Looking for minimal graded Betti numbers

Abstract

We consider $O$-sequences that occur for arithmetically Cohen-Macaulay (ACM) schemes $X$ of codimension three in ${\pp}^n$. These are Hilbert functions $\varphi$ of Artinian algebras that are quotients of the coordinate ring of $X$ by a linear system of parameters. Using suitable decompositions of $\varphi$, 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 $\varphi$ of codimension $3$ such that the poset of all graded Betti sequences of the Artinian Gorenstein algebras with Hilbert function $\varphi$ 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.