Numerical properties of isotrivial fibrations

Details Numerical properties of isotrivial fibrations Numerical properties of isotrivial fibrations

2008-10-23

arxiv.org/abs/0810.4195v3

Geometriae Dedicata, Volume 147, Number 1 (2010), 323-355

Mathematics

Algebraic Geometry

30 pages. Final version, to appear in Geometriae Dedicata

Scientific paper

10.1007/s10711-010-9457-z

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.

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