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’))$ ?