By analogy with the representation theory of quivers, one says that a projective variety X is of finite type if its homogeneous coordinate ring R has finitely many maximal Cohen-Macaulay (CM) indecomposable modules; also X is tame if these modules vary in families of dimension 1 at most, or wild if the dimension of these families is unbounded.
I will show that, if R is CM and X is not a cone, then X is wild except for a number of completely classified cases. If time allows I will describe CM modules on a few tame varieties.
[In collaboration with J. Pons-Llopis].