On Q-conic bundles, II

Abstract

A $\mathbb Q$-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 $\mathbb Q$-conic bundle germs when the base surface germ is singular. This is a generalization of our previous paper math/0603736, which further assumed that the fiber over $o$ is irreducible.