Canonical Models of Adjoint Foliated Structures on Surfaces

In this paper, we study the adjoint foliated structures of the form $K_{\mathcal{F}}+D$ on algebraic surfaces, with particular focus on their minimal and canonical models. We investigate the effective behavior of the multiple linear system $\mathrm{\mid }m(K_{\mathcal{F}}+D)\mathrm{\mid }$ for sufficiently divisible integers $m>0$. As an application, we provide an effective answer to a boundedness problem for foliated surfaces of general type, originally posed by Hacon and Langer.