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Some well-known arithmetically Cohen-Macaulay configurations of linearvarieties in $\mathbb{P}^r$ as $k$-configurations, partial intersections andstar configurations are generalized by introducing tower schemes. Tower schemesare reduced schemes that are finite union of linear varieties whose support setis a suitable finite subset of $\mathbb{Z}_+^c$ called tower set. We prove thatthe tower schemes are arithmetically Cohen-Macaulay and we compute theirHilbert function in terms of their support.  Afterwards, since even in codimension 2 not every arithmeticallyCohen-Macaulay squarefree monomial ideal is the ideal of a tower scheme, weslightly extend this notion by defining generalized tower schemes (incodimension 2) and we show that the support of these configurations (thegeneralized tower set) gives a combinatorial characterization of the primarydecomposition of the arithmetically Cohen-Macaulay squarefree monomial ideals.

Tower sets and other configurations with the Cohen-Macaulay property