miyaoka-Yau type ineqaulity $\chi(\sO_S)\ge\frac{p^2-4p-1}{4(3p+1)(p-3)}K_S^2$ for all minimal surface $S$ of general type in characteristic $p\ge 5$ and the equality holds for Raynaud's examples. Some similar inequalities are also established for $p=2,3$, which answer completely a question of Shepherd-Barron on the positivity of $\chi(\sO_S)$.


Introduction
Let S be a smooth projective surface over an algebraically closed field k of characteristic p > 0. By a fibration of S over a smooth projective curve C, we mean a flat morphism f : S → C such that f * O S = O C . In particular, the general fibre of f : S → C could be singular in case of positive characteristic. Let ω S/C := ω S ⊗ f * ω −1 C be the relative canonical sheaf of f , and K S/C := K S − f * K C be the relative canonical divisor. Then K 2 S/C = K 2 S − 8(g − 1)(b − 1) and χ f := deg(f * ω S/C ) = χ(O S ) − (g − 1)(b − 1) + l where l := dim k (R 1 f * O S ) tor is the torsion dimension of the direct image sheaf R 1 f * O S . The fibration f is called relatively minimal if S contains no (−1)-curve in fibres. Then we have the so called slope inequality.
Theorem 1. Let f : S → C be a relatively minimal fibration, then When char.(k) = 0, this inequality was proved by Xiao (see [14])and independently by Cornalba-Harris for semi-stable fibration (see [2]). When char.(k) = p > 0, there exist a few approaches to this inequality (see [10], [15], etc). We point out that all of them require the condition that the generic fibre of f : S → C is smooth. In a recent paper [13], the authors have generalized Xiao's approach to cover the case of characteristic p > 0. Although their theorem contains the assumption that f has smooth generic fibre, their method can be generalized to cover arbitrary relatively minimal fibration f .
For a minimal smooth projective surface S of general type over an algebraically closed field of characteristic p > 0, one still has K 2 S > 0 by Bombieri-Mumford's classification, and the Noether formula  [12,Theorem 8]) that there is a fibration f : S → C with singular general fibres of arithmetic genus 2 ≤ g ≤ 4 and p ≤ 7. But the question if there exist surfaces S of general type with χ(O S ) < 0 ? remains unsolved (see the Remark at page 268 of [12]). Shepherd-Barron also suggested that the most obvious place to look for such examples would be in the case where (p, g) = (2, 2). Indeed, it is proved in [3] that χ(O S ) > 0 when p ≥ 3. As an application of Theorem 1, we have Theorem 2. Let S be a minimal smooth projective surface of general type over an algebraically closed field k of characteristic p > 0. Then (1) if p ≥ 5, then we have the optimum inequality S . An example is given at the end of § 5 where the equality holds. In particular, χ(O S ) > 0 holds for all smooth projective surfaces S of general type.
The above application shows that characteristic p version of slope inequalities may be useful tools for the study of algebraic surfaces in positive characteristic. Thus it is deserved to generalize the following result for non-hyperelliptic fibration to positive characteristic by using the characteristic p version of Xiao's approach and some ideas similar to null characteristic.
(1) f is non-hyperelliptic and g = 3: K 2 S/C ≥ 3χ f ; (2) f is non-hyperelliptic and g = 4: K 2 S/C ≥ 24 7 χ f ; (3) f is non-hyperelliptic and g = 5: K 2 S/C ≥ 40 11 χ f . These inequalities over complex field C was proved by: (1) is by Horikawa (see [4]) and Konno (see [5]). (2) & (3) was proved by Konno (see [6]) and (2) is independently proved by Chen (see [1]). We would like to remind the readers that in characteristic p > 0, both sides of all these inequalities along with Xiao's can be negative. As a consequence and for example, in case f is non-hyperelliptic and g = 3, one can not say that (1) is better than Xiao's. This paper is organized as follows. Some notations are assumed and Clifford's Theorem for Gorenstein curve is recalled in § 2. In § 3, we recall the characteristic p version of Xiao's approach and point out that smoothness of its generic fibre is unnecessary. In § 4, we prove several slope inequalities for non-hyperelliptic fibrations in positive characteristic. In § 5, we apply slope inequality to study algebraic surfaces S of general type with c 2 (S) < 0. As a consequence, we prove that χ(O S ) > 0 for any characteristic p > 0 which answer a question of Shepherd-Barron. Finally, Raynaud's examples are presented in § 6 and we show that if p ≥ 5, the equality of holds when S is one of Raynaud's examples.

Notations
Let us assume the following notions: • k: an algebraically closed field of characteristic p > 0; • C: a proper nonsingular algebraic curve of genus b := g(C) over k; • S: a proper nonsingular algebraic surface over k; particular, all geometric fibres are connected; • K := K(C) the function field of C; • c: a general closed point of C ; • F := S c a general closed fibre of f , in particular, it is integral; • ω S/C : the relative dualizing sheaf and K S/C the relative canonical divisor; • g := p a (F ) the arithmetic genus of fibre F ; Definition 1. An irreducible, reduced and Gorenstein curve over k is called hyperelliptic if it admits a double cover to P 1 k . The above fibration f : S → C is called hyperelliptic if the general fibre F is so.
Theorem 3 (Clifford's Theorem, see [8]). Let D be a special Cartier divisor on an irreducible, reduced and Gorenstein curve V , then and the equality holds only if either D = 0 or D = K V , or V is hyperelliptic.

Remark 1.
(1) We have l = 0 unless there is some multiple fibre of multiplicity divided by p.
(2) We do not assume the generic fibre F is smooth unless otherwise stated in this paper.

Xiao's approach of slope inequality in positive characteristic
In the paper [14], Xiao introduces an approach of slope inequality for surface fibrations by studying the Harder-Narasimhan filtration of the direct image f * ω S/C . For readers' convenience, we briefly recall the idea.
For a vector bundle E on a smooth projective curve C, let where rk(E) and deg(E) denote the rank and degree of E respectively. E is called semi-stable (resp., stable) if for any nontrivial subbundle E ′ E, one has µ(E ′ ) ≤ µ(E) (resp., <). If E is not semi-stable, one has the following well-known theorem.
which is the so-called Harder-Narasimhan filtration, such that By using Harder-Narasimhan filtration of f * ω S/C , Xiao constructs a sequence of effective divisors Then he use the following elementary lemma to get a lower bound of K 2 S/C .
and a sequence of rational numbers It is important that semi-stability of E i /E i−1 implies nefness of where Γ i is a fibre of P(E i ) → C when char.(k) = 0. A key observation of [13] is that strongly semi-stability of E i / i−1 implies nefness of Frobenius morphism, a bundle E on C is called strongly semi-stable (resp., stable) if its pullback by k-th power F k C is semi-stable (resp., stable) for any integer k ≥ 0.

Lemma 3. For each sub-bundle E i in the Harder-Narasimhan filtration
Proof. The proof follows a slight modification of its null characteristic counterpart in [9, Theorem 3.1] with the notification that pull-backs of strongly semi-stable bundles under a finite morphism are still strongly semi-stable.
Then Xiao's approach applies in positive characteristic, the assumption that f : S → C has smooth general fibre in the previous work [13, Theorem 6] is unnecessary. Indeed, we have Theorem 5. Let f : S → C be a relatively minimal fibration over an algebraically closed field of positive characteristic, then Proof. One can prove it by following the same argument in the proof of [13,Theorem 6], and we give a sketch here. Take the following commutative diagram (for any integer k ≥ k 0 ): which is a torsion-free sheaf of rank 1 and is locally free on an open set U i ⊂ S of codimension at least 2. Thus there is a morphism (over C) Then we get a sequence of effective divisors Z 1 ≥ Z 2 ≥ · · · ≥ Z n ≥ 0 and a sequence of rational numbers µ 1 > µ 2 > · · · > µ n such that since ω S/C | F is generated by global sections and Z n is supported on fibres of f : which and Lemma 1 imply the slope inequality.

Remark 2. From above proof, we see that the equality holds only if
• either the general fibre F is hyperelliptic; • or f * ω S/C is strongly semistable.
Proposition 1. Assume χ f > 0, then the slope inequality holds without equality unless f is hyperelliptic.
Proof. From above remark, we see that Thus there is no equality when both χ f > 0 and f is non-hyperelliptic.
Remark 3. The RHS of (3.2) in [13, Proposition 1] should be replaced by where l is the torsion dimension of R 1 f * O S . However this mistake does not affect the proof and its conclusion.

Slope inequalities for non-hyperelliptic fibrations
Over C, after Xiao's approach, another technique named counting relative hyperquadrics is used in developing the slope inequality. In the paper [6], such a method is systematically studied and the following slope inequalities for non-hyperelliptic fibrations are proved.
(1) f is non-hyperelliptic and g = 3: K 2 S/C ≥ 3χ f (see also [4], [5]); (2) f is non-hyperelliptic and g = 4: (3) f is non-hyperelliptic and g = 5: The method in [6] can however be modified to work in positive characteristic to obtain the same slope inequalities. For the restriction of context, we shall not go through all the detailed arguments for above three inequalities. Instead, we mainly take the case g = 3, 4 as an example to show how Konno's method in [6] is modified and applied in positive characteristic.
In the following we assume that f : S → C is a non-hyperelliptic fibration of genus g defined over an algebraically closed field k of characteristic p > 0. The main result in this section is the following generalization of slope inequalities.
Theorem 6. Let f : S → C be a non-hyperelliptic fibration over an algebraically closed field k of positive characteristic. We have

Remark 4.
In the original statement of the three slope inequalities above, the gonality of F is considered. Here for simplicity, we shall not go that much further.
Recall, for non-hyperelliptic fibration f : S → C, the second multiplication map: 4.1. Genus 3. When g = 3, the second multiplication map is an isomorphism outside finitely many points on C, so we have: Combining this inequality with (1), we immediately have χ f in this case is proved for complex field C in both [1] and [6]. We shall here mainly follow the idea of [6] which can shed more lights on g = 5.
Given a vector bundle E on C, we shall use the following notations: • π : P(E) → C is the projective bundle associated to E; • D is a general fibre of π, it is a projective variety of dimension rank(E) − 1; • a relative hyperquadric Q associated to δ ∈ Pic(C) is an effective divisor in the linear system |O(2) ⊗ π * δ| on P(E). • for a relative hyperquadric Q, its rank is defined to be rank of the smallest subbundle E ′ ⊆ E such that Q is defined over E ′ .
In the paper [6], the author firstly gives several key lemmas (Lemma 1.1-1.6) on relative hyperquadrics in characteristic zero. By using of the existence of Harder-Narasimhan filtrations with strongly semi-stable sub-quotients (see Lemma 2), we can generalize these lemmas to the case of positive characteristic.
Proof of Theorem 6 when g = 4. We may firstly assume χ f > 0, otherwise, Xiao's inequality (see Theorem 5) K 2 S/C ≥ 3χ f is already stronger than the desired one.
Take E = f * ω S/C for short. Consider the second multiplication map, we have the following exact sequence: where K is an invertible sheaf and C is a skyscraper sheaf. It therefore gives a relative hyperquadric Q ∈ |O(2) ⊗ π * K −1 |. Let x := deg K, (3). In particular, We can next assume that E is not strongly semi-stable, otherwise by [13,Proposition 1]. Using µ i , ν j introduced above, we can still apply the trick of Xiao as in § 3. Indeed, we have and where d i = N i · F is defined as in § 3. Here we remark that the inequality (6) is obtained using the similar trick of Xiao with filtration 0 E 1 E. Now, combining inequalities (4), (5) and (6), we can prove the desired inequality after a case-by-case discussion as below.
Proof. By [6, Lemma 1.4], we have r n−1 ≥ rank(Q) = 3, hence r n−1 = 3. In this case d n−1 < 6 only if the general fibre F pass through the vertex of the cone Q ∩ D. This is however not the case.
Case n = 3. Since r 2 = 3, d 2 = 6 by the previous lemma, we have On the other hand we have x ≤ µ 1 +µ 2 by Lemma 4. Hence x ≤ 1 6 K 2 S/C again holds. The same trick then gives K 2 S/C ≥ 24 7 χ f .
Case n = 4. In this case, we have d 1 = 0, d 2 ≥ 3 (see Clifford's Theorem) and d 3 = 6 by the previous lemma. Hence we have . Again, we have Lemma 4 that x ≤ min(µ 1 + µ 3 , 2µ 2 ). In particular x ≤ 1 6 K 2 S/C holds again and the same trick then applies.
4.3. Genus 5. One can again follow the argument of Konno (see [6]) to obtain K 2 S/C ≥ 40 11 χ f with his key lemmas modified. For the restriction of context, we shall not go through the details here.

Application of Xiao's slope inequality: Miyaoka-Yau type inequality in positive characteristic
Over the field of complex numbers C, the celebrated Miyaoka-Yau inequality asserts c 2 1 (S) ≤ 3c 2 (S) (7) holding for all complex algebraic surface S of general type. Using Noether's formula: 12χ(O S ) = c 2 1 (S) + c 2 (S), Miyaoka-Yau inequality can be rewritten as: in terms of χ(O S ) and K 2 S for all complex algebraic surface S of general type.
In positive characteristic, there are minimal algebraic surface S of general type with c 2 (S) < 0 (see Raynaud's examples in § 6), while c 2 1 (S) > 0 by Bombieri-Mumford's classification of algebraic surfaces. Henceforth, the original form (7) of Miyaoka-Yau inequality does not hold in positive characteristic. As Miyaoka-Yau inequality is such a powerful tool over C, one would naturally like to find a certain modified inequality of similar form as (8) in positive characteristic. In the paper [12], Shepherd-Barron also asked whether χ(O S ) > 0 holds true for all algebraic surface S of general type in any characteristic p > 0 and later the first author of the present paper (see [3]) raised the question whether we can find an inequality of the same form as Miyaoka-Yau inequality (8): S , κ p > 0. in each characteristic. Once such an inequality exists, Shepherd-Barron's question is solved. Following the work of Shepherd-Barron (see [12]), such inequalities are given for each p ≥ 3 in [3] and meanwhile a conjecture of the optimum inequality for p ≥ 5 is given: holds for any algebraic surface S over an algebraically closed field of characteristic p ≥ 5.
In this section, we solve Shepherd-Barron's question and prove above conjecture completely by applying the characteristic-p version of Xiao's inequality (see Theorem 5). From now on, we always assume that S is a minimal algebraic surface of general type over an algebraically closed field k of characteristic p > 0.
To proceed, we firstly recall a fundamental theorem on the structure of algebraic surfaces of general type with negative c 2 due to Shepherd-Barron. • the geometric generic fibre is a singular rational curve with cusp singularity.
Let ∆ be the divisorial part of the nonsmooth locus of f : S → C, which has be studied in [3]. An important fact is • the horizontal part ∆ h of ∆ is nonempty and consisting of divisors inseparable over the base C. As a consequence, we see that Proposition 2. If f has a multiple fibre, then its multiplicity is divided by p.
By abuse of language, we call such f : S → C as the Albanese fibration of S. As an application of Theorem 5, we have Theorem 8. Let S be a minimal algebraic surface of general type over an algebraically closed field k of characteristic p > 0 with c 2 (S) < 0. Then we have where g is defined as in Theorem 7.
Before our proof, we recall the following numerical relations.
where l : Proof. Applying Theorem 5 to the Albanese fibration f , one has which is equivalent to After simplification, the inequality (13) turns into 2g + 1 3g and it implies that by the inequality (see [3, (3.3)]): Now, it is clear that or equivalently, Colloary 1. Let S be a minimal algebraic surface of general type over an algebraically closed field k of characteristic p > 0. Then (1) if p ≥ 5, The equality holds for Raynaud's examples (see § 6).
Proof. In order to prove our results, note that if c 2 (S) ≥ 0 Noether's formula implies χ(O S ) ≥ 1 12 K 2 S , it is enough to consider that c 2 (S) < 0.
Since we always have g ≥ When p = 2, 3, the inequality follows immediately from Theorem 8 by that g ≥ 2. We are done.
In the rest of this section, we concentrate into the case p = 2, 3.
Proof. According to the previous corollary and theorem, the equality holds only if: (1) c 2 (S) < 0; (2) the Albanese fibration (see Theorem 7) f : S → C is a genus 2 fibration; (3) when p = 3, each fibre of f is irreducible and reduced (by equality (14) and Proposition 2). We shall show that above three conditions lead to a contradiction. Note that the relative canonical map gives a morphism: π : S → P(f * ω S/C ) since each fibre of f is irreducible and reduced, and such morphism π is necessarily a separable flat double cover (see [3, § 2]). Let M P(f * ω S/C ) be the branch divisor of π, which satisfies: • M is a smooth, horizontal divisor and [M : C] = 6; • each component of M is inseparable over C; • for each point c ∈ C, its inverse image in M has exactly two points as a set. In fact, otherwise there is some c has one inverse image. Then the fibre of f at c by construction is a flat double cover of P 1 k branching at a single point of multiplicity 6. Such fibre is clearly not irreducible.
where F M is the frobenius morphism and v is anétale double cover.
Indeed we only need to consider the case A), since replacing C by the base change v above which is anétale double cover, the case B) can be turned into A).
Finally we go to exclude case A). Let Σ be the divisor class O(1) of P(f * ω S/C ), and M i ∼ num 3Σ + u i F for i = 1, 2. Recall that Σ 2 = deg f * ω S/C = χ f , and we have Thus u 1 = u 2 and b = 1, which is a contradiction. (1) f : S → C is a genus 2 fibration; (2) any fibre of f is irreducible, singular and rational.  Finally, we go to construct an explicit surface fibration f : S → C satisfying conditions (1) and (2) of Lemma 8. Define C to be the quintic plane curve defined by homogeneous equation: over an algebraically closed field k of characteristic p = 2. One can easily check that C is a smooth curve of genus b := g(C) = 6. There are two affine subset C i (i = 0, 1) of C as below: For simplicity, we introduce the following notations: • ∞ is the point (0, 1, 0) which is the complement of C 0 in C;

One can check immediately that
• dx (resp., dx ′ ) is a generator of Ω C over C 0 (resp., C 1 ); • C 10 is the open subset where x and x ′ is invertible; • C ′ 1 is the open subset where 1 + z ′3 is invertible. Over C 0 : S is defined as Here the superscript on each element is its homogeneous degree. Over C ′ 1 : S is defined as in the weighted projective space . The homegeneous translation relation is given by and this construction makes sense because that Moreover, one can check that S is a non-singular surface as following: (Over C 0 ) : When T 0 = 1, take s = S 0 T 0 , then the function defining S is It is even smooth over C 0 by Jacobian criterion. When S 0 = 1, take t = T 0 S 0 , then the function defining S is Since dx is a generator of Ω C on C 0 , S is smooth over k by Jacobian criterion. (Over C ′ 1 ) : When T 1 = 1, s 1 = S 1 T 1 , then the function defining S is It is even smooth over C ′ 1 by Jacobian criterion. When S 1 = 1, As 1 1 + z ′6 is invertible on C ′ 1 and dx ′ is a generator of Ω C , S is smooth over k by Jacobian criterion.
On each fibre, the unique singularity lies on the infinity T 0 = 0 or T 1 = 0, and up toétale equivalence the singularity is as the following cusp One can also find out that each fibre F of the fibration f : S → C is irreducible, rational and p a (F ) = 2. Thus, by Lemma 8 we have In fact, we have χ(O S ) = 1 and K 2 S = 32 in above example. We would also like to indicate that S is given from C × P 1 by taking quotient of the foliation D = s 6 ∂ ∂s + ∂ ∂x , where s is the parameter of P 1 .

Raynaud's examples
In the paper [11], Raynaud constructed a class of pairs (S, L), where S is a smooth projective algebraic surface in positive characteristic and L is an ample line bundle on S such that H 1 (S, L) = 0. These pairs then give counterexamples to Kodaira's vanishing theorem. Honestly, Raynaud's example is so special that it does not only violates Kodaira's vanishing theorem, but also leads to many pathologies in positive characteristic.
We now briefly recall his example, and one can also refer to [3, § 4]. Fixing an algebraically closed field k of characteristic p > 2. Let (C, P = P(E), Σ) be a triple satisfying: • C is a projective nonsingular curve over k of genus b; • π : P = P(E) → C is the P 1 -bundle over C associated to a rank-2 locally free sheaf E; • Σ P is a reduced horizontal divisor consisting of two irreducible components Σ i (i = 1, 2) such that X 1 ) π : Σ 1 → C is an isomorphism, or equivalently Σ 1 is a rational section of π; X 2 ) π : Σ 2 → C is isomorphic to the Frobenius morphism; According to the computations in [11], one can easily check that such data (C, P, Σ) is equivalent to the pair (C, f ) with the same curve C and a rational function f ∈ K(C)\K(C) p with div(df ) = pD for some divisor D on C. As a (non-direct) consequence, one can also prove that Σ is an even divisor on P .
Example 1 (Artin-Schreier curve). Let C be the complete normal curve associated to the following plane equation: where ∞ ∈ C is the unique point at infinity. In particular, such example of (C, f ) or equivalently (C, P, Σ) satisfying above properties does exist in any characteristic p > 2.
Definition 2. Let S be a smooth projective surface over k and σ : S → P be any flat double cover with branch divisor Σ where (C, P, Σ) satisfies above properties. We call such a fibration f : S → C as one of Raynaud's examples.
One can directly check that such a fibration f satisfies the following properties: a) the surface S itself is nonsingular over k; b) every geometric fibre of f is a singular rational curve of arithmetic genus p a = p − 1 2 ; c) every geometric fibre is hyperelliptic and integral (irreducible & reduced).
Lemma 9. Suppose f : S → C is any surface fibration with above properties a), b) and c). Then S must be one of Raynaud's examples.
Proof. Let ρ be the hyperelliptic involution, and σ : S → P := S/ρ be the quotient map. Then condition c) implies that the canonical homomorphism π : P → C has integral fibres. Note that π : P → C is birational to a ruled surface (recall that K(C) is C1 by Tsen's Theorem, hence a smooth plane conic over K(C) is P 1 K(C) ) and P is normal with integral π-fibres, we see that P is exactly a smooth minimal ruled surface over C. Thus the quotient map σ : S → P is a flat double cover with some branch divisor Σ P , and Σ itself is smooth over k (see [3, § 2.2]). on the other hand, it can be deduced from [3, § 2.1 & 2.2] that Σ K(C) := Σ × π,C K(C) is a divisor of P K(C) := P × π,C K(C) ≃ P 1 K(C) with • deg K(C) Σ K(C) = p + 1 (by p a = p − 1 2 ); • Σ K(C) contains a point inseparable over K(C) (by the fact all fibres are singular). By degree counting, it in all concludes that Σ consists of two components with property X 1 ), X 2 ) &X 3 ) as above. We are done.
According to the proof of Corollary 1, when p ≥ 5 the equality of (15) holds only if c 2 (S) < 0. Let f : S → C be the Albanese fibration, then moreover we have that g = p − 1 2 and both of the inequalities (12) and (14) are equalities, thus we have • the fibration f is of arithmetic genus p − 1 2 ; • each geometric fibre of f is irreducible and reduced. In fact, the equality (14) gives that each fibre of f is irreducible. On the other hand, since the genus is p − 1 2 , we can have no multiple fibre by Proposition 2. As a direct consequence of Lemma 9, we have Colloary 2. Let S be a minimal algebraic surface of general type over an algebraically closed filed k of characteristic p ≥ 5. Assume that the equality of (15) holds and its Albanese fibration is hyperelliptic, then S is one of Raynaud's examples.
We end this section with a remark about some numerial facts of Raynaud's examples. Let f : S → C be one of Raynaud's examples associated to the triple (C, P, Σ), then we have (see [3, § 4