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Defect of irreducible plane curves with simple singularities

Piotr Pokora

Abstract

In this note we focus on the defect of singular plane curve that was recently introduced by Dimca. Roughly speaking, the defect of a reduced plane curve measures the discrepancy from the property of being a free curve. We find some lower-bound on the defect for certain classes of irreducible plane curves admitting nodes, ordinary cusps and ordinary triple points. The main result of the note tells us that reduced simply singular plane curves with sufficiently high Arnold exponents are never free.

Defect of irreducible plane curves with simple singularities

Abstract

In this note we focus on the defect of singular plane curve that was recently introduced by Dimca. Roughly speaking, the defect of a reduced plane curve measures the discrepancy from the property of being a free curve. We find some lower-bound on the defect for certain classes of irreducible plane curves admitting nodes, ordinary cusps and ordinary triple points. The main result of the note tells us that reduced simply singular plane curves with sufficiently high Arnold exponents are never free.
Paper Structure (1 section, 3 theorems, 44 equations)

This paper contains 1 section, 3 theorems, 44 equations.

Table of Contents

  1. Acknowledgement

Key Result

Theorem 2

Let $C \, = \, \{ f=0 \}$ be a reduced plane curve of degree $d$ and $r= {\rm mdr}(C)$. Then the following hold.

Theorems & Definitions (17)

  • Definition 1
  • Theorem 2: Dimca1
  • Conjecture 3
  • Definition 4
  • Definition 5
  • Conjecture 6
  • Definition 7
  • Remark 8
  • Theorem 9: Dimca-Sernesi
  • Theorem 10
  • ...and 7 more