, Volume 147, Issue 1, pp 323-355
Date: 20 Jan 2010

Numerical properties of isotrivial fibrations

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Abstract

In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations \({\varphi : X \longrightarrow C}\) , where X is a smooth, projective surface and C is a curve. In particular we prove that, if g(C) ≥ 1 and X is neither ruled nor isomorphic to a quasi-bundle, then \({K_X^2 \leq 8 \chi(\mathcal{O}_X)-2}\) ; this inequality is sharp and if equality holds then X is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that K X is ample, we obtain \({K_X^2 \leq 8\chi(\mathcal{O}_X)-5}\) and the inequality is also sharp. This improves previous results of Serrano and Tan.