Geodesic
Singularities on toric fibrations
Sbornik. Mathematics, Tome 212 (2021) no. 3, pp. 288-304

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In this paper we investigate singularities on toric fibrations. In this context we study a conjecture of Shokurov (a special case of which is due to M\textsuperscript{c}Kernan) which roughly says that if $(X,B)\to Z$ is an $\varepsilon$-lc Fano-type log Calabi-Yau fibration, then the singularities of the log base $(Z,B_Z+M_Z)$ are bounded in terms of $\varepsilon$ and $\dim X$ where $B_Z$ and $M_Z$ are the discriminant and moduli divisors of the canonical bundle formula. A corollary of our main result says that if $X\to Z$ is a toric Fano fibration with $X$ being $\varepsilon$-lc, then the multiplicities of the fibres over codimension one points are bounded depending only on $\varepsilon$ and $\dim X$. Bibliography: 20 titles.
Keywords: toric varieties, Shokurov's conjecture, singularities of pairs. 
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     title = {Singularities on toric fibrations},
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C. Birkar; Y. Chen. Singularities on toric fibrations. Sbornik. Mathematics, Tome 212 (2021) no. 3, pp. 288-304. http://geodesic.mathdoc.fr/item/SM_2021_212_3_a2/