As we saw in the last chapter, we can determine if a set of points X in (Formula Presented) is ACM directly from a combinatorial description of the points. In this chapter we show that this combinatorial description, in particular, the tuples αX and βX, 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 αX and βX or from the set SX. We conclude this chapter by answering the interpolation question introduced in Chapter 1 Specifically, we classify what functions H: ℕ 2→ ℕ are the Hilbert functions of ACM reduced sets of points in (Formula Presented).
Homological invariants
Guardo E.;Van Tuyl A.
2015-01-01
Abstract
As we saw in the last chapter, we can determine if a set of points X in (Formula Presented) is ACM directly from a combinatorial description of the points. In this chapter we show that this combinatorial description, in particular, the tuples αX and βX, 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 αX and βX or from the set SX. We conclude this chapter by answering the interpolation question introduced in Chapter 1 Specifically, we classify what functions H: ℕ 2→ ℕ are the Hilbert functions of ACM reduced sets of points in (Formula Presented).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
