Optimal bound for singularities on Fano type fibrations of relative dimension one

Abstract

Let $\pi:X\rightarrow Z$ be a Fano type fibration with $\dim X-\dim Z=d$ and let $(X,B)$ be an $\epsilon$-lc pair with $K_X+B\sim_{\RR} 0/Z$. The canonical bundle formula gives $(Z,B_Z+M_Z)$ where $B_Z$ is the discriminant divisor and $M_Z$ is the moduli divisor which is determined up to $\RR$-linear equivalence. Shokurov conjectured that one can choose $M_Z\geq 0$ such that $(Z,B_Z+M_Z)$ is $\delta$-lc where $\delta$ only depends on $d,\epsilon$. Very recently, this conjecture was proved by Birkar \cite{Bir23}. For $d=1$ and $\epsilon=1$, Han, Jiang and Luo \cite{HJL22} gave the optimal value of $\delta=1/2$. In this paper, we give the optimal value of $\delta$ for $d=1$ and arbitrary $0<\epsilon\leq 1$.