LOOKING FOR MINIMAL GRADED BETTI NUMBERS

摘要

We consider O-sequences that occur for arithmetically Cohen-Macaulay (ACM) schemes X of codimension three in P~n. These are Hilbert functions φ of Artinian algebras that are quotients of the coordinate ring of X by a linear system of parameters. Using suitable decompositions of φ, we determine the minimal number of generators possible in some degree c for the defining ideal of any such ACM scheme having the given O-sequence. We apply this result to construct Artinian Gorenstein O-sequences φ of codimension 3 such that the poset of all graded Betti sequences of the Artinian Gorenstein algebras with Hilbert function φ admits more than one minimal element. Finally, for all 3-codimensional complete intersection O-sequences we obtain conditions under which the corresponding poset of graded Betti sequences has more than one minimal element.