Numerical Reduction and Sharp Thresholds for 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 associated with the adjoint divisor \[ K_{\mathcal F}+\epsilon K_X,\qquad \epsilon>0. \]
Using a numerical reduction procedure for negative definite exceptional configurations, we classify \(\epsilon\)-adjoint log canonical singularities for \(0<\epsilon<1/3\). The reduction detects negative vertices, peels off the special chains generated by them, and reduces the classification to the residual intersections left after peeling. In this form, the first stability threshold is \(\epsilon=1/5\): for \(0<\epsilon<1/5\), every \(\epsilon\)-adjoint log canonical singularity is foliated log canonical, while at \(\epsilon=1/5\) a new boundary configuration appears.
Imposing the stronger \(\epsilon\)-adjoint canonical condition gives a second classification below \(\epsilon=1/4\). The same residual mechanism detects the wall \(1/4\), which gives the sharp canonical-to-log-canonical stability interval. Both thresholds are sharp and are realized by explicit examples.
As an application, we describe the negative part of the Zariski decomposition of \(K_{\mathcal F}+\epsilon K_X\) below the wall \(1/4\), and obtain the corresponding stability range for the adjoint minimal model program.