The Picard group of a quasi-bundle

Abstract

A quasi-bundle is defined to be a morphism from an algebraic surface onto a curve having all smooth fibres connected and isomorphic, and allowing as only singular fibres multiples of smooth curves. When no multiple fibre occurs it is called a fibre bundle. Toe general fibre F of a quasi-bundle is said to be divisible by an integer k if (I/k)F is still the numerical class of an integral divisor. This paper focuses on the relationship between the divisibility properties of F and the torsion of H 2 ( S, Z). For fibre bundles, the link between those two notions is established by means of Serre spectral sequence. As for general quasi-bundles, a suitable base change leads back to the fibre bundle case. Toe results become most explicit for elliptic quasibundles, where the action of the monodromy can be fully computed. For any prime number p, the paper contains examples of fibre bundles whose fibre is divisible by p.