ACM vector bundles on projective surfaces of nonnegative Kodaira dimension

In this paper, we contribute to the construction of families of arithmetically Cohen-Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces (X,OX(1)) for OX(1) an ample line bundle. In many cases, we show that for every positive integer r there exists a family of indecomposable aCM vector bundles of rank r, depending roughly on r parameters, and in particular they are of wild representation type. We also introduce a general setting to study the complexity of a polarized variety (X,OX(1)) with respect to its category of aCM vector bundles. In many cases we construct indecomposable vector bundles on X which are aCM for all ample line bundles on X.