On $Q$-conic Bundles

Abstract

A $Q$-conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of $Q$-conic bundles near their singular fibers. One corollary to our main results is that the base surface of every $Q$-conic bundle has only Du Val singularities of type A (a positive solution of a conjecture by Iskovskikh). We obtain the complete classification of $Q$-conic bundles under the additional assumption that the singular fiber is irreducible and the base surface is singular.