On the slope of relatively minimal fibrations on rational complex surfaces

Details On the slope of relatively minimal fibrations on rational complex surfaces On the slope of relatively minimal fibrations on rational complex surfaces

2009-12-31

arxiv.org/abs/1001.0177v3

Mathematics

Algebraic Geometry

Final version. Typos and small inaccurancies corrected. A new, more motivated introduction. To appear in Collectanea Math

Scientific paper

Given a relatively minimal fibration $f: S \to \Bbb P^1$ on a rational surface $S$ with general fiber $C$ of genus $g$, we investigate under what conditions the inequality $6(g-1)\le K_f^2$ occurs, where $K_f$ 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 $11\le g\le 49$ we prove the inequality: $$ 6(g-1) +4 -4\sqrt g \le K_f^2.$$

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