Shokurov's conjecture on conic bundles with canonical singularities

TL;DR

This paper proves Shokurov's conjecture on the singularities of bases in conic bundles with canonical singularities, establishing bounds on fiber multiplicities and the nature of the base's singularities.

Contribution

It confirms Shokurov's conjecture for conic bundles with canonical singularities, providing sharp bounds and extending to lc-trivial fibrations of relative dimension one.

Findings

Base $Z$ is always $rac{1}{2}$-lc.

Fiber multiplicities over codimension 1 points are bounded by 2.

Results apply to more general lc-trivial fibrations.

Abstract

A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov which predicts that, if $X\to Z$ is a conic bundle such that $X$ has canonical singularities and $Z$ is $\mathbb{Q}$-Gorenstein, then $Z$ is always $\frac{1}{2}$-lc, and the multiplicities of the fibers over codimension $1$ points are bounded from above by $2$. Both values $\frac{1}{2}$ and $2$ are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension $1$ with canonical singularities.