Low dimensional orders of finite representation type

In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427-452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten-Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups G subset of GL2, explicitly computingH2(G,k*), and then matching these up with Artin's list of ramification data and Reiten-Van den Bergh's AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83-115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let B = k zeta[x,y] be the skew power series ring where zeta is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form A = B/(f) where f is an element of Z(B) which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.