Toric modifications of free toric varieties

Geometric properties of strict transforms of certain toric complete intersection varieties under toric blowups are studied. Let V=V(f_1,…,f_n−r) be a free complete intersection in A^n which is a simplicial affine toric variety with weight semigroup generated by n vectors [see J. C. Rosales and P. A. García-Sánchez, Semigroup Forum 58 (1999), no. 3, 367–385]. With the set {f_1 ,…,f_n−r} one associates a fan Σ(f_1,…,f_ n−r) with support R^n+. Each cone in Σ(f_1 ,…,f_ n−r) is defined as the set of all vectors a∈R^n+ such that the inner product (a,⋅) achieves its minimum on a given face of the Newton polyhedron of each f_i. The fan Σ(f_1 ,…,f_n−r) can be refined to a regular fan Σ. The latter fan defines the blowup morphism π:X=X(Σ)→A^n, which is isomorphic outside π^−1(V). The main result of the paper says that the strict transform of V is non-singular and transversal to the strata of the stratification of π^−1(0). This result can be viewed as a generalization of previous constructions of embedded resolutions of curve singularities [R. Goldin and B. Teissier, in Resolution of singularities (Obergurgl, 1997), 315–340, Progr. Math., 181, Birkhäuser, Basel, 2000] and of quasi-ordinary singularities.