Numerical properties of isotrivial fibrations

Abstract

In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations \varphi \colon X \lr C, where $X$ is a smooth, projective surface and $C$ is a curve. In particular we prove that, if $g(C) \geq 1$ 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 two ordinary 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 is also sharp. This improves previous results of Serrano and Tan.Comment: 30 pages. Final version, to appear in Geometriae Dedicat