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.
Citation
Alfio Ragusa. Giuseppe Zappalà. "Looking for minimal graded Betti numbers." Illinois J. Math. 49 (2) 453 - 473, Summer 2005. https://doi.org/10.1215/ijm/1258138028
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