On Q-conic bundles, II

TL;DR

This paper classifies Q-conic bundle germs over singular base surfaces, extending previous work by removing the irreducibility assumption of the fiber, thus broadening the understanding of these geometric structures.

Contribution

It provides a complete classification of Q-conic bundle germs with singular base surfaces, generalizing prior results that assumed irreducible fibers.

Findings

Complete classification of Q-conic bundle germs over singular bases

Extension of previous classification results to reducible fibers

Broader understanding of threefold morphisms with terminal singularities

Abstract

A $\mathbb Q$-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. We obtain the complete classification of $\mathbb Q$-conic bundle germs when the base surface germ is singular. This is a generalization of our previous paper math/0603736, which further assumed that the fiber over $o$ is irreducible.