Numerical properties of isotrivial fibrations

Abstract

In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations $φ\colon X \lr C$, where $X$ is a smooth, projective surface and $C$ is a curve. In particular we prove that, if $g(C)\ge 1$ and $X$ is neither ruled nor isomorphic to a quasi-bundle, then $K_X^2 \leq 8 χ(\mO_X)-2$; 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 $K_X^2 \leq 8 χ(\mO_X)-5$ and the inequality is also sharp. This improves previous results of Serrano and Tan.

30 pages. Final version, to appear in Geometriae Dedicata

References in corpus (6)

• Standard isotrivial fibrations with p_g=q=1
• On the Milnor fibers of cyclic quotient singularities
• The classification of surfaces with p_g = q = 0 isogenous to a product of curves
• The classification of surfaces with p_g=q=1 isogenous to a product of curves
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Cited by in corpus (5)

• The classification of minimal product-quotient surfaces with $p_g=0$
• A note on surfaces with $p_g=q=2$ and an irrational fibration
• Product-Quotient Surfaces: Result and Problems
• Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves
• Isotrivially fibred surfaces and their numerical invariants