Geodesic
Optimal bound for singularities on Fano type fibrations of relative dimension one
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e193

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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 $K_X+B\sim _{\mathbb {R}} 0/Z$. 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 $\mathbb {R}$-linear equivalence. Shokurov conjectured that one can choose $M_Z\geqslant 0$ 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 [8]. For $d=1$ and $\epsilon =1$, Han, Jiang, and Luo [13] gave the optimal value of $\delta =1/2$. In this paper, we give the optimal value of $\delta $ for $d=1$ and arbitrary $0<\epsilon \leqslant 1$. 
Chen, Bingyi. Optimal bound for singularities on Fano type fibrations of relative dimension one. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e193. doi: 10.1017/fms.2025.10141
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