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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.