Numerical properties of isotrivial fibrations

In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations phi : X -> 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) <= 8 chi(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) <= 8 chi(O(X)) - 5 and the inequality is also sharp. This improves previous results of Serrano and Tan.