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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2512.20744 |
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| _version_ | 1866914363951022080 |
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| author | Xu, Shi |
| author_facet | Xu, Shi |
| contents | Let $(X,\mathcal{F})$ be a foliated surface over the complex numbers. We study the variation of $ε$-adjoint singularities, defined by the adjoint divisor $K_{\mathcal{F}}+εK_X$ ($ε>0$), and analyze their stability as $ε$ varies.
We prove that a sharp first stability threshold occurs at $ε=1/5$: for $ε\in (0,1/5)$, every $ε$-adjoint log canonical singularity is foliated log canonical, while at $ε=1/5$ a boundary configuration enters the admissible region. In the adjoint canonical setting, the maximal stability interval is $ε\in (0,1/4)$. Both thresholds are optimal and arise from explicit extremal configurations.
These results are obtained via a complete classification of $ε$-adjoint log canonical singularities for $ε\in (0,1/3)$ in terms of negative definite exceptional configurations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Thresholds for $ε$-Adjoint Singularities of Foliated Surfaces Xu, Shi Algebraic Geometry Let $(X,\mathcal{F})$ be a foliated surface over the complex numbers. We study the variation of $ε$-adjoint singularities, defined by the adjoint divisor $K_{\mathcal{F}}+εK_X$ ($ε>0$), and analyze their stability as $ε$ varies. We prove that a sharp first stability threshold occurs at $ε=1/5$: for $ε\in (0,1/5)$, every $ε$-adjoint log canonical singularity is foliated log canonical, while at $ε=1/5$ a boundary configuration enters the admissible region. In the adjoint canonical setting, the maximal stability interval is $ε\in (0,1/4)$. Both thresholds are optimal and arise from explicit extremal configurations. These results are obtained via a complete classification of $ε$-adjoint log canonical singularities for $ε\in (0,1/3)$ in terms of negative definite exceptional configurations. |
| title | Sharp Thresholds for $ε$-Adjoint Singularities of Foliated Surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.20744 |