The affine structure on $\partial\Delta_{\lambda}^{\vee}\setminus\widetilde{Sing}$ can be often extended to a subset of $\partial\Delta_{\lambda}^{\vee}$ with codimension 2 complement. This in general involves a somewhat ad hoc choice of the singular locus. In the Fermat family case, due to the discrete symmetry, the barycentric subdivision provides a canonical choice. (cf. section 3.5).