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On cubic-line arrangements with simple singularities

Przemysław Talar

Abstract

In the present note we study combinatorial and algebraic properties of cubic-line arrangements in the complex projective plane admitting nodes, ordinary triple and $A_{5}$ singular points. We deliver a Hirzebruch-type inequality for such arrangement and we study the freeness of such arrangements providing an almost complete classification.

On cubic-line arrangements with simple singularities

Abstract

In the present note we study combinatorial and algebraic properties of cubic-line arrangements in the complex projective plane admitting nodes, ordinary triple and singular points. We deliver a Hirzebruch-type inequality for such arrangement and we study the freeness of such arrangements providing an almost complete classification.
Paper Structure (3 sections, 4 theorems, 39 equations)

This paper contains 3 sections, 4 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. On the weak combinatorics of cubic-line arrangements
  3. Freeness of cubic-line arrangements

Key Result

Theorem 1.2

A reduced curve $C \subset \mathbb{P}^{2}_{\mathbb{C}}$ given by $f \in S_{d}$ with ${\rm mdr}(f)\leqslant (d-1)/2$ is free if and only if where $\tau(C)$ denotes the total Tjurina number of $C$.

Theorems & Definitions (13)

  • proof
  • Definition 1.1
  • Theorem 1.2: du Plessis-Wall
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 3 more