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Numerical Macaulification

An unpublished example due to Joe Harris from 1983 (or earlier) gave two smooth space curves with the same Hilbert function, but one of the curves was arithmetically Cohen-Macaulay (ACM) and the other was not. Starting with an arbitrary homogeneous ideal in any number of variables, we give two constructions, each of which produces, in a finite number of steps, an ideal with the Hilbert function of a codimension two ACM subscheme. We call the subscheme associated to such an ideal "numerically ACM." We study the connections between these two constructions, and in particular show that they produce ideals with the same Hilbert function. We call the resulting ideal from either construction a "numerical Macaulification" of the original ideal. Specializing to the case where the ideals are unmixed of codimension two, we show that (a) every even liaison class, $\mathcal L$, contains numerically ACM subschemes, (b) the subset, $\mathcal M$, of numerically ACM subschemes in $\mathcal L$ has, by itself, a Lazarsfeld-Rao structure, and (c) the numerical Macaulification of a minimal element of $\mathcal L$ is a minimal element of $\mathcal M$. Finally, if we further restrict to c