On the slope of relatively minimal fibrations on rational complex surfaces

TL;DR

This paper studies inequalities involving the canonical sheaf of fibrations on rational surfaces, providing conditions based on genus, gonality, and exceptional curves, and establishing new bounds for certain genus ranges.

Contribution

It offers new sufficient conditions for the inequality $6(g-1) \\le K_f^2$ to hold, depending on genus, gonality, and exceptional curves, and proves a specific inequality for genus 11 to 49.

Findings

Established conditions for the inequality $6(g-1) \\le K_f^2$ based on geometric properties.

Proved a new inequality $6(g-1) +4 -4\\sqrt g \\le K_f^2$ for genus between 11 and 49.

Provided methods for constructing fibrations satisfying these inequalities.

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: $$ 6(g-1) +4 -4\sqrt g \le K_f^2.$$