We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka's maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.Moreover, if there is an equivalence φ between two support fabrics F 1 and F 2 , then for every J ∈ H 1 , the blow-ups π J (F 1 ) and π φ(J) (F 2 ) are also equivalent.There is a surjective map π # J : H ′ → H between the strata sets of π J (F ) and F respectively, given by π # J (H ′K ∞ ) = {K}, for each K ∈ {J} ∩ H and by π #Proposition 1. The map π # J : H ′ → H is continuous.Proof. Remark that for every J ′ ∈ H ′ , we have π # J P(J ′ ) ⊂ P π # J (J ′ ) . Corollary 1. Given an open setwhere M is a complex analytic variety and E is a strong normal crossings divisor of M . That is, E is the union of a finite family {E i } i∈I of irreducible smooth hypersurfaces E i , where we fix an order in the index set I and the next properties hold:Let us consider a polyhedra system D = (F ; {∆ J } J∈H , d) and a stratum T ∈ H. We introduce now a new polyhedra system D T , using Hironaka's projection of D from T , that plays an important role in Section 7.3.Let F T = (I T , H T ) be the support fabric obtained by projection of F from T . Given J * ∈ H T , let us take the stratum J = J * ∪ T ∈ H and let us consider the subset4 Definition 2. The total transform Λ 0 J (D) of a polyhedra system D = (F ; {∆ J } J∈H , d) centered in a stratum J ∈ H is the polyhedra system Λ 0The characteristic transform Λ J (D) is a polyhedra system again, because the center of the blow-up is singular and then δ(∆ 0 {∞} ) ≥ 1. Let N = (F ; {N J } J∈H , 1) be a Newton polyhedra system and take an integer number d ≥ 1. Consider the polyhedra system D = N /d and a singular stratum J ∈ Sing(D). The d-moderatedRemark 4. If there is an equivalence φ between two polyhedra systems D 1 and D 2 , then for all J ∈ Sing(D 1 ), the characteristic transforms Λ J (D 1 ) and Λ φ(J) (D 2 ) are also equivalent. Reduction of Singularities.In this section we prove Theorem 2 for Hironaka quasi-ordinary polyhedra systems.Given a Hironaka quasi-ordinary system D = (F ; {∆ J } J∈H , d) and a singular stratum J,Lemma 1. The set A of decreasing sequences of natural numbers is well-ordered for the lexicographical order.Proof. Given a decreasing sequence ϕ 1 ≥ ϕ 2 ≥ · · · ≥ ϕ k ≥ · · · in A, we take k 0 = 0 andLet us consider the decreasing sequence ϕ : N → N given by ϕ(n) = ϕ kn (n). There is a positive number l ∈ N such that ϕ(n) = ϕ(l), for all n ≥ l. As a consequence we have ϕ j = ϕ k l , for all j ≥ k l .Proposition 3. Every Hironaka quasi-ordinary polyhedra system has reduction of singularities.Proof. Choose a bijective map s D : {1, 2, . . . , #H} → H such thatTake the decreasing sequence ϕ : N → N given by ϕ(j) = dδ(∆ sD (j) ) if j ≤ #H and ϕ(j) = 0 if j > #H. We choose ...
A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C 3 , 0) without saddle-nodes has invariant surface. We extend the argument of Cano-Cerveau for the nondicritical case to the compact dicritical components of the exceptional divisor. These components are projective toric surfaces and the isolated invariant branches of the induced foliation extend to closed irreducible curves. We build the invariant surface as a germ along the singular locus and those closed irreducible invariant curves. The result of Ortiz-Bobadilla-Rosales-Gonzalez-Voronin about the distribution of invariant branches in dimension two is a key argument in our proof.
A codimension one singular holomorphic foliation is Newton nondegenerate if it satisfies the classical conditions of Kouchnirenko and Oka, in terms of its Newton polyhedra system. In this paper we prove that a foliation is Newton non-degenerate if and only if it admits a logarithmic reduction of singularities of a combinatorial nature.
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