Surfaces of minimal degree of tame representation type and mutations of Cohen–Macaulay modules

We provide two examples of smooth projective surfaces of tame CM type, by showing that the parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in P5 is either a single point or a projective line. These turn out to be the only smooth projective ACM varieties of tame CM type besides elliptic curves, [1]. For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For F0 and F1, embedded as quintic or sextic scrolls, a complete classification of rigid ACM bundles is given.