Let $S = \mathbb{C}[z_0, \dots, z_n]$, and let $X$ be a set of points in $\mathbb{P}^n$. I'm looking for references concerning results for the graded Betti numbers $\beta_{n,j}(S/I(X))$, i.e., the last column in the Betti diagram for a minimal graded free resolution of $S/I(X)$.

In particular, I'm wondering if anyone has studied the relation between the graded Betti numbers $\beta_{n,j}(S/I(X))$ and $\beta_{n,k}(S/I(X_P'))$, where $X_P' = X \setminus \{P\}$, $P \in X$. (Perhaps for some specific class of sets X.)

Some experimentation with Macaulay2 suggests that for any fixed $j$ with $\beta_{n,j}(S/I(X)) \geq 1$, there exists a point $P$ such that $\beta_{n,j}(S/I(X_P')) = \beta_{n,j}(S/I(X))-1$. Is there a proof or counterexample for this?


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