Singularities on toric fibrations

TL;DR

This paper studies singularities in toric fibrations, proving bounds on fiber multiplicities over codimension one points for epsilon-lc toric Fano fibrations, advancing understanding of singularity behavior in algebraic geometry.

Contribution

It establishes bounds on fiber multiplicities in toric Fano fibrations with epsilon-lc singularities, confirming a special case of a conjecture related to singularity boundedness.

Findings

Bounded fiber multiplicities depending only on epsilon and dimension.

Verification of a special case of Shokurov's conjecture for toric fibrations.

Enhanced understanding of singularity control in toric Fano fibrations.

Abstract

In this paper we investigate singularities on toric fibrations. In this context we study a conjecture of Shokurov (a special case of which is due to M$^\rm{c}$Kernan) which roughly says that if $(X,B)\to Z$ is an $\epsilon$-lc Fano type log Calabi-Yau fibration, then the singularities of the log base $(Z,B_Z+M_Z)$ are bounded in terms of $\epsilon$ and $\dim X$ where $B_Z,M_Z$ are the discriminant and moduli divisors of the canonical bundle formula. A corollary of our main result says that if $X\to Z$ is a toric Fano fibration with $X$ being $\epsilon$-lc, then the multiplicities of the fibres over codimension one points are bounded depending only on $\epsilon$ and $\dim X$.