Hypersurfaces of bounded Cohen-Macaulay type

Let R=k[[x0,…,xd]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring k[[x0,…,xd]]. We investigate the question of which rings of this form have bounded Cohen-Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen-Macaulay modules. As with finite Cohen-Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen-Macaulay type if and only if R≅k[[x0,…,xd]]/(g+x22+⋯+xd2), where g∈k[[x0,x1]] and k[[x0,x1]]/(g) has bounded Cohen-Macaulay type. We determine which rings of the form k[[x0,x1]]/(g) have bounded Cohen-Macaulay type.