Abstract
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta–Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolovtype inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section, see Theorem 3.1. This gives an affirmative answer in this case to a problem posed by Peternell [17]. Concerning ruled surfaces with a negative section, we prove a similar result for generically nef vector bundles, with nef and balanced general fiber restriction and with a numerical condition on first Chern class, which is satisfied, for instance, if in its class there is a reduced divisor, see Theorem 3.5. Finally, we use such results to bound the invariants of curve fibrations, which factor through finite covers of ruled surfaces.
Introduction
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces, see Definition 2.1. By Mehta–Ramanathan Restriction Theorem 2.2 and by Miyaoka characterization of semistable vector bundles on a curve, Theorem 2.3, the generic nefness can be considered as a weak form of semistability, see Corollary 2.4.
A question about a possible relation between generic nefness and Bogomolovtype inequality has been posed by Th. Peternell in the paper [17, remarks after Theorem 3.8]. By considering the Harder–Narasimhan filtration of a vector bundle \(\mathcal {E}\) on a projective surface, Miyaoka proved the inequality \( c_2 (\mathcal {E}) \ge 0, \) provided that \(\mathcal {E}\) is generically nef and \(c_1(\mathcal {E})\) is nef.
In the present paper, under the hypotheses of generic nefness and the nefness of the generic fiber restriction, we give an affirmative answer to Peternell’s question for ruled surfaces with invariant \(e =C_0^2 \le 0\) (see Theorem 3.1):
Theorem 1.1
Let Y be a ruled surface on a smooth curve B with invariant \(e=C_0^2\le 0\). Let \(\mathcal {E}\) be a generically nef vector bundle of rank r on Y with nef generic fiber restriction.
Then
where \(c_1(\mathcal {E})\equiv aC_0+\delta L\) and \((a_1,\ldots ,a_r)\) is the generic splitting type of \(\mathcal {E}\).
In the \(e>0\) case, we prove a similar bound under two additional assumptions, namely that the general fiber restriction is balanced, and \( c_1(\mathcal {E})\cdot C_0 \ge {e\over 2} \), see Theorem 3.5. The last condition is satisfied, for instance, if \(c_1(\mathcal {E})\) is nef or if \(c_1(\mathcal {E})\) is effective and \(C_0\) is not contained in the base locus of \(2\ c_1(\mathcal {E})  C_0\); this assumption is typically satisfied by the Tschirnhausen sheaf associated with a finite cover of smooth surfaces with reduced branch divisor.
Our results allow us to obtain some bounds on the invariants of fibered surfaces factoring through finite covers. In these specific cases, the bounds found are better than the recent bounds found by Lu and Zuo [13]. Moreover, in the case of primitive cyclic covers \( \pi :S \rightarrow Y\), we obtain the same bound \(\lambda _{g,0,n}\) given in [9, Remark 4.4], see Theorem 5.7.
The techniques involved concern vector bundles and algebraic surfaces techniques. We believe that our approach can be the starting point for further research in the theory of vector bundles on fibered surfaces in general.
Notation and preliminaries
Let us introduce the definitions involved in the notion of generic nefness, see [17, Definition 3.1]).
Definition 2.1
A vector bundle \(\mathcal {E}\) on a smooth curve is nef if the tautological divisor of \(\mathbb {P}(\mathcal {E})\) is nef.
A vector bundle \(\mathcal {E}\) on a projective variety X of dimension \(n\ge 2\) is called generically nef with respect to an ample divisor H if the restriction \(\mathcal {E}_{\vert C}\) is nef for a general curve \(C=D_1 \cap \cdots \cap D_{n1}\), where \(D_i\in \vert m_i \ H\vert \) are general and \(m_i>>0\); such a curve C is said to be MRgeneral, which means general in the sense of Mehta–Ramanathan, with respect to H (w.r.t. H).
A vector bundle \(\mathcal {E}\) is generically nef if for every ample divisor H on X, the restriction \(\mathcal {E}_{\vert C}\) is nef for a MRgeneral curve w.r.t. H.
We now recall Mehta–Ramanathan Restriction Theorem and Miyaoka characterization of semistable vector bundles on a curve, which imply that the generic nefness can be considered as a weak form of semistability.
Theorem 2.2
(Mehta–Ramanathan Restriction Theorem) A locally free sheaf \(\mathcal {E}\) on a projective variety X is semistable w.r.t. an ample divisor H if and only \(\mathcal {E}_{\vert C}\) is semistable for C MRgeneral w.r.t. H.
Theorem 2.3
(Miyaoka) Let C be a smooth curve and E a vector bundle on C. Then E is semistable if and only if the \(\mathbb {Q}\)bundle \(E \otimes {det \ E^\vee \over {\mathrm{rk}}\ E}\) is nef, where \(E^\vee =Hom(E,\mathcal {O}_C)\) is the dual vector bundle.
Corollary 2.4
If \(\mathcal {E}\) is semistable w.r.t. H and if \(c_1(\mathcal {E}) \cdot H \ge 0\), then \(\mathcal {E}\) is generically nef w.r.t. H.
Finally, since we shall consider vector bundles on ruled surfaces, we can talk of the general splitting type.
Definition 2.5
Let \(p:Y \rightarrow B\) be a ruled surface over a smooth curve B, and let \(\mathcal {E}\) be a rank r vector bundle on Y. We say that \((a_1,\ldots ,a_r)\), with \(a_1 \le \cdots \le a_r\) is the generic splitting type of \(\mathcal {E}\) if for a general fiber L of p we have
We say that a fiber L is a jumping line if
Moreover, we say that \(\mathcal {E}\) is uniform if it has no jumping lines.
Finally, let us recall that for a tensor product \(\mathcal {V}\otimes \mathcal {L}\), where \(\mathcal {V}\) is a rank r vector bundle and \(\mathcal {L}\) is a line bundle on a smooth surface, we have:
Bogomolovtype inequalities
Theorem 3.1
Let Y be a ruled surface on a smooth curve B with invariant \(e=C_0^2\le 0\). Let \(\mathcal {E}\) be a generically nef vector bundle of rank r on Y with nef generic fiber restriction.
Then
where \(c_1(\mathcal {E})\equiv aC_0+\delta L\) and \((a_1,\ldots ,a_r)\) is the generic splitting type of \(\mathcal {E}\).
Proof
We prove the statement by induction on \(r\ge 2\). If \(r=2\), the claimed inequality has been proved in [5, Theorem 2.8], in the slightly different setting of blown up ruled surfaces, and under the assumption on \(\mathcal {E}\) to be weakly positive and with nef general fiber restriction. We observe, however, that the weak positivity of a vector bundle implies its generic nefness. Moreover, since the weak positivity is preserved for any quotient bundle, the same properties hold for such quotients. Then it is simple to check that the proof of the statement in the rank 2 case can be done with some little adaptations.
So let us assume \(r \ge 3\) and suppose that the claim holds for any vector bundle of rank \(q \le r1\) satisfying the assumptions of the statement.
We can consider the push–pull map
such a map is generically injective; hence, it is an injective map of locally free sheaves. Moreover, the quotient sheaf is locally free outside a subscheme Z of codimension 2.
It follows that we have a Brosiustype sequence (see [4]):
By setting \(\mathcal {A}:= p_\star (\mathcal {E}(a_r \ C_0))\), \(A:= c_1 (\mathcal {A})=\) and \(\alpha :={{\mathrm{deg}}}(A)\), and by tensoring by \(\mathcal {O}_Y(a_r \ C_0)\), we get:
where \(\mathcal {G}\) is a vector bundle of rank \(q\le r1\) and \(\mathcal {A}\) is a vector bundle of rank \(rq\ge 1\). We observe that \(rq\) is equal to the number of integers in the general splitting type \((a_1,\ldots ,a_r)\), which are equal to \(a_r\), so that we have
The sequence (3.2) gives
Let us compute each term appearing in (3.4). By the first relation in (2.1), we have \(c_1(p^\star A\ (a_r \ C_0))\equiv p^\star A + (rq)a_r\ C_0\). Next, we set \(\mathcal {M}:=\mathcal {G}(a_r \ C_0)\). From the sequence (3.2), we get
where we have set
and
By taking into account the second relation (2.1), we get
Moreover, since \(\mathcal {M}\) is a quotient of \(\mathcal {E}\) away from the zerodimensional scheme Z, \( \mathcal {M}\) is a generically nef vector bundle of rank \(q < r\). Now we analyze the general fiber restriction of \(\mathcal {M}\). Since \(\mathcal {E}\) has nef general fiber restriction, we have that the general splitting type \((a_1,\ldots ,a_r)\) satisfies
Moreover, by construction, the generic splitting type of \(\mathcal {M}\) is
where the integers \(a_i\) are the first q integers appearing in (3.8); hence, also \(\mathcal {M}\) is nef on the generic fiber restriction.
It follows that \(\mathcal {M}\) satisfies the assumptions of the theorem and we can apply the induction hypothesis, which gives the inequality
By (3.5), we get
and by observing that
the inequality (3.9) becomes
By taking into account (3.6), (3.7), (3.11) and the fact that Z is effective, from (3.4), we get
We can rewrite the inequality above in the form
where
In particular, the coefficient of \(\alpha \) in the expression (3.13) is
Now we shall bound the integer \(\alpha \). Recall that a generically nef vector bundle has a generically nef first Chern class, since exterior products of nef vector bundles are nef (see, for instance, [12, Theorem 6.2.12, (iv)]). Hence, \(c_1(\mathcal {M})\) is generically nef. So for any nef divisor \(xC_0+yL\) with \(x\ge 1\) and \(y \ge {1\over 2} x\ C_0^2\), we have
that is
which gives
and since this holds for any \(x\ge 1\) and any \(y\ge {1\over 2} x\ C_0^2\), we get
By substituting the right hand expression in (3.13), we get
Next we shall suitably express the integer \(\delta \). As \(c_1 (\mathcal {E})^2 =a^2 C_0^2 + 2a \delta \), we can write
Then the inequality (3.14) becomes
By computing the coefficients in (3.15), we get that the coefficient of \(C_0^2\) is zero and we get
which is the bound in the statement. \(\square \)
As a particular case, we can consider generically nef vector bundles \(\mathcal {E}\) with nef and balanced general fiber restriction, that is the restriction of \(\mathcal {E}\) to a general fiber of Y is a balanced vector bundle with splitting type \( (m,\ldots ,m,m+1,\ldots ,m+1)\).
Corollary 3.2
Let Y be a ruled surface on a smooth curve B with invariant \(e=C_0^2\le 0\). Let \(\mathcal {E}\) be a generically nef vector bundle of rank r on Y, such that the restriction of \(\mathcal {E}\) to a general fiber of Y is a nef and balanced vector bundle with splitting type \( (m,\ldots ,m,m+1,\ldots ,m+1)\), and set \(c_1(\mathcal {E})\equiv aC_0+\delta L\), \(a =mr+k\) and \(1\le k \le r1\).
Then
Moreover, the equality holds if and only if \(\mathcal {E}\) is uniform and \(\mathcal {E}\) is an extension sitting in an exact sequence of the type
where \(\mathcal {B}\) is a rank k vector bundle on B satisfying \( {\mathrm{deg}}\ c_1(\mathcal {B})= \delta , \) and \(\mathcal {V}\) is a rank \(rk\) vector bundle on B satisfying \( c_1(\mathcal {V})\equiv 0. \)
Proof
The bound is a direct consequence of the general bound. The characterization of the vector bundles attaining the equality can be directly obtained by imposing the equalities in all the bounds considered in the proof of Theorem 3.1. \(\square \)
We remark that vector bundles with balanced general fiber restriction are the natural generalization of vector bundles with semistable general fiber restriction, in which case the Bogomolov discriminant is nonnegative by Moriwaki’s Theorem [14, Theorem 2.2.1], which we recall.
Theorem 3.3
Let \(\varphi :Z \rightarrow C\) be a fibration from a smooth surface Z to a smooth curve C. Let \(\mathcal {E}\) be a torsionfree sheaf on Z such that the restriction of \(\mathcal {E}\) to a general fiber \(F\subset Z\) is a \(\mu \)semistable locally free sheaf. Then the Bogomolov discriminant \(\Delta (\mathcal {E})\) satisfies
Since in Moriwaki Theorem the only assumption is the semistability of the general fiber restriction, one can wonder, if the generic balancedness condition could be sufficient in order to have a Bogomolovtype inequality. A negative answer is given in the following example.
Example 3.4
On \(Y=\mathbb {P}^1 \times \mathbb {P}^1\), consider the rank two split vector bundle
We have
so we see that \(c_2(\mathcal {E})\) can be arbitrarily lowered by adjusting \(b_1\), even with \(c_1(\mathcal {E})^2\) fixed.
Let us conclude this section with a result in the \(e>0\) case. We will consider only the balanced case, and we will need to assume \(c_1(\mathcal {E})\cdot C_0 \ge {e\over 2}\), which is satisfied, for instance, if \(c_1(\mathcal {E})\) is nef, or if \(c_1(\mathcal {E})\) is effective and \(C_0\) is not contained in the base locus of \(2\ c_1(\mathcal {E})  C_0\). The last condition is typically satisfied by Tschirnhausen sheaves associated with surface covers with reduced branch divisor.
Theorem 3.5
Let Y be a ruled surface on a smooth curve B with invariant \(e=C_0^2> 0\). Let \(\mathcal {E}\) be a generically nef vector bundle of rank r on Y, such that the restriction of \(\mathcal {E}\) to a general fiber of Y is a nef and balanced vector bundle with splitting type \( (m,\ldots ,m,m+1,\ldots ,m+1)\), \(c_1(\mathcal {E})\equiv aC_0+\delta L\), \(a =mr+k\) and \(1\le k \le r1\).
Assume, moreover, that \( c_1(\mathcal {E})\cdot C_0 \ge {e\over 2} \).
Then
Proof
The proof is similar to that of Theorem 3.1. By assumption, the restriction of \(\mathcal {E}\) to a general fiber L of \(p:Y\rightarrow \mathbb {P}^1\) is balanced. Since by hypothesis \(c_{1}(\mathcal {E})\cdot L=a=mr+k\), the general fiber restriction of \(\mathcal {E}\) is of the type
Then we have again a Brosiustype exact sequence:
where \(p^\star p_\star \mathcal {E}((m+1)C_0) \) has rank k and \(\mathcal {G}\) has rank \((rk)\).
Set \(\mathcal {A}:= p_\star \mathcal {E}((m+1)C_0)\), \(A:= c_1 (\mathcal {A})=\), \(\alpha :={{\mathrm{deg}}}(A)\) and \(\mathcal {M}:=\mathcal {G}((m+1)C_0)\), so that (3.17) becomes
and
The main difference in the present proof is the bound on \(c_2(\mathcal {M})\). Since the restriction of the Brosius exact sequence (3.17) to the general fiber \(L\cong \mathbb {P}^1\) gives
the restriction of \(\mathcal {G}\) to the general fiber of Y is \(\mu \)semistable. Since \(\mathcal {M}\) is a twist of \(\mathcal {G}\), the same holds for the general fiber of \(\mathcal {M}\). Hence, \(\mathcal {M}\) is Bogomolov semistable by Moriwaki Theorem 3.3, and we have
By observing that with the notations of the proof of Theorem 3.1, we have
the relation (3.12) becomes
which simplifies as
Next we use the generic nefness of \(c_1(\mathcal {E})\) to bound \(\alpha ={{\mathrm{deg}}}(A)\). Let H be a very ample divisor of Y which avoids the points of Z arising in the Brosius sequence (3.17). Since \(\mathcal {E}\) is generically nef, and since \(\mathcal {M}\otimes \mathcal {I}_Z\) is a quotient of \(\mathcal {E}\), \(\mathcal {G}((m+1)C_0)\otimes \mathcal {O}_{m_0H}\) is nef for \(m_0>>0\), hence \(c_1(\mathcal {M})\otimes \mathcal {O}_{m_0H})\ge 0\).
An ample divisor on a ruled surface admitting a negative section \(C_0\) is of the type \(H\in xC_{0}+yL\), with \(x>0\) and \(y> x\ e\). The condition \(c_1 (\mathcal {M}) \cdot m_0H\ge 0\) gives
in particular \(\alpha \le \delta +m(rk){1\over x}\) for any \(x>0\), so
Moreover, using again the trick
we get
that is
The last bound is not satisfactory, since \(C_0^2<0\), so we finally use the assumption that \( c_1(\mathcal {E})\cdot C_0 \ge {C_0^2\over 2}, \) which gives \( \delta \ge ({1\over 2}a)\ C_0^2. \) The expression (3.22) yields \( C_0^2 \ge {c_1(\mathcal {E})^2 \over a(a1)}, \) and by (3.23), we get
\(\square \)
The normalized relative canonical divisor
In this section, we shall apply the Bogomolovtype inequalities to the Tschirnhausen sheaf of a finite cover of a Hirzebruch surface. Indeed, by the Viehweg Weak Positivity Theorem [21], the Tschirnhausen sheaf is weakly positive away from the branch locus, and hence nef on the complement of the branch locus (see also [11]^{Footnote 1}), so it is in particular generically nef. This will allow us to bound the relative Euler characteristic \(\chi _f\) of a fibration factoring through a finite cover.
Moreover, we shall introduce the normalized relative canonical divisor of a finite morphism \(\pi \) and we shall show that its selfintersection is related with the slope.
We first recall how to determine the invariants and the slope of a fibration factoring through a finite cover.
Definition 4.1
Let \(\pi :S \rightarrow Y\) be a finite cover of degree n between smooth surfaces. Then the sheaf \(\pi _\star \mathcal {O}_S\) is locally free of rank n. Similarly to the argument given in [15], we can consider the natural injective map \(\mathcal {O}_Y \rightarrow \pi _\star \mathcal {O}_S\), which admits a splitting by 1 / n times the trace map. Let \(\mathcal {E}_1\) be the cokernel of such a map; \(\mathcal {E}_1\) is a locally free sheaf of rank \(n1\) on Y and we have
Following [15], it is customary to call the dual sheaf \(\mathcal {E}:=\mathcal {E}_1^\vee \) the Tschirnhausen sheaf of the finite morphism \(\pi \).
Finally, if \(\omega _{S/Y}\) denotes the relative canonical sheaf, we have \((\pi _\star \omega _{S/Y})^\vee \cong \pi _\star \mathcal {O}_S\) by relative duality, hence
Lemma 4.2
Let S, Y be smooth surfaces, and let \(\pi :S \rightarrow Y\) be a finite cover of degree n with relative canonical divisor \(K_{S/Y}\). Then in the rational Chow ring \(A(Y) \otimes \mathbb Q\), we have

(1)
\(\pi _\star K_{S/Y}\equiv 2 c_1 (\mathcal {E})\),

(2)
\( \chi (\mathcal {O}_{S})= n \chi (\mathcal {O}_Y) +{1\over 2} c_1 (\mathcal {E}) \cdot K_Y +{1\over 2} c_1 (\mathcal {E})^2  c_2 (\mathcal {E})\);
Proof
The first relation is well known in the case of flat finite morphisms. The result follows from Grothendieck–Riemann–Roch Theorem applied to the morphism \(\pi :S\rightarrow Y\) and the sheaf \(\omega _{S/Y}\).
The Grothendieck–Riemann–Roch Theorem asserts that for a proper morphism \(\pi \) of smooth varieties we have
As \(R^1 \pi _\star \omega _{S/Y}=0\) since \(\pi \) is finite, we have
This yields
The divisorial part satisfies
As \(K_S \sim \pi ^\star K_Y + K_{S/Y}\), we have \(\pi _\star K_S \equiv nK_Y + \pi _\star K_{S/Y}\) and the first claim follows.
The equality between the codimension two cycles gives formula (2). \(\square \)
Definition 4.3
A fibration\(f:S\rightarrow B\) is a flat surjective morphisms between a smooth surface S and a smooth curve B with connected fibers, such that if \(x\in B\) is general then \(F_x:=f^{1}(x)\) is a smooth curve.
Following Xiao [20], we can associate with \(f:S \rightarrow B\) a rational number s(f), called the slope of f, defined as:
where \(K_f=K_S f^\star K_{B}\) is the relative canonical divisor, \(\chi _f : = {\mathrm{deg}}\ f_\star \omega _f \), and \(\omega _f := {\mathcal {O}}_S(K_f)\).
Remark 4.4
We recall the wellknown relations:
Corollary 4.5
Let \(f:S\rightarrow B\) be a fibration, which factorizes through a finite cover \(\pi : S \rightarrow Y\) of a ruled surface Y. Then
Proof
We can write \(K_{S}^{2}=K_{S/Y}^{2} +2K_{S/Y}\cdot \pi ^{\star }K_{Y}+nK_{Y}^{2}\), then by projection formula and Lemma 4.2\(K_{S}^{2}=K_{S/Y}^2+nc_{1}(\mathcal {E})\cdot K_{Y}+nK_{Y}^{2}\). As \(K_f ^2 = K_S ^2  8 (g1)(b1)\), where \(b =g(B)\), we have
Finally, by choosing the generators of the NeronSeveri group of Y to be the classes \([C_{0}]\) and [L] where \(L\in {\mathrm{NS}}(Y)\) is the class of a ruling and \(C_{0}\in {\mathrm{NS}}(Y)\) is the class of a section of minimal selfintersection, we may write
and the first formula follows.
Taking into account that \(\chi _f= \chi (\mathcal {O}_S)(g1)(b1)\), the formula for \(\chi _f\) follows from Lemma 4.2, (2). \(\square \)
Now we introduce the normalized relative canonical divisor of a finite cover, and we shall see that it is closely related to the slope of the induced fibration. Such a connection is not surprising, as a similar argument has already been used in such a context.
For instance, the Cornalba–Harris theory for bounding the slope of any fibration f relies on the study of the normalized relative canonical divisor of a fibration \(f:S \rightarrow B\)
and on the normalized Hodge bundle
Indeed, the pseudoeffectivity of \(f_\star (\mathbb {K}_f ^2)\), proved by Cornalba and Harris in [7, Theorem 1.1, Proposition 2.9 and Section 4], under the assumption that the Hilbert points of the general fiber are semistable, is a crucial step in their proof of the classical bound on the slope
A similar task has been used by Fedorchuk and Jensen [10], who obtained as a straight consequence of the positivity of \(c_1 (f_\star {\widetilde{\omega }} _{f} ^{\otimes 2})\) the result that if \(S\rightarrow B\) is a flat family of Gorenstein curves with the generic fiber a canonically embedded curve whose 2nd Hilbert point is semistable (e.g., with the generic fiber a general trigonal curve), then the slope satisfies the inequality \(s(f) \ge 5  {6\over g}\).
Also in the context of projective vector bundles \(\pi : \mathbb {P}(\mathcal {G}) \rightarrow Y\) fibered in \(\mathbb {P}^{r1}\) over a variety Y a similar divisor is studied, namely the socalled normalized tautological divisor
and the nefness of such a divisor has been investigated by Nakayama [16]. More precisely, Nakayama proved the following result:
Theorem 4.6
Let \(\mathcal {G}\) be a rank r vector bundle on a smooth complex projective variety Y of dimension \(d\ge 2\). Then the following conditions are equivalent:

\(\mathbb {T}_\mathcal {G}\) is nef;

\(\mathcal {G}\) is \(\mu \)semistable and \(\left( c_2(\mathcal {G}){(r1)\over 2r} c_1(\mathcal {G})^2\right) \cdot A^{d2}=0\) for an ample divisor A.
It turns out that in our context, as we are dealing with a finite morphism and a rank \((n1)\) torsionfree sheaf \(\mathcal {E}\) with \(c_1 (\mathcal {E})=c_1 (\pi _\star \omega _{S/Y})\), it is natural to give the following definition:
Definition 4.7
Let \(\pi :S \rightarrow Y\) be a finite morphism of degree n. The \(\mathbb {Q}\)divisor
is called the normalized relative canonical divisor of \(\pi \).
To explain the reason which leads to the definition of \(\Lambda _\pi \), we need to recall the following wellknown result (see [6]).
Theorem 4.8
Let Y be an integral surface and let \( \pi :S \rightarrow Y\) be a Gorenstein cover of degree \(n\ge 3\). There exists a unique \(\mathbb {P}^{n2}\)bundle \(\pi _Y:\mathbb {P}\rightarrow Y\) and an embedding \(i:S \rightarrow \mathbb {P}\) such that \(\pi =\pi _Y \circ i\). Moreover, \(\mathbb {P}\cong \mathbb {P}(\mathcal {E})\) and the ramification divisor R satisfies:
Remark 4.9
Consider the embedding \(S\subseteq \mathbb {P}(\mathcal {E})\). Since \((T_{\mathbb {P}(\mathcal {E})})_{\vert S}=K_{S/Y}\), we get that
The connection between the slope and \(\Lambda _\pi \) is given by the following:
Proposition 4.10
Let \(f:S \rightarrow B\) be a fibration, with general fiber F a smooth curve of genus g.
Assume that f factorizes through a finite degree n cover \(\pi :S \rightarrow Y\), where Y is a ruled surface, and assume that the general fiber restriction of the Tschirnhausen sheaf \(\mathcal {E}\) is a twist of the trivial sheaf.
By setting
we have
and the following equality holds:
where
is the conjectural bound of Stankova (see [19, Conjecture 13.3]).
In particular,
Proof
By Theorem 3.3, we have \(c_2(\mathcal {E}) \ge \frac{n2}{2(n1)}c_{1} (\mathcal {E})^{2}\), and since by Corollary 4.5 it holds \(\chi _f = {(g+n2)\over 2 (g+n1)} c_1 ^2(\mathcal {E})  c_2 (\mathcal {E})\), it follows that \(\chi _f\le {(g+n2)\over 2 (g+n1)} c_1 ^2(\mathcal {E})\frac{n2}{2(n1)}c_{1} (\mathcal {E})^{2}=\chi _0^{\max }\).
With a direct computation, one can obtain the equality (4.6), and the inequality (4.7) follows immediately. \(\square \)
As a consequence, the problem of bounding the slope can be rephrased in a problem of bounding \(\Lambda _\pi ^2\).
The Bogomolovtype inequalities given in Corollary 3.2 and Theorem 3.5 allow us to derive a similar result also in the nondivisible case \((n1) \not  g\). Indeed, the Tschirnhausen sheaf is nef outside the branch locus by the Weak Positivity Theorem of Viehweg [21, 3.4]. If we assume that the general fiber restriction of \(\mathcal {E}\) is balanced, which can be rephrased with the assumption that the general fiber of the fibration corresponds to a point outside the Maroni locus of a suitable Hurwitz scheme or of the moduli space, and that the branch divisor is reduced, which corresponds to impose the open condition that \(\pi \) has generically simple ramifications, then Corollary 3.2 and Theorem 3.5 apply.
Therefore, recalling that \(\chi _f= {(g+n2)\over 2 (g+n1)}c_1 ^2(\mathcal {E})  c_2 (\mathcal {E})\) and setting \(g+n1=(n1)m+k\), we have
that is
Then we can write
where
This shows that the function \(\mathcal {F}(n,g,k)\) can be used to replace the function \(\mathcal {F}(n,g)\) in the nondivisible case; more precisely, we have:
Proposition 4.11
Let \(f:S\rightarrow B\) a semistable fibration over a rational curve B. Assume that f factorizes through a finite cover of degree n of ruled surface and that the Tschirnhausen sheaf is balanced on the general fiber. If the genus of the general fiber of f is \(g=(m1)(n1)+k\), where \(1\le k\le n2\), then
Remark 4.12
The bounds (4.7) and (4.9) given in Propositions 4.10 and 4.11 hold also for fibrations \(f:S \rightarrow B\), which are the relatively minimal model of fibrations satisfying the given hypotheses. Indeed, it is enough to observe that the selfintersection of the relative canonical divisor of a relatively minimal model of a given fibration cannot decrease, and the relative Euler characteristic is unchanged.
Positivity results on \(\Lambda _\pi \)
From the results of the previous section, it follows that a bound on \(\Lambda _\pi ^2\) gives a bound also on the slope. Therefore, we are going to establish some conditions, under which the normalized relative canonical divisor has nonnegative selfintersection. Some very similar problems have been studied in [2, 3], but their results do not apply to our case.
We first analyze the restriction of the divisor \(\Lambda _\pi \) to the general fiber.
Proposition 5.1
Let F be a general fiber of the fibration \(f:S \rightarrow B\). If \( g=(n1)(m1) \) and F is not contained in the Maroni locus, the restriction of \(\Lambda _\pi \) to F satisfies:

(1)
\((\Lambda _\pi ) _{F}\sim K_F  (m2)\Gamma _F, \) where \(\Gamma _F\in g^1_n\) is a gonal divisor;

(2)
\(h^0(\mathcal {O}_F( K_F  (m2)\Gamma _F))=n1\);

(3)
the linear system \( K_F  (m1)\Gamma _F\) is base point free.
Proof
We have
where \(\Gamma _F\in g^1_n\) is a gonal divisor, which proves (1).
Let us compute \(h^0( \mathcal {O}_F( (m2)\Gamma _F))\) using the Geometric Riemann Roch Theorem:
where \(\langle (m2)\Gamma _F\rangle \subset \mathbb {P}^{g1}\) is the linear span on the canonical model of the curve F. Now recall that since F is Maroni general, the canonical model of F lies on \(W\cong \mathbb {P}(\oplus ^{n1} \mathcal {O}_{\mathbb {P}^1}(m2))\) embedded in \(\mathbb {P}^{g1}\) by the tautological linear system. It follows that
hence by the Geometric Riemann Roch Theorem we have
hence by Riemann Roch
which proves (2).
Finally, assume by contradiction that P is a base point of the linear system \(K_F  (m2)\Gamma _F\). Then by the Geometric Riemann Roch Theorem, we would have
and
We claim that the last equality cannot be satisfied. Indeed, the subspace \(\langle (m2)\Gamma _F\rangle \) cuts on W exactly \(m2\) fibers; indeed, since the minimum degree of a unisecant curve on W is \(m2\), the subspace \(\langle (m2)\Gamma _F\rangle \) contains no horizontal component. It follows that the divisor cut out by \(\langle (m2)\Gamma _F\rangle \) on the canonical model of F is exactly \((m2)\Gamma _F\). \(\square \)
Corollary 5.2
Let \(f:S \rightarrow B\) be a fibration in irreducible curves, with general fiber F a smooth curve of genus g. Assume that f factorizes through a finite degree n cover \(\pi :S \rightarrow Y\), where Y is a ruled surface, with Tschirnhausen sheaf generically a twist of the trivial sheaf.
If the restriction map \( H^0 (\mathcal {O}_S(\Lambda _\pi )) \rightarrow H^0(\mathcal {O}_F(\Lambda _\pi )) \) is surjective, then \( \Lambda _\pi ^{2} \ge 0. \)
Proof
We claim that the linear system \( \Lambda _\pi \) has no horizontal base locus.
Assume by contradiction that \( \Lambda _\pi \) has a horizontal component \(\varkappa \) in its base locus. Then \(\varkappa _{ F}\) is contained in the base locus of \((\Lambda _\pi ) _{F}\). But the latter linear system is base point free by Proposition 5.1, (3). This proves that \(\Lambda _\pi \) has no horizontal base locus.
Finally, since all the fibers of f are irreducible, \(\Lambda _\pi \) has no vertical base locus.
Summing up, as \(\Lambda _\pi \) is effective and \(\Lambda _\pi \) has no positive dimensional base locus, we have \( \Lambda _\pi ^2 \ge 0. \)\(\square \)
Rational fibrations with uniform Tschirnhausen sheaf
In the following proposition, we shall prove that in the divisible case, the fibrations over a rational curve, with uniform and generically balanced Tschirnhausen sheaf and with semistable unisecant restriction, satisfy the assumption of Proposition 5.1. We remark that by a recent result given in [8], a sufficiently general curve in the Hurwitz scheme of degree n covers of curves of given genus \(p \ge 0\) has a \(\mu \)semistable Tschirnhausen sheaf. Therefore, the assumption of semistability on unisecant restrictions can be read as a generality assumption concerning the family of pullback curves of a family of general unisecant curves.
Proposition 5.3
Let \(f:S \rightarrow \mathbb {P}^1\) be a fibration with irreducible fibers, with general fiber F a smooth curve of genus g and gonality n, where \(n \ge 5\), such that \((n1)g\) and with balanced reduced gonal direct image sheaf. Assume that f factorizes through a finite morphism \(\pi :S \rightarrow Y\), where \(p:Y \rightarrow \mathbb {P}^1\) is a Hirzebruch surface.
If the restriction of \(\mathcal {E}\) to some unisecant ample divisor is semistable and if \(\mathcal {E}\) is uniform, then the restriction map \(H^0 (\mathcal {O}_S(\Lambda _\pi )) \rightarrow H^0(\mathcal {O}_F(\Lambda _\pi ))\) is an isomorphism.
In particular, \(s(f) \ge \mathcal {F}(n,g)\).
Proof
We set \( c_1 (\mathcal {E}) = (g+n1) C_0 + \delta L, \) where \(C_0\) is a section with \(C_0 ^2\le 0\) and L a fiber of ruling on \(Y=\mathbb {F}_e\).
Since \(\mathcal {E}\) is uniform, the restriction to any fiber L of the ruling satisfies \( \mathcal {E}_{L} \cong \bigoplus _{i=1}^{n1} \mathcal {O}_{\mathbb {P}^1} (m), \) where \(m = {g\over n1} +1\). Then the injective map of sheaves \( p^\star p_\star \mathcal {E}(m C_0) \rightarrow \mathcal {E}(m C_0) \) is an isomorphism, so
By the assumption that \(\mathcal {E}\) is semistable with respect to an ample divisor \(H \sim C_0 + (e+a)L\), we get \( \mathrm{\deg A_i} = k, \) for any i, and \( k= {\delta \over (n1)} \ge 0, \) so that
We finally get
On the other hand,
Hence the restriction exact sequence
determines the isomorphism of the statement. \(\square \)
Remark 5.4
We observe that in the case when \(\mathcal {E}\) is a uniform vector bundle, that is \(\Delta (\mathcal {E})=0\), and if \(\mathcal {E}\) is \(\mu \)semistable with respect to some ample divisor, then the normalized tautological divisor\(T_{\mathbb {P}(\mathcal {E})}  {1\over (n1)} p^\star c_1 (\mathcal {E})\) where \(p:\mathbb {P}(\mathcal {E}) \rightarrow Y\) is nef by Nakayama’s Theorem 4.6. It follows that \(\left( T_{\mathbb {P}(\mathcal {E})}  {1\over (n1)} p^\star c_1 (\mathcal {E}) \right) _{S} =K_{S/Y}  {1\over (n1)} \pi ^\star c_1 (\mathcal {E})=\Lambda _\pi \) is also nef. Since the restriction map is surjective by Proposition 5.3, all this implies directly that in such a case \(\Lambda _\pi ^2 \ge 0\).
Upper bounds on \(\Lambda _\pi ^2\) and primitive cyclic covers
We recall that for finite covers, we have the following upper bound on \(R^2\) in terms of \(c_1(\mathcal {E})^2\), which is a consequence of the Hodge Index Theorem applied to the \(\mathbb {Q}\)  divisor \(R\pi ^\star {2\over n} c_1(\mathcal {E})\):
Lemma 5.5
Let \(\pi :S \rightarrow Y\) be a Gorenstein cover degree n, and let \(\mathcal {E}\) be the Tschirnhausen sheaf. Then
Proof
See [5, Lemma 3.12]. \(\square \)
Corollary 5.6
With the assumptions of Lemma 5.5, we have
Proof
Since \(\Lambda _\pi = K_{S/Y}  {1\over n1}\ \pi ^\star c_1(\mathcal {E})\), we have
and by projection formula and by Lemma 4.2, (1), it follows
\(\square \)
We remark that the Hodge Index Theorem also implies that the equality holds in (5.2) if and only if
Such a condition is satisfied, for instance, when all ramification points of \(\pi \) are total ramification points, that is of maximal ramification index n.
A typical context, when this happens, is the one of primitive cyclic covers\( \pi :S \rightarrow Y\), that is covers such that there exist an effective divisor \(A \subset Y\) and an effective divisor \(D\subset S\) such that
and such that \(\pi :S\rightarrow Y\) does not factorize through two covers of smaller degree. In this case, the following holds:

(1)
\(B_\pi \sim nA\) and \(\pi ^\star B_\pi = n D\);

(2)
\(K_{S/Y} \sim (n1)D\);

(3)
\(\pi _\star \mathcal {O}_S(K_{S/Y})\cong \bigoplus _{i=0}^{n1} \mathcal {O}_Y(iA)\);

(4)
\(c_1(\mathcal {E})={n(n1)\over 2} A\), \(c_2(\mathcal {E})={n(n1)(n2)(3n1)\over 12} A^2\).
From this, we obtain that
Since \(c_1(\mathcal {E})={n(n1)\over 2} A\), if \(A^2 \ge 0\), we get \(\Lambda _\pi ^2 \ge 0\). This gives a bound, which is exactly the bound \(\lambda _{g,0,n}\) given in [9, Remark 4.4]:
Theorem 5.7
Let \(f:S \rightarrow B\) be the relatively minimal model of a finite cyclic cover \(\pi : \widetilde{S} \rightarrow Y\) of a ruled surface Y.
Then
Proof
Recall that the selfintersection of the relative canonical divisor of a relatively minimal model of a given fibration cannot decrease, and the relative Euler characteristic is unchanged. Then we can apply formula (4.6). \(\square \)
Remark 5.8
We observe that for cyclic covers the Tschirnhausen sheaf is uniform.
A Beniamino Segre’s construction
Following closely [1, Chapter 21 Section 12], we shall show that for a general \([C]\in {\overline{{\mathcal M}^{1}_{g,n}}}\), where \({\overline{{\mathcal M}^{1}_{g,n}}}\) is the closure of the ngonal locus in the moduli space \(\overline{\mathcal M}_g\) of curves of genus g, if \(p:C\rightarrow \mathbb {P}^{1}\) is the gonal covering, then the corresponding Tschirnhausen sheaf \(\mathcal {E}_{C}\) is balanced.
Theorem 5.9
Let \(g\ge 3\). For any integer n such that \(3\le n\le \frac{g}{2}+1\), there exists a smooth curve C of genus g admitting a complete \(g^{1}_{n}\) without base points, and such that the corresponding Tschirnhausen sheaf \(\mathcal {E}_{C}\) is balanced.
Proof
Let \(\mathbb F_{e}\) be a Hirzebruch surface with invariant \(e =C_0^2 \ge 0\). Consider the complete linear system
Assume that \(h>\frac{ne}{2}\). Then \(\Sigma _{n,h}\) is very ample and the image of the morphism \(\phi _{\Sigma _{n,h}}:\mathbb F_{e}\rightarrow \mathbb {P}^{N}\) associated with \(\Sigma _{n,h}\) is a smooth surface; hence, by Bertini’s theorem, the general member of \(\Sigma _{n,h}\) is smooth. Then, by adjunction, the genus of the general element \(\Gamma \in \Sigma _{n,h}\) is
Moreover, by Riemann–Roch, we have
Following [1, Theorem 12.16 see page 870], we immediately see that, given any fixed integer \(n\ge 3\) and any fixed integer \(e\ge 0\), the intervals
cover the half line \([0,\infty )\). Then there exist an integer \(\delta \) and an integer h such that \(0\le \delta \le h+n1\frac{ne}{2}\) and
A simple computation shows that
By Castelnuovo’s theorem applied to \(\Sigma _{n,h}\), see for instance [1, Theorem 12.6 page 865], it follows that given \(\delta \) general points \(a_{1},\ldots , a_{\delta }\in \mathbb F_{e}\), there exists an irreducible curve \(\Gamma \in \Sigma _{n,h}=nC_{0}+hl\) having \(\delta \) nodes at \(a_{1}\),..., \(a_{\delta }\) and no other singularities. Let \(\nu :Z\rightarrow \mathbb F_{e}\) be the blowup at \(a_{1}\),..., \(a_{\delta }\) and let \(E_{i}:=\nu ^{1}(a_{i})\), \(i=1,\ldots ,\delta \). The normalization C of \(\Gamma \) is contained in Z and \(\nu _{C}:C\rightarrow \Gamma \) is the normalization morphism. Let \(H_{0}:=\nu ^{1}(C_{0})\) and \(\widetilde{L}:=\pi ^{1}(L)\). The smooth curve C has a \(g^{1}_{n}\) induced by the ruling of \(\mathbb F_{e}\). Let us denote by D a divisor of the \(g^{1}_{n}\). Then
By standard surface theory, we get
Then \((n2)H_{0}+(he2v)L\sum _{i=1}^{\delta }E_{i}\sim K_{Z}+CvL\) and by adjunction theory on surfaces we have the following exact sequence
Notice now that by projection formula and by Serre duality we have
since \(v\ge 0\). As \(h^{0}(Z,\mathcal {O}_{Z}(K_{Z}vL))=0\), by considering the long cohomology sequence associated with the sequence 5.4, we obtain that the restriction morphism
is an isomorphism. In particular \(h^{0}(C,\omega _{C}(vD))=h^{0}(Z,\mathcal {O}_{Z}(K_{Z}+CvL))\).
Since \(a_{1},\ldots , a_{\delta }\) are general points, we have
Hence if we assume that \(h(e+2+v)\ge \frac{(n2)e}{2}\), then \({\mathrm{dim}}\ \Sigma _{n2,h(e+2+v)} \delta =g(n1)v\).
We have shown that if \(h(e+2+v)\ge \frac{(n2)e}{2}\) and if k is the unique integer such that \(1\le k\le n2\) and \(g=m(n1)+k\) then
if \(m\ge v\).
Finally, let \(p:C\rightarrow \mathbb {P}^{1}\) be the gonal morphism; then \(p_{\star }\omega _{C}=\omega _{\mathbb {P}^{1}}\oplus \mathcal {E}_{C}(2)\), where
with \(m_{1}\le m_{2}\le \cdots \le m_{n1}\). By projection formula and by Eq. 5.5, it follows that in fact \(m_{1}=\cdots =m_{n1k}=m1\) and \(m_{nk}=\cdots =m_{n1}=m\), that is \(\mathcal {E}_{C}\) is balanced. \(\square \)
Sharpness results
In this section, we construct some examples that realize the bound on the slope given in Proposition 5.3.
Existence of fibrations with the required properties: rational basis
Let \(Y=\mathbb {P}^1\times \mathbb {P}^1\) and \(T=Y\times \mathbb {P}^1\). Let \(\pi _i:T\rightarrow \mathbb {P}^1\) be the projection with respect to the ith factor and set \(\mathcal {L}_i:=\pi _{i}^{\star }\mathcal {O}_{\mathbb {P}^1}(1)\) where \(i=1,2,3\). Let
be a general element, where \(n\ge 3\). If \(\langle x_0,x_1\rangle =H^0(T,\mathcal {L}_{3})\), \(\langle y_0,y_1\rangle =H^0(T,\mathcal {L}_{2})\) and \(\langle z_0,z_1\rangle =H^0(T,\mathcal {L}_{1})\), then
Thus if \(a_i((y_0:y_1),(z_0:z_1))\in H^0(Y,\mathcal {O}_Y(n,n))\) for \(i=0,\ldots ,n\) are general, the morphism \(\pi :S\rightarrow Y\) is finite of degree n. Consider now the composition \(f:S\rightarrow \mathbb {P}^1\) of the inclusion \(j:S\hookrightarrow T\) with the natural morphism \(\rho :T\rightarrow Y\) followed by the projection on the first factor \(\pi '_1:Y\rightarrow \mathbb {P}^1\). The fiber over \(z_0=a,z_1=b\) is the curve \(C_{[a:b]}=V(F_{[a:b]})\,\ {\mathrm{where}}\,\, F_{[a:b]}=\sum _{i=0}^{n}a_i((y_0:y_1),(a:b))x_0^{ni}x_1^{i}\) inside \(\mathbb {P}^1\times \mathbb {P}^1\). In particular, \(f:S\rightarrow \mathbb {P}^1\) is a fibration in curves of genus \(g=(n1)^2\) and gonality n such that
Moreover \(\Lambda _{\pi }\) is effective and it is induced on S by the linear system \((0,0,n2)\), hence \(\Lambda _{\pi }^2=0\). Finally by performing the pushforward of the standard exact sequence
by projection formula and by relative duality it follows that the Tschirnhausen sheaf satisfies \(\mathcal {E}\cong \mathcal {O}_{Y}(n,n)^{\oplus n1}\), so it is a uniform and balanced vector bundle which is also semistable on the (0, 1)sections of the projection \(\pi '_1:Y\rightarrow \mathbb {P}^1\). This shows that the bound given in Proposition 5.3 is sharp. Note that instead of \(S\in nL_1+nL_2+nL_3\) we can take \(S\in n_1L_1+n_2L_2+n_3L_3\) where \(n_i\ge 1\) to obtain similar results.
Existence of fibrations with the required properties: other cases
Let \(T=C_1\times \mathbb {P}^1\times \mathbb {P}^1\), where \(C_1\) is a smooth curve of genus \(g_1 >0\), and let \(\mathcal {L}:=\pi ^{\star }\mathcal {O}_{C_1}(L_1)\otimes \pi _{2}^{\star }\mathcal {O}_{\mathbb {P}^1}(n)\otimes \pi _{3}^{\star }\mathcal {O}_{\mathbb {P}^1}(n)\). As above we obtain a semistable ngonal fibration over \(C_1\) of genus \(g=(n1)^2\) such that:
Also in this case \(\Lambda _\pi ^2\) is zero and the Tschirnhausen sheaf is uniform and with balanced fiber restriction.
Notes
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Acknowledgements
This research is supported by national MIUR funds, PRIN project Geometria delle varietà algebriche (2015). Beorchia Valentina is also supported by national MIUR funds FINANZIAMENTO ANNUALE INDIVIDUALE DELLE ATTIVITÀ BASE DI RICERCA  2018. Zucconi Francesco is supported by Università degli Studi di Udine  DIMA project Geometry PRIDZUCC2017.
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Beorchia Valentina and Zucconi Francesco are members of GNSAGA of INdAM. The examples contained in Sect. 6.1 give an answer to a question posed by Ciro Ciliberto at the conference New Trends in Algebraic Geometry, Università della Calabria, June 12–14 (2013), about the sharpness of some slope estimates.
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Valentina, B., Francesco, Z. Generically nef vector bundles on ruled surfaces. Annali di Matematica 198, 443–463 (2019). https://doi.org/10.1007/s1023101807815
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Keywords
 Vector bundles
 Chern classes
 Fibrations
 Finite covers
Mathematics Subject Classification
 14J60
 14D06