标题： 环上挠率纤维的有效界限 显示英文标题

标题： Effective bound for singularities on toric fibrations

摘要： M\textsuperscript{c}Kernan和Shokurov猜想，对于任何相对维数为$r$的Fano收缩$f:X \to Z$，其中$X$是$\epsilon$-lc，存在一个仅依赖于$r,\epsilon$的正数$\delta$，使得$Z$是$\delta$-lc，并且$f$在$Z$的余维一点上的纤维的重数被$1/\delta$从上方有界。 最近，这个猜想被Birkar \cite{Bi23}确认。 在本文中，我们在环面情况下给出了$\delta$关于$\epsilon,r$的显式值，这属于$O(\epsilon^{2^r})$中的$\epsilon\rightarrow 0$。 该阶$O(\epsilon^{2^r})$在某种意义上是最优的。 显示英文摘要

摘要： 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.