D-critical locus structure on the Hilbert schemes of some local toric Calabi-Yau threefolds

The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [6]. In this paper we show that there is a d-critical locus structure on the Hilbert scheme of dimension zero subschemes on some local toric Calabi-Yau 3-folds. We also show that using this d-critical locus structure and a choice of orientation data, the resulting motivic invariants agree with the definition given by the previous work of [2].