Optimal bound for singularities on Fano type fibrations of relative dimension one
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866912134048251904 |
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| author | Chen, Bingyi |
| author_facet | Chen, Bingyi |
| contents | 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 $0<ε\leq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_08469 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Optimal bound for singularities on Fano type fibrations of relative dimension one Chen, Bingyi Algebraic Geometry 14E30 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 $0<ε\leq 1$. |
| title | Optimal bound for singularities on Fano type fibrations of relative dimension one |
| topic | Algebraic Geometry 14E30 |
| url | https://arxiv.org/abs/2210.08469 |