Classes of simplicial complexes which admit nontrivial Cohen-Macaulay modifications

The set of Cohen-Macaulay monomial ideals with a given radical contains the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. We present classes of Cohen-Macaulay squarefree monomial ideals with infinitely many nontrivial Cohen-Macaulay modifications.