Tower sets and other configurations with the Cohen–Macaulay property

Abstract

Some well-known arithmetically Cohen–Macaulay configurations of linear varieties in Pr as k-configurations, partial intersections and star configurations are generalized by introducing tower schemes. Tower schemes are reduced schemes that are a finite union of linear varieties whose support set is a suitable finite subset of Z+c called tower set. We prove that the tower schemes are arithmetically Cohen–Macaulay and we compute their Hilbert function in terms of their support. Afterwards, since even in codimension 2 not every arithmetically Cohen–Macaulay squarefree monomial ideal is the ideal of a tower scheme, we slightly extend this notion by defining generalized tower schemes (in codimension 2). Our main result consists in showing that the support of these configurations (the generalized tower set) gives a combinatorial characterization of the primary decomposition of the arithmetically Cohen–Macaulay squarefree monomial ideals.