Numerical Properties of Isotrivial Fibrations

A smooth, projective surface S is called a standard isotrivial fibration if there exists a finite group G which acts faithfully on two smooth projective curves C1 and C2, so that S is isomorphic to the minimal desingularization of T := (C1 × C2)/G, where G acts diagonally on the product. When the action of G is free, then S = T is called a quasi-bundle. In this paper we analyse several numerical properties of standard isotrivial fibrations. In particular we prove that if q(S) ≥ 1 and S is neither ruled nor a quasi-bundle, then its minimal model b S satisfies K2b S ≤ 8χ(O b S) − 2; moreover, equality holds if and only if the singularities of T are exactly two ordinary double points, and in this case S is a minimal surface of general type. This improves previous results of Serrano. Under the same assumptions on S, we provide a lower bound for K S in terms of pg(S) and q(S) only. Using this bound one should be able, in principle, to achieve the complete classification of standard isotrivial fibrations with fixed birational invariants. 0. Introduction One of the most useful tools in the study of algebraic surfaces is the analysis of fibrations, that is morphisms with connected fibres from a surface X onto a curve C. When all smooth fibres of a fibration φ : X −→ C are isomorphic to each other, we call φ an isotrivial fibration. As far as we know, there is hitherto no systematic study of minimal models of isotrivial fibrations; the aim of the present paper is to shed some light on this problem. A smooth, projective surface S is called a standard isotrivial fibration if there exists a finite group G acting faithfully on two smooth projective curves C1 and C2, so that S is isomorphic to the minimal desingularization of T := (C1 ×C2)/G, where G acts diagonally on the product. When the action of G is free, then S = T is called a quasi-bundle. These surfaces have been investigated in [Se90], [Se96], [Ca00], [BaCaGr06], [Pol07], [CarPol07], [MiPol08], [BaCaGrPi08]. A monodromy argument shows that every isotrivial fibration φ : X −→ C is birationally isomorphic to a standard one ([Se96, Section 2]); this means that there exist T = (C1 × C2)/G, a birational map T 99K X and an isomorphism C2/G −→ C such that the diagram T // _ _ _ _ X