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Motivic local density of isolated surface singularities

Sidonie Ratajczak

Abstract

The goal of this paper is to compute the motivic local density of an isolated algebraic surface singularity, in order to explain its link with algebraic multiplicity. In this context, we can use an additional data: the inner rate related to the bilipschitz geoemtry of the singularity, as studied by A. Belotto da Silva, L. Fantini and A. Pichon.

Motivic local density of isolated surface singularities

Abstract

The goal of this paper is to compute the motivic local density of an isolated algebraic surface singularity, in order to explain its link with algebraic multiplicity. In this context, we can use an additional data: the inner rate related to the bilipschitz geoemtry of the singularity, as studied by A. Belotto da Silva, L. Fantini and A. Pichon.

Paper Structure

This paper contains 6 sections, 20 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Recap on motivic integration
  3. Introduction to the inner rates of a complex surface
  4. Discrepancies and the Mather canonical divisor
  5. Properties of the motivic local density
  6. Proof of the formula for surfaces

Key Result

Proposition 1.1

Let $f \in \mathbb{C}[x,y]$ be a power series without square factor. Let $f=f_1\dots f_r$ the decomposition in $\mathbb{C}[\![x,y]\!]$ of $f$ into irreducible factors. Denote by $N_i$ the multiplicity of $f_i$ and $C$ the curve defined by $f$. Then the motivic density of $C$ at the origin is given b

Theorems & Definitions (53)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 3.1
  • ...and 43 more