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Lipschitz saturation of toric singularities in any dimension

François Bernard, Enrique Chávez-Martínez, Arturo E. Giles Flores

Abstract

We describe the semigroup of the Lipschitz saturation of a complex analytic toric singularity in arbitrary dimension. We give a necessary and sufficient condition for a monomial in the normalization to belong to the Lipschitz saturation, in terms of Newton polyhedra and lattice conditions, and deduce a finite algorithm to compute it. We also show that, in dimension greater than two, Campillo's notion of presaturation differs from the Lipschitz saturation, even for complex singularities.

Lipschitz saturation of toric singularities in any dimension

Abstract

We describe the semigroup of the Lipschitz saturation of a complex analytic toric singularity in arbitrary dimension. We give a necessary and sufficient condition for a monomial in the normalization to belong to the Lipschitz saturation, in terms of Newton polyhedra and lattice conditions, and deduce a finite algorithm to compute it. We also show that, in dimension greater than two, Campillo's notion of presaturation differs from the Lipschitz saturation, even for complex singularities.

Paper Structure

This paper contains 6 sections, 12 theorems, 45 equations, 2 figures.

Table of Contents

  1. Acknowledgement
  2. Campillo's criterion
  3. Newton polyhedrons and support functions
  4. A criterion for the elements which are not in the Lipschitz saturation
  5. Description of the Lipschitz saturation of a toric singularity
  6. A generating set of the semigroup $\Gamma^s$

Key Result

Proposition 1.1

Let $A\hookrightarrow B\hookrightarrow \kappa(A)$ be an extension of $R$-algebras. Let $p\in A$ and $p_1,\dots, p_r, q_1, \dots q_s \in A^{\times}$ such that Then $p(p_1\dots p_r)/(q_1\dots q_s) \in A^s_B$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Illustration of Proposition \ref{['Prop_mPlusGamma_m']}
  • Figure 2: Illustration of Example \ref{['Ex_KeyEx']}

Theorems & Definitions (39)

  • Proposition 1.1: Campillo's criterion
  • proof
  • Definition 1.2
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Example 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 1.8: Rockafellar1997ConvexAnalysis, Section 13
  • ...and 29 more