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On the slope of relatively minimal fibrations on rational complex surfaces. (English) Zbl 1221.14012

Let \(f : S \rightarrow T\) be a relatively minimal projective morphism between a smooth projective surface \(S\) and a smooth curve \(T\) such that the genus \(g\) of the general fiber of \(f\) is positive. The slope of \(f\), \(\lambda_f\), is defined to be the ratio \(K_{S/T}^2/\deg (f_{\ast}K_{S/T})\), where \(K_{S/T}\) is the relative canonical sheaf of \(f\). Part of its significance is that if \(T=\mathbb{P}^1\), \(g \geq 2\), and \(f\) is semistable, then \(\lambda_f\) is related to the the number \(\sigma\) of the singular fibers of \(f\) by the relation \(K_{S/T}^2<(\sigma-2)(2g-2)\).
In this paper the authors study the slope of a morphism \(f : S \rightarrow T\) when \(S\) is a rational surface and \(T=\mathbb{P}^1\). In particular they obtain lower bounds for it under restrictions on the genus \(g\) and the gonality of the general fiber of \(f\). Their method is the following. It is known that in the case of the paper, \(\deg (f_{\ast}K_{S/T})=g\) and hence in order to bound the slope it suffices to bound \(K_{S/T}^2\). In order to do that the authors show that under certain restrictions on the genus \(g\) and the gonality of the general fiber \(C\) of \(f\), \(C+nK_S\) is effective for \(n=2,3\). Then there is a Zariski decomposition \(C+nK_S=P+N\) where \(P\) is a nef divisor and \(N\) an effective divisor. They calculate explicitely the divisors \(P\) and \(N\) and then the inequalities for \(K_{X/T}^2\) are obtained from the fact that \(P^2\geq 0\).

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14J26 Rational and ruled surfaces

Keywords:

fibration; minimal; slope
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References:

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