On the slope of relatively minimal fibrations on rational complex surfaces

Claudia R. Alcantara; Abel Castorena; Alexis G. Zamora

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arXiv · 1001.0177

On the slope of relatively minimal fibrations on rational complex surfaces

Claudia R. Alcantara·Abel Castorena·Alexis G. Zamora

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Abstract

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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BibTeX RIS

Claudia R. Alcantara, Abel Castorena, Alexis G. Zamora. 2010-08-17. On the slope of relatively minimal fibrations on rational complex surfaces. https://arxiv.org/abs/1001.0177

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