= Multiplicity descent in toric desingularization
{title2=$\operatorname{mult}(\sigma)=[N\cap\operatorname{span}_{\mathbb R}\sigma:\sum_i\mathbb Zv_i]$}
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> $w=\sum_i\alpha_iv_i$, 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>.
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