Numerical Macaulification in Arbitrary Codimension

An ideal J is said to be numerically c-ACM (NACM) if R/J has the Hilbert function of some codimension c ACM subscheme of P^n. We exhibit algorithm which takes an arbitrary ideal and produces, via a finite sequence of basic double links, an ideal which is numerically c-ACM. An immediate consequence of this result is that every even liaison class of equidimensional codimension c subschemes of P^n contains elements which are NACM. This was first proved for the codimension two case by Migliore and Nagel, but was an open question in higher codimension.

Let L denote the even liaison class of three skew lines in P^4, and let L_S be denote the even liaison class of three skew lines on a smooth hypersurface S in P^4 of degree at least two. We also give a complete description of the sequences of basic double links which (in S) produce curves which are NACM. We conclude by showing that the subset of L_S consisting of NACM subschemes, which we denote by M_S, fails to have the Lazarsfeld-Rao property.