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

1. Improve the bound in the main result (Theorem 1.2). 2. Add Example 1.4 to indicate that the order of the bound is optimal