Numerical properties of fat schemes with special support

Given a finite subset S of N^c we construct a standard -scheme whose defining ideal is a square-free monomial ideal, and, more generally, we construct -schemes with respect to families of generic linear forms. For these schemes we give a characterization on ⊂ 2 in order to be arithmetically Cohen–Macaulay. Moreover, for fat schemes with support on a standard -scheme, we produce a minimal free resolution. Using this result, we furnish the graded Betti sequences for such fat schemes and, under some numerical conditions, for fat -schemes where has some combinatorial property.