Effective bound for singularities on toric fibrations

Abstract

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 $\epsilon$-lc, there is a positive $\delta$ depending only on $r,\epsilon$ such that $Z$ is $\delta$-lc and the multiplicity of the fiber of $f$ over a codimension one point of $Z$ is bounded from above by $1/\delta$. Recently, this conjecture was confirmed by Birkar \cite{Bi23}. In this paper, we give an explicit value for $\delta$ in terms of $\epsilon,r$ in the toric case, which belongs to $O(\epsilon^{2^r})$ as $\epsilon\rightarrow 0$. The order $O(\epsilon^{2^r})$ is optimal in some sense.