Let V be a smooth quasi-projective complex surface such that the first three logarithmic plurigenera $\overline P_1(V)$ , $\overline P_2(V)$ and $\overline P_3(V)$ are equal to 1 and the logarithmic irregularity $\overline q(V)$ is equal to $2$ . We prove that the quasi-Albanese morphism $a_V\colon V\to A(V)$ is birational and there exists a finite set S such that $a_V$ is proper over $A(V)\setminus S$ , thus giving a sharp effective version of a classical result of Iitaka [12].

Let be an infinitesimal group scheme, defined over an algebraically closed field of characteristic p. We employ rank varieties of -modules to study the stable Auslander-Reiten quiver of the distribution algebra of . As in case of finite groups, the tree classes of the AR-components are finite or infinite Dynkin diagrams, or Euclidean diagrams. We classify the components of finite and Euclidean type in case is supersolvable or a Frobenius kernel of a smooth, reductive group.