On Cohen–Macaulay modules on surface singularities

We study Cohen–Macaulay modules over normal surface singularities. Using the method of Kahn and extending it to families of modules, we classify Cohen–Macaulay modules over cusp singularities and prove that a minimally elliptic singularity is Cohen–Macaulay tame if and only if it is either simple elliptic or cusp. As a corollary, we obtain a classification of Cohen–Macaulay modules over log-canonical surface singularities and hypersurface singularities of type ${\rm T}_{pqr}$ especially they are Cohen–Macaulay tame. We also calculate the Auslander–Reiten quiver of the category of Cohen–Macaulay modules in the considered cases.