Shokurov's conjecture on conic bundles with canonical singularities. (9th June 2022) - Han, Jingjun; Jiang, Chen; Luo, Yujie
Abstract: A conic bundle is a contraction $X→ Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov that predicts that if $X→ Z$ 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 fibres 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.