Effective bound for singularities on toric fibrations

It was conjectured by M\textsuperscript{c}Kernan and Shokurov that for any Fano contraction $f:X \to Z$ of relative dimension $r$ with $X$ being $ε$-lc, there is a positive $δ$ depending only on $r,ε$ such that $Z$ is $δ$-lc and the multiplicity of the fiber of $f$ over a codimension one point of $Z$ is bounded from above by $1/δ$. Recently, this conjecture was confirmed by Birkar \cite{Bi23}. In this paper, we give an explicit value for $δ$ in terms of $ε,r$ in the toric case, which belongs to $O(ε^{2^r})$ as $ε\rightarrow 0$. The order $O(ε^{2^r})$ is optimal in some sense.

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