## Graded Betti numbers $\beta_{n,j}$ for points in $\mathbb{P}^n$

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 free graded resolution of $$S/I(X)$$.

In particular, I’m interested in the following question. Set $$X_P’ = X \setminus \{P\}$$, $$P \in X$$. Does there exist some class of sets $$X$$ such that given the graded Betti numbers $$\beta_{n,j}(S/I(X))$$, one can infer something about $$\beta_{n,j}(S/I(X_P’))$$?