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EFFECTIVE BOUND FOR SINGULARITIES ON TORIC FIBRATIONS

Part of: Local theory Birational geometry Surfaces and higher-dimensional varieties Special varieties

Published online by Cambridge University Press:  17 September 2025

BINGYI CHEN Open the ORCID record for BINGYI CHEN [Opens in a new window]
BINGYI CHEN*
Affiliation:
Department of Mathematics https://ror.org/0064kty71 Sun Yat-sen University Guangzhou P. R. China
*
chenby253@mail.sysu.edu.cn
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Abstract

It was conjectured by McKernan 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 [9]. In this article, 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.

Keywords

toric varietiessingularities of pairsminimal model program

MSC classification

Primary: 14B05: Singularities 14J17: Singularities 14M25: Toric varieties, Newton polyhedra 14E30: Minimal model program (Mori theory, extremal rays)

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Type
Article
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Nagoya Mathematical Journal , First View , pp. 1 - 17
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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