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On the lower bound of the slope of a non-hyperelliptic fibration of genus 4. (English) Zbl 0816.14006
In 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, $K^ 2_ S \geq {7 \over 2} \chi ({\mathcal O}_ S) + {27 \over 2} (g(C) - 1).$

##### MSC:
 14D99 Families, fibrations in algebraic geometry 14L24 Geometric invariant theory
##### Keywords:
slope of fibration
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