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Superisolated singularities and friends

Enrique Artal Bartolo

Abstract

Superisolated surface singularities in $(\mathbb{C}^3,0)$ were introduced by I. Luengo to prove that the $μ$-constant stratum may be singular. The main feature of this family is that it can bring information from the projective plane curves (global setting but smaller dimension) into surface singularities. They are simple enough to allow to retreive information and complicated enough to offer a variety of properties. The so-called Lê-Yomdin singularities are a generalization which offers a wider catalog of examples. We study some properties of these singularities, mainly topological and related with the monodromy, and we introduce another family which exploits the same properties but in the quasi-homogeneous setting.

Superisolated singularities and friends

Abstract

Superisolated surface singularities in were introduced by I. Luengo to prove that the -constant stratum may be singular. The main feature of this family is that it can bring information from the projective plane curves (global setting but smaller dimension) into surface singularities. They are simple enough to allow to retreive information and complicated enough to offer a variety of properties. The so-called Lê-Yomdin singularities are a generalization which offers a wider catalog of examples. We study some properties of these singularities, mainly topological and related with the monodromy, and we introduce another family which exploits the same properties but in the quasi-homogeneous setting.

Paper Structure

This paper contains 13 sections, 36 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and first properties
  3. Combinatorics and Zariski pairs
  4. Abstract topology
  5. On the monodromy
  6. Embedded resolutions
  7. Embedded Q-resolutions
  8. Monodromy over Z
  9. SIS, k-LYS, and Zariski pairs not of Alexander type
  10. Weighted Lê-Yomdin singularities
  11. Quotient singularities
  12. Weighted blow-ups
  13. Weight filtration of a nilpotent morphism

Key Result

Proposition 2.2

Let us suppose that $V$ is the zero locus of $F(x,y,z)\in\mathbb{C}\{x,y,z\}$. Let be the decomposition in homogeneous forms (the subscript stands for the degree). Let $C_m:=V_\mathbb{P}(F_m)\subset\mathbb{P}^2$ be the projective zero locus of $F_m$ (a maybe non-reduced curve if $F_m\neq 0$). Then, $V$ is SIS if and only if $\mathop{\mathrm{Sing}}\nolimits(C_d)\cap C_{d+1}=\emptys

Figures (4)

  • Figure 1: Curves $C^1, C^2$ with common (unmarked) graph. The self-intersection of the undecorated black vertices is $0$; the self-intersection of the gray vertices is $-1$.
  • Figure 2: Paths from the base value to the critical values, $\mu=4$.
  • Figure 3: $\mathop{\mathrm{Comb}}\nolimits_4$ (right) and $\mathop{\mathrm{Comb}}\nolimits_4^{\text{Irr}}$ (left)
  • Figure 4: $\rho^{-1}(Y)$

Theorems & Definitions (91)

  • Definition 2.1: Luengo Luengo87
  • Proposition 2.2: Luengo87
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 81 more