# 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 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?