Combinatorial aspects of classical resolution of singularities

Abstract

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 ...

Handbook of Geometry and Topology of Singularities I

“…It is the combinatorial skeleton of resolution of singularities that appears, implicitly or explicitly in every desingularization algorithm that consists of a sequence of blowings up along nonsingular subvarieties of the ambient regular variety. We refer the reader to [70], [173], [145] and [128] for various solutions of this problem.…”

Newton Non-degenerate Foliations and Blowing-ups

“…When each polyhedron has a single vertex, the foliated space is Newton non-degenerate if and only if it is logarithmically desingularized. We end the proof of the theorem by noting that there is a reduction of singularities for the polyhedra system, as it is shown in [8].…”

“…following the definitions in [8]. We associate to each J ∈ H M,E a positively convex polyhedron N J ⊂ R J ≥0 as follows.…”

“…The construction of the polyhedra N J is compatible with the natural projections pr : R J ′ → R J , when J ⊂ J ′ , in the sense that N J = pr(N J ′ ). This "coherence" property is the essential condition considered in the theory of polyhedra systems in [8].…”

Foliaciones de codimensión uno Newton no degeneradas

“…En este capítulo hablamos de un resultado publicado en [29]. Aquí expondremos las definiciones, enunciaremos los resultados y daremos el hilo conductor de sus pruebas, dejando los detalles de éstas que se podrán consultar en el Anexo I.…”