Effective bound for singularities on toric fibrations

TL;DR

This paper provides an explicit bound for singularities on toric fibrations, confirming a conjecture in the toric case and showing the bound's optimality in terms of epsilon and relative dimension.

Contribution

It explicitly determines the bound delta for singularities on toric fibrations, refining previous conjectures and results in the toric setting.

Findings

Explicit bound delta in terms of epsilon and r for toric fibrations

The bound belongs to O(epsilon^{2^r}) as epsilon approaches zero

The bound's order is proven to be optimal in some sense

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.