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arXiv 2311.00985math.AG
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toric fibrations上奇点的有效界

Effective bound for singularities on toric fibrations

Sun Yat-sen University（中山大学）

Bingyi Chen

备注：
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
发表信息：
Nagoya Math. J. 260 (2025) 779-795
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AI总结：

本文在toric情况下给出了Fano收缩中奇点有效界δ的显式表达式，其规模为O(ε^{2^r})

摘要：

M\ extsuperscript{c}Kernan和Shokurov曾猜想，对于任何相对维数为r的Fano收缩f:X→Z，当X为ε-lc时，存在一个仅依赖于r和ε的正数δ，使得Z为δ-lc，并且f在Z的codimension一 点处的纤维的乘数被上界限制在1/δ以内。最近，Birkar在Bi23中证实了这一猜想。在本论文中，我们给出了δ的显式值，该值以ε和r为参数，在ε→0时属于O(ε^{2^r})的顺序。这个顺序在某种意义上是最佳的。

英文摘要：

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 $ε$-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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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