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

Let π :X→ Z be a Fano type fibration with X- Z=d and let $(X,B)$ be an ε -lc pair with KX+B~ R 0/Z . The canonical bundle formula gives (Z,BZ+MZ) where BZ is the discriminant divisor and MZ is the moduli divisor which is determined up to R -linear equivalence. Shokurov conjectured that one can choose MZ 0 such that (Z,BZ+MZ) is δ -lc where δ only depends on d,ε . Very recently, this conjecture was proved by Birkar [8]. For $d=1$ and ε =1 , Han, Jiang, and Luo [13] gave the optimal value of δ =1/2 . In this paper, we give the optimal value of δ for $d=1$ and arbitrary 0<ε 1 .