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Blurred combinatorics in resolution of singularities: (a little) beyond the characteristic polytope

Helena Cobo, M. J. Soto, José M. Tornero

Abstract

We introduce a variation of the well-known Newton-Hironaka polytope for algebroid hypersurfaces. This combinatorial object is a perturbed version of the original one, parametrized by a real number. For well-chosen values of the parameter, the objects obtained are very close to the original, while at the same time presenting more (hopefully interesting) information in a way which does not depend on the choice of parameter.

Blurred combinatorics in resolution of singularities: (a little) beyond the characteristic polytope

Abstract

We introduce a variation of the well-known Newton-Hironaka polytope for algebroid hypersurfaces. This combinatorial object is a perturbed version of the original one, parametrized by a real number. For well-chosen values of the parameter, the objects obtained are very close to the original, while at the same time presenting more (hopefully interesting) information in a way which does not depend on the choice of parameter.
Paper Structure (6 sections, 11 theorems, 69 equations, 8 figures)

This paper contains 6 sections, 11 theorems, 69 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Precedents: The Newton-Hironaka polytope
  3. The perturbed Newton-Hironaka polytope
  4. Faces of $\Delta_\varepsilon (F)$
  5. Comparing perturbed polytopes
  6. Some useful information to be found in $\Delta_\varepsilon(F)$

Key Result

Lemma 1

Let $(i_1,\ldots,i_m,k) \in N^\ast(F)$. Then

Figures (8)

  • Figure 1: Newton-Hironaka polygon of $F = Z^4 + (Y-X)^4Z^2 + (Y+3X)^8$
  • Figure 2: The distance between $\rho(P)$ and $\rho_\varepsilon(P)$ can vary wildly.
  • Figure 3: When $\varepsilon$ is not sufficiently small, the perturbed polygon may lose faces. Left to right, the classic polygon of $F=Z^3+(X^2+XY^2)Z+X^2Y$ and the perturbed polygon of the same surface with $\varepsilon=3$.
  • Figure 4: Every point representing a monomial in the initial form of $F$ lies in the shaded area.
  • Figure 5: The ideal $(Z,X)$ is permissible if and only if $\rho_{\varepsilon}\bigl(N(F)\bigr)$ lies in the shaded region. Note the half open segment on the horizontal axis.
  • ...and 3 more figures

Theorems & Definitions (43)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Example 1
  • Lemma 1
  • proof
  • Remark 3
  • Example 2
  • Lemma 2
  • ...and 33 more