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On maximizing curves of degree $7$

Izabela Czarnota

Abstract

In the present paper we investigate the question concerning the existence of maximizing curves of degree $7$ with some prescribed ${\rm ADE}$ singularities. We give a result proving the non-existence of such maximizing septics and we provide new examples of conic-line arrangements with some ${\rm ADE}$ singularities that are free but not maximizing.

On maximizing curves of degree $7$

Abstract

In the present paper we investigate the question concerning the existence of maximizing curves of degree with some prescribed singularities. We give a result proving the non-existence of such maximizing septics and we provide new examples of conic-line arrangements with some singularities that are free but not maximizing.
Paper Structure (4 sections, 4 theorems, 39 equations)

This paper contains 4 sections, 4 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constraints on the existence of maximizing curves
  4. Examples of free conic-line arrangements in degree $7$

Key Result

Theorem 2.5

Let $C$ be a plane curve of degree $n=2m \geqslant 4$ having only ${\rm ADE}$ singularities. Then $C$ is maximizing if and only if $C$ is a free curve with the exponents $(d_{1},d_{2}) = (m-1,m)$.

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Freeness
  • Definition 2.4
  • Theorem 2.5: Dimca-Pokora
  • Definition 2.6
  • Theorem 3.1
  • Theorem 3.2: Dimca-Sernesi
  • proof
  • Proposition 3.3
  • ...and 9 more