论文标题
在相对尺寸的Fano型纤维上，最佳界限
Optimal bound for singularities on Fano type fibrations of relative dimension one
论文作者
论文摘要
让$π：x \ rightarrow z $为$ \ dim x- \ dim z = d $，让$（x，b）$是$ε$ -lc对，带有$ k_x+b \ sim _ {\ rr} 0/z $。规范捆绑包配方给出$（z，b_z+m_z）$，其中$ b_z $是判别分数，而$ m_z $是moduli除数，该模量最高可达$ \ rr $ lineAr-linear等价。 Shokurov猜想可以选择$ m_z \ geq 0 $，这样$（z，b_z+m_z）$是$δ$ -LC，其中$δ$仅取决于$ d，ε$。最近，这个猜想是由Birkar \ cite {bir23}证明的。对于$ d = 1 $和$ε= 1 $，Han，Jiang和Luo \ cite {hjl22}给出了$δ= 1/2 $的最佳值。在本文中，我们给出了$ d = 1 $和任意$ 0 <ε\ leq 1 $的$δ$的最佳值。
Let $π:X\rightarrow Z$ be a Fano type fibration with $\dim X-\dim Z=d$ and let $(X,B)$ be an $ε$-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 $δ$-lc where $δ$ only depends on $d,ε$. Very recently, this conjecture was proved by Birkar \cite{Bir23}. For $d=1$ and $ε=1$, Han, Jiang and Luo \cite{HJL22} gave the optimal value of $δ=1/2$. In this paper, we give the optimal value of $δ$ for $d=1$ and arbitrary $0<ε\leq 1$.
