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Numerical properties of isotrivial fibrations

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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.

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Correspondence to Francesco Polizzi.

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Polizzi, F. Numerical properties of isotrivial fibrations. Geom Dedicata 147, 323–355 (2010). https://doi.org/10.1007/s10711-010-9457-z

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