Homological invariants

Abstract

As we saw in the last chapter, we can determine if a set of points X in $\mathbb{P}^{1} \times \mathbb{P}^{1}$ is ACM directly from a combinatorial description of the points. In this chapter we show that this combinatorial description, in particular, the tuples α _X and β _X , also allows us to determine the bigraded Betti numbers in the bigraded minimal free resolution of I(X) when X is ACM. Consequently, the Hilbert function of X when X is ACM can also be computed directly from α _X and β _X or from the set S _X . We conclude this chapter by answering the interpolation question introduced in Chapter 1 Specifically, we classify what functions $H: \mathbb{N}^{2} \rightarrow \mathbb{N}$ are the Hilbert functions of ACM reduced sets of points in $\mathbb{P}^{1} \times \mathbb{P}^{1}$.