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

TL;DR

This paper investigates the stability thresholds of $oldsymbol{ ext{epsilon}}$-adjoint singularities on foliated surfaces, identifying sharp bounds at $oldsymbol{1/5}$ and $oldsymbol{1/4}$, with explicit classifications of extremal configurations.

Contribution

It establishes the precise stability thresholds for $oldsymbol{ ext{epsilon}}$-adjoint singularities on foliated surfaces and provides a complete classification of these singularities for certain epsilon ranges.

Findings

Sharp stability threshold at $oldsymbol{1/5}$ for $oldsymbol{ ext{epsilon}}$-adjoint singularities.

Maximal stability interval is $oldsymbol{(0,1/4)}$ in the canonical case.

Explicit extremal configurations characterize the thresholds.

Abstract

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 $\epsilon=1/5$: for $\epsilon \in (0,1/5)$, every $\epsilon$-adjoint log canonical singularity is foliated log canonical, while at $\epsilon=1/5$ a boundary configuration enters the admissible region. In the adjoint canonical setting, the maximal stability interval is $\epsilon \in (0,1/4)$. 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 $\epsilon \in (0,1/3)$ in terms of negative definite exceptional configurations.