Geodesic
Shokurov’s conjecture on conic bundles with canonical singularities
Forum of Mathematics, Sigma, Tome 10 (2022)

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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 that 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 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. 
@article{10_1017_fms_2022_32,
     author = {Jingjun Han and Chen Jiang and Yujie Luo},
     title = {Shokurov{\textquoteright}s conjecture on conic bundles with canonical singularities},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {10},
     year = {2022},
     doi = {10.1017/fms.2022.32},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.32/}
}
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Jingjun Han; Chen Jiang; Yujie Luo. Shokurov’s conjecture on conic bundles with canonical singularities. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.32

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