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Abstract: In this paper we investigate the numerical properties of relatively minimalisotrivial fibrations $\varphi \colon X \lr C$, where $X$ is a smooth,projective surface and $C$ is a curve.
In particular we prove that, if $gC\geq 1$ and $X$ is neither ruled nor isomorphic to a quasi-bundle, then $K X^2\leq 8 \chi\mO X-2$; this inequality is sharp and if equality holds then $X$is a minimal surface of general type whose canonical model has precisely twoordinary double points as singularities.
Under the further assumption that$K X$ is ample, we obtain $K X^2 \leq 8 \chi\mO X-5$ and the inequality isalso sharp.
This improves previous results of Serrano and Tan.



Autor: Francesco Polizzi

Fuente: https://arxiv.org/



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