 Multiplicity descent in toric desingularization

First make a toric fan simplicial by compatible rational subdivisions. In a nonregular simplicial cone, choose a nonregular face minimal under inclusion. Its half-open fundamental parallelepiped contains a nonzero point of the toric lattice $w={\sum }_i{\alpha }_i{v}_i$, with every $0<{\alpha }_i<1$: a zero coefficient would put it in a regular proper face and force all coefficients to be integers. A star subdivision at the primitive point on this ray replaces a generator by $w$ and multiplies the affected lattice determinant by a number strictly between zero and one. The maximum multiplicity and the number of maximal cones realizing it decrease lexicographically, giving a finite toric resolution of singularities.