On the slope of relatively minimal fibrations on rational complex surfaces

Abstract

Given a relatively minimal fibration \({f:S \to \mathbb{P}^1}\), defined on a rational surface S, with a general fiber C of genus g, we investigate under what conditions the inequality \({6(g-1)\le K_f^2}\) occurs, where Kf is the canonical relative sheaf of f. We give sufficient conditions for having such inequality, depending on the genus and gonality of C and the number of certain exceptional curves on S. We illustrate how these results can be used for constructing fibrations with the desired property. For fibrations of genus 11 ≤ g ≤ 49 we prove the inequality: $$6(g-1) +4 -4\sqrt g \le K_f^2.$$