Toric embedded resolutions of quasi-ordinary hypersurface singularities

We build two toric embedded resolutions procedures of a reduced quasiordinary hypersurface singularity (S, 0) of dimension d . The first one provides an embedded resolution as hypersurface of (C, 0) as a composition of toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection (S, 0) → (C, 0) . This gives a positive answer to a a question of Lipman (see [L5]). The second method applies to the analytically irreducible case; if g ≥ 1 denotes the number of characteristic monomials we re-embed the germ (S, 0) in the affine space (C, 0) by using certain approximate roots of a suitable Weierstrass polynomial defining the embedding (S, 0) ⊂ (C, 0) . We build a toric morphism which is a simultaneous toric embedded resolution of the irreducible germ (S, 0) ⊂ (C, 0) , and of an affine toric variety Z obtained from (S, 0) ⊂ (C, 0) by specialization and defined by a rank d semigroup Γ generalizing the classical semigroup of a plane branch. Finally we compare both resolutions and we prove that the first one is the restriction of the second to a smooth (d+1) -variety containing the strict transform of S .