Note on the 3-dimensional log canonical abundance in characteristic $>3$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866929232115924992 |
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| author | Xu, Zheng |
| author_facet | Xu, Zheng |
| contents | In this paper, we prove the non-vanishing and some special cases of the abundance for log canonical threefold pairs over an algebraically closed field $k$ of characteristic $p > 3$. More precisely, we prove that if $(X,B)$ be a projective log canonical threefold pair over $k$ and $K_{X}+B$ is pseudo-effective, then $κ(K_{X}+B)\geq 0$, and if $K_{X}+B$ is nef and $κ(K_{X}+B)\geq 1$, then $K_{X}+B$ is semi-ample.
As applications, we show that the log canonical rings of projective log canonical threefold pairs over $k$ are finitely generated and the abundance holds when the nef dimension $n(K_{X}+B)\leq 2$ or when the Albanese map $a_{X}:X\to \mathrm{Alb}(X)$ is non-trivial. Moreover, we prove that the abundance for klt threefold pairs over $k$ implies the abundance for log canonical threefold pairs over $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_04039 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Note on the 3-dimensional log canonical abundance in characteristic $>3$ Xu, Zheng Algebraic Geometry In this paper, we prove the non-vanishing and some special cases of the abundance for log canonical threefold pairs over an algebraically closed field $k$ of characteristic $p > 3$. More precisely, we prove that if $(X,B)$ be a projective log canonical threefold pair over $k$ and $K_{X}+B$ is pseudo-effective, then $κ(K_{X}+B)\geq 0$, and if $K_{X}+B$ is nef and $κ(K_{X}+B)\geq 1$, then $K_{X}+B$ is semi-ample. As applications, we show that the log canonical rings of projective log canonical threefold pairs over $k$ are finitely generated and the abundance holds when the nef dimension $n(K_{X}+B)\leq 2$ or when the Albanese map $a_{X}:X\to \mathrm{Alb}(X)$ is non-trivial. Moreover, we prove that the abundance for klt threefold pairs over $k$ implies the abundance for log canonical threefold pairs over $k$. |
| title | Note on the 3-dimensional log canonical abundance in characteristic $>3$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2212.04039 |