On some properties of partial intersection schemes
In an earlier paper [\textit{R. Ragusa} and \textit{G. Zappalà}, Beitr. Algebra Geom.\ 44, No.1, 285--302 (2003; Zbl 1033.13004)], the authors introduced the notion of ``partial intersection schemes'' in projective space \(\mathbb P^r\). These schemes are \(c\)-codimensional, reduced, arithmetically Cohen-Macaulay unions of linear varieties, obtained by starting with a partially ordered subset of \(\mathbb N^c\) and carrying out a certain technical procedure. As the authors point out in the current paper (remark 1.7), their configurations are precisely the pseudo-liftings of Artinian monomial ideals, a special case of a construction by \textit{J. C. Migliore} and \textit{U. Nagel} [Commun.\ Algebra 28, No. 12, 5679--5701(2000; Zbl 1003.13005)]. Nevertheless, the authors' combinatorial approach provides a fresh and useful way of viewing these objects. They first show that partial intersection schemes are not necessarily in the linkage class of a complete intersection (i.e.\ they are not necessarily licci). Then they give a large class of partial intersection schemes that nonetheless are licci. They complete the picture by showing that every partial intersection is in the Gorenstein linkage class of a complete intersection (i.e.\ glicci). This latter result had been shown earlier from the point of view of pseudo-liftings [\textit{J. C. Migliore} and \textit{U. Nagel}, Compos. Math.\ 133, No. 1, 25--36 (2002; Zbl 1047.14034)]. The last part of the paper gives interesting bounds and connections between the first and last graded Betti numbers of partial intersections, especially in codimension 3. A nice statement (among others) is that if \(X\) is a 3-codimensional partial intersection having \(p\) minimal last syzygies, and if \(\nu (I_X)\) is the number of minimal generators, then \(\lceil {{p+5} \over 2} \rceil \leq \nu(I_X) \leq 2p+1\). The authors also show that all possibilities in this range can occur. As they point out, such a bound is impossible in the case of arithmetically Cohen-Macaulay subschemes in general.
scientific article; zbMATH DE number 2064201
scientific article