Ruled fibrations on normal surfaces (Q1109092)

Let Y be a complex normal projective surface. A ruled fibration on Y is a surjective morphism \(p: Y\to B\) over a smooth curve, whose general fibre is isomorphic to \({\mathbb{P}}^ 1.\) A ruled fibration is called minimal if its fibres contain no exceptional curve C of the first kind (i.e. an irreducible curve C with \(K_ Y\cdot C<0\) and \(C^ 2<0\), where \(K_ Y\) stands for a canonical divisor). The structure of normal surfaces admitting a minimal ruled fibration p is studied. The author shows that all singular fibres of p are multiple fibres containing one or two points of Sing(Y). In particular this allows him to correct a statement in one of his previous papers [J. Reine Angew. Math. 366, 121-128 (1986; Zbl 0582.14011)]. By using a suitable invariant related to the amount of Sing(Y), the author extends to the normal case some results, which are known for smooth ruled surfaces, e.g. the ampleness criterion and the Nagata inequality for the self-intersection numbers of all sections of p. Moreover the classification of Y in terms of its anti-Kodaira dimension and the numerical type of \(-K_ Y\) are given. As a last thing the author proves that \(-K_ Y\) is ample if and only if Y either admits another minimal ruled fibration or contains an exceptional curve of the first kind.