On the lower bound of the slope of a non-hyperelliptic fibration of genus 4.

*(English)*Zbl 0816.14006In this paper, the base field is the complex number field \(\mathbb{C}\). Let \(f : S \to C\) be a fibration of a smooth projective surface \(S\) onto a smooth curve \(C\), i.e. a morphism with connected fibres. Suppose that the fibration \(f\) is relatively minimal, that is, \(S\) has no \((-1)\)-curves contained in a fibre of \(f\). The fibration \(f\) is called of genus \(g\) if the genus of its general fibre equals \(g\). Similarly a fibration is said to be hyperelliptic or non-hyperelliptic according to the type of its general fibre. The invertible sheaf \(\omega_{S/C} = \omega_ S \otimes f^* \omega_ C^ \vee\) is called the relative canonical sheaf of the fibration \(f\). Any divisor corresponding to \(\omega_{S/C}\) is said to be a relative canonical divisor and denoted by \(K_{S/C}\). We have the following basic relative numerical invariants of \(f\): \(K^ 2_{S/C} = K^ 2_ S - 8(g - 1) (g(C) - 1)\), \(\chi_ f = \chi ({\mathcal O}_ S) - (g - 1) (g(C) - 1) = \deg f_ * \omega_{S/C}\). Whenever \(\chi_ f \neq 0\), the slope of the fibration \(f\) can be defined as \(\lambda_ f = K^ 2_{S/C}/ \chi_ f\). The slope \(\lambda_ f\) is an important invariant for a fibration. Hence the determination of its bound is an interesting problem. In 1987, G. Xiao [Math. Ann. 276, 449-466 (1987; Zbl 0596.14028)] proved that for a relatively minimal fibration \(f\) which is not locally trivial with \(g \geq 2\), one has \(4 - 4/g \leq \lambda_ f \leq 12\), and \(\lambda_ f = 12\) if and only if every fibre of \(f\) is smooth and reduced. There were examples with \(\lambda_ f = 4 - 4/g\) and G. Xiao [loc. cit.] conjectured that if \(\lambda_ f = 4 - 4/g\) then \(f\) must be hyperelliptic. Since the hyperelliptic fibrations have been investigated in detail, the non-hyperelliptic fibrations become the next object of study.

In this paper we will show that for a relatively minimal non- hyperelliptic fibration \(f\) with \(g = 4\) which is not locally trivial, one has \(\lambda_ f \geq {24 \over 7}\), or equivalently, \(K^ 2_ S \geq {24 \over 7} \chi ({\mathcal O}_ S) + {96 \over 7} (g(C) - 1)\). This bound is exact. At the end of this paper, we will give an explicit example. If the general fibre of \(f\) has two different \(g^ 1_ 3\)’s (this is the general case), then \(\lambda_ f \geq {7 \over 2}\), or equivalently, \[ K^ 2_ S \geq {7 \over 2} \chi ({\mathcal O}_ S) + {27 \over 2} (g(C) - 1). \]

In this paper we will show that for a relatively minimal non- hyperelliptic fibration \(f\) with \(g = 4\) which is not locally trivial, one has \(\lambda_ f \geq {24 \over 7}\), or equivalently, \(K^ 2_ S \geq {24 \over 7} \chi ({\mathcal O}_ S) + {96 \over 7} (g(C) - 1)\). This bound is exact. At the end of this paper, we will give an explicit example. If the general fibre of \(f\) has two different \(g^ 1_ 3\)’s (this is the general case), then \(\lambda_ f \geq {7 \over 2}\), or equivalently, \[ K^ 2_ S \geq {7 \over 2} \chi ({\mathcal O}_ S) + {27 \over 2} (g(C) - 1). \]