On $ℚ$-Conic Bundles, III

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. Building upon our previous paper [MP08a], we prove the existence of a Du Val anti-canonical member under the assumption that the central fiber is irreducible. §1. IntroductionThe present paper is a continuation of a series of papers [MP08a], [MP08b]. Recall that a Q-conic bundle is a projective morphism f : X → Z from an (algebraic or analytic) threefold with terminal singularities to a surface that satisfies the following properties:For f : X → Z as above and for a point o ∈ Z, we call the analytic germ (X, f −1 (o) red ) a Q-conic bundle germ.In this paper we complete the proof of the following