Subsets of complete intersections and the EGH conjecture (Q2895434)

Let consider the following question:NEWLINENEWLINEFix integers \(1\leq d_1\leq d_2\leq \dots\leq d_n\) and let \({\mathcal H}\) be the Hilbert function of some finite set of distinct points in \({\mathbb P}^n\). Do there exist finite sets of distinct, reduced points \({\mathbb X},{\mathbb Y}\), such that \((i) {\mathbb X}\subset {\mathbb Y}\); (ii) the Hilbert function of \({\mathbb X}\) is \({\mathcal H}\) and (iii) \({\mathbb Y}\) is a complete intersection of type \( \{d_1, d_2, \dots, d_n\}\).NEWLINENEWLINEIn this paper the author conjectures that it is enough to solve the above question for a class of complete intersections, namely rectangular complete intersections, and prove it for some special cases in \({\mathbb P}^2 \) and \({\mathbb P}^3\). A rectangular complete intersection of type \( \{d_1, d_2, \dots, d_n\}\), with \(d_1\geq 2\) is the subset \({\mathbb Y}\subset {\mathbb P}^n\) of \(d_1 d_2 \dots d_n\) distinct points with integer coordinates: NEWLINE\[NEWLINE\{ [1:b_1:\dots:b_n]/ b_i\in {\mathbb Z}, 0\leq b_1\leq d_n-1, \dots, 0\leq b_n\leq d_1-1\}.NEWLINE\]NEWLINE Finally as an application the author gives a family of points which has the Cayley-Bacharach Property.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13005].