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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.20744 |
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Table of 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.