Adjoint log canonical foliated singularities on surfaces

Abstract: Let $(X,\mathcal{F})$ be a foliated surface over $\mathbb{C}$. We study the singularities of the adjoint foliated divisor $K_{\mathcal{F}}+εK_X$. We provide a complete classification of $ε$-adjoint log canonical singularities of foliated surfaces for $ε\in(0,1/3)$. Moreover, we prove that for any $ε\in(0,1/5)$, every $ε$-adjoint log canonical singularity is log canonical for $\mathcal{F}$, and that for any $ε\in(0,1/4)$, every $ε$-adjoint canonical singularity is log canonical for $\mathcal{F}$. Finally, we present examples showing that both bounds are sharp.