On Q-conic bundles

Abstract

A $\mathbb{Q}$-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 $\mathbb{Q}$-conic bundles near their singular fibers. One corollary to our main results is that the base surface of every $\mathbb{Q}$-conic bundle has only Du Val singularities of type A (a positive solution of a conjecture by Iskovskikh). We obtain the complete classification of $\mathbb{Q}$-conic bundles under the additional assumption that the singular fiber is irreducible and the base surface is singular.

54 pages, LaTeX