On $Q$-conic Bundles, II

Abstract

A $Q$-conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ $(Z\ni o)$ of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We obtain the complete classification of $Q$-conic bundle germs when the base surface germ is singular. This is a generalization of [MP08], which further assumed that the fiber over $o$ is irreducible.