Resolving Singularities of Plane Analytic Branches with one Toric Morphism

Abstract

Let (C, 0) be an irreducible germ of complex plane curve. Let Γ ⊂ ℕ be the semigroup associated to it and C Γ ⊂ ℂ g+1 the corresponding monomial curve, where g is the number of Puiseux exponents of (C, 0). We show, using the specialization of (C 0) to (C Γ, 0), that the same toric morphisms ZΣ→ℂ g+1 which induce an embedded resolution of singularities of (C Γ, 0) also resolve the singularities of (C, 0) ⊂ (⊂ ℂ g+1, 0), the embedding being defined by elements of the analytic algebra O C, 0 whose valuations generate the semigroup Γ.KeywordsComplete IntersectionExceptional DivisorSemi GroupSimplicial ConeNewton PolyhedronThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.