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On type three complex plane curves

Alexandru Dimca, Gabriel Sticlaru

TL;DR

The paper develops a systematic theory for type ${3}$ complex plane curves by introducing and exploiting the invariant ${t(C)=d_1+d_2+1-d}$ in relation to Jacobian syzygies, the Jacobian module ${N(f)}$, and the freeness defect ${\nu(C)}$. It proves that type ${3}$ curves partition into four subclasses ${3A},{3B},{3B'},{3C}$ with explicit formulas for the total Tjurina number ${\tau(C)}$ and the freeness defect in terms of the exponents, and it establishes converse results. The authors then explore a rich array of geometric examples—including maximal Tjurina curves, nodal curves, Zariski and Ziegler–Yuzvinsky line arrangements, and symmetry-rich families—while also addressing low-degree phenomena (${d\le 5}$) and the structure of line-arrangement types up to ${d=8}$. These results reveal intricate interactions between syzygies, singularities, and combinatorial data beyond mere intersection lattices, and they provide concrete constructions and classifications that illuminate the landscape of type-3 curves in the plane.

Abstract

The type of a complex projective plane curve has been recently introduced by T. Abe, P. Pokora and the first author. In the same paper they have studied the type two curves. In this paper we study plane curves of type three, with special attention to low degree curves and line arrangements.

On type three complex plane curves

TL;DR

The paper develops a systematic theory for type complex plane curves by introducing and exploiting the invariant in relation to Jacobian syzygies, the Jacobian module , and the freeness defect . It proves that type curves partition into four subclasses with explicit formulas for the total Tjurina number and the freeness defect in terms of the exponents, and it establishes converse results. The authors then explore a rich array of geometric examples—including maximal Tjurina curves, nodal curves, Zariski and Ziegler–Yuzvinsky line arrangements, and symmetry-rich families—while also addressing low-degree phenomena () and the structure of line-arrangement types up to . These results reveal intricate interactions between syzygies, singularities, and combinatorial data beyond mere intersection lattices, and they provide concrete constructions and classifications that illuminate the landscape of type-3 curves in the plane.

Abstract

The type of a complex projective plane curve has been recently introduced by T. Abe, P. Pokora and the first author. In the same paper they have studied the type two curves. In this paper we study plane curves of type three, with special attention to low degree curves and line arrangements.
Paper Structure (9 sections, 24 theorems, 116 equations)

This paper contains 9 sections, 24 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Basic facts related to Jacobian syzygies of curves
  3. Tjurina numbers and freeness defects
  4. Geometric examples of curves of type 3
  5. Maximal Tjurina curves
  6. Nodal curves
  7. Other curves of type 3
  8. On the curves of type 3 and low degree
  9. On the line arrangements of type 3

Key Result

Theorem 2.1

For positive integers $d$ and $d_1$, define two new integers by If $C:f=0$ is a reduced curve of degree $d$ in $\mathbb{P}^2$ and $d_1$ is the first exponent of $f$, then Moreover, for $d_1 \geq d/2$, the stronger inequality $\tau(C) \leq \tau'(d,d_1)_{max}$ holds, where

Theorems & Definitions (41)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 31 more