We obtain local equations for the toric Hilbert scheme, which parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal. We also prove a conjecture of Sturmfels’ providing a criterion for an ideal to have such a Hilbert function.

We give a new short proof of the fact that the CastelnuovoMumford regularity of a strongly stable ideal is the highest degree of a minimal monomial generator. Our proof depends on results due to D. Maclagan and G. Smith on multigraded regularity. More precisely, we construct a Stanley filtration for strongly stable ideals which provides a bound for the Castelnuovo-Mumford regularity.

The paper continues work on monomial ideals in system reliability began by Giglio and Wynn [GW04] following work in discrete tube theory by Naiman and Wynn [NW92, NW97]. The key component is that of multigraded Betti numbers, and an algorithm using MayerVietoris trees by the first author [dC06] is the main tool. First a mapping must be made between the states of a multistate system and a monomi...