Sharp Thresholds for $\epsilon$-Adjoint Singularities of Foliated Surfaces

Let $(X,\mathcal{F})$ be a foliated surface over the complex numbers. We study the variation of $\epsilon$-adjoint singularities, defined by the adjoint divisor $K_{\mathcal{F}}+\epsilon K_X$ ($\epsilon >0$), and analyze their stability as $\epsilon$ varies. We prove that a sharp first stability threshold occurs at $\epsilon =1\mathrm{/}5$: for $\epsilon \in (0,1\mathrm{/}5)$, every $\epsilon$-adjoint log canonical singularity is foliated log canonical, while at $\epsilon =1\mathrm{/}5$ a boundary configuration enters the admissible region. In the adjoint canonical setting, the maximal stability interval is $\epsilon \in (0,1\mathrm{/}4)$. Both thresholds are optimal and arise from explicit extremal configurations. These results are obtained via a complete classification of $\epsilon$-adjoint log canonical singularities for $\epsilon \in (0,1\mathrm{/}3)$ in terms of negative definite exceptional configurations.