Blowing up monomial ideals

Let k be a field. Spivakovskyʹs theorem on the solution of Hironakaʹs polyhedral game has been extended by Bloch to show that a morphism f : Z→S of finite type k-schemes can be put in good position with respect to a normal crossing divisor ∂S on S by taking the proper transform with respect to an iterated blowing up of faces of ∂S. We extend these results to schemes of finite type over a regular scheme of dimension one, including the case of mixed characteristic.