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_∌_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.