Surfaces of minimal degree of tame and wild representation type

We provide two examples of smooth projective surfaces of tame CM type, by showing that any parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in projective 5-space is either a single point or a projective line. For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For the rational normal scrolls S(2,3) and S(3,3), a complete classification of rigid ACM bundles is given in terms of the action of the braid group in three strands.