On Q-conic bundles, III

TL;DR

This paper proves the existence of a Du Val anti-canonical member in Q-conic bundle germs with an irreducible central fiber, extending previous work on threefolds with terminal singularities.

Contribution

It establishes the existence of a Du Val anti-canonical divisor in Q-conic bundle germs with irreducible central fibers, building on prior classifications.

Findings

Existence of Du Val anti-canonical members proven

Applicable to Q-conic bundles with irreducible central fibers

Extends previous classification results

Abstract

A 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. Building upon our previous paper [math/0603736], we prove the existence of a Du Val anti-canonical member under the assumption that the central fiber is irreducible.