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A new hierarchy for complex plane curves

Takuro Abe, Alexandru Dimca, Piotr Pokora

TL;DR

This work introduces the type t(C) of a reduced plane curve C via the initial degree of its Bourbaki ideal, connecting it to the two smallest degrees of Jacobian-syzygy generators and to freeness notions. It develops exact sequences for unions and for line/conic-line arrangements, enabling precise control of exponents under unions and perturbations, and proves that type 0 corresponds to free, type 1 to plus-one generated, and type 2 encompasses a broad and richly structured class. The authors construct extensive families of line and conic-line arrangements realizing all types and, in particular, provide detailed criteria and constructions yielding type 2 (subtypes 2A and 2B) with explicit exponent patterns and Tjurina numbers. They also establish lower bounds for the type in line arrangements via Arnold exponents, linking singularity theory with combinatorial geometry. Overall, the paper presents a systematic hierarchy for complex plane curves and deepens the interplay between syzygies, Bourbaki ideals, and the geometry of curve unions.

Abstract

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves of type $1$ are the plus-one generated curves. In this paper, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type $2$ and construct families of line arrangements and conic-line arrangements of this type.

A new hierarchy for complex plane curves

TL;DR

This work introduces the type t(C) of a reduced plane curve C via the initial degree of its Bourbaki ideal, connecting it to the two smallest degrees of Jacobian-syzygy generators and to freeness notions. It develops exact sequences for unions and for line/conic-line arrangements, enabling precise control of exponents under unions and perturbations, and proves that type 0 corresponds to free, type 1 to plus-one generated, and type 2 encompasses a broad and richly structured class. The authors construct extensive families of line and conic-line arrangements realizing all types and, in particular, provide detailed criteria and constructions yielding type 2 (subtypes 2A and 2B) with explicit exponent patterns and Tjurina numbers. They also establish lower bounds for the type in line arrangements via Arnold exponents, linking singularity theory with combinatorial geometry. Overall, the paper presents a systematic hierarchy for complex plane curves and deepens the interplay between syzygies, Bourbaki ideals, and the geometry of curve unions.

Abstract

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type are precisely the free curves, while curves of type are the plus-one generated curves. In this paper, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type and construct families of line arrangements and conic-line arrangements of this type.

Paper Structure

This paper contains 16 sections, 19 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. Some key exact sequences
  4. Exact sequences for line arrangements
  5. The proofs of Theorems \ref{['thm1']} and \ref{['thm10']} and of Corollaries \ref{['cor10']} and \ref{['cor11']}
  6. Proof of Theorem \ref{['thm1']}
  7. Proof of Theorem \ref{['thm10']} and Corollaries \ref{['cor10']} and \ref{['cor11']}
  8. The proof of Propositions \ref{['propB']}, \ref{['propCA']} and \ref{['propCB']}
  9. Proof of Proposition \ref{['propB']}
  10. Proof of Proposition \ref{['propCA']} and \ref{['propCB']}
  11. Proof of Theorem \ref{['thm0']}, Theorem \ref{['thm4']}, Proposition \ref{['prop2']} and Theorem \ref{['thm2']}
  12. Proof of Theorem \ref{['thm0']}
  13. Proof of Theorem \ref{['thm4']}
  14. Proof of Proposition \ref{['prop2']}
  15. Proof of Theorem \ref{['thm2']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $C:f=0$ be a reduced plane curve of degree $d$ and let $d_1$ and $d_2$ be the minimal degrees of a minimal system of generators for the module of Jacobian syzygies $D_0(f)$ as above. Then the following conditions hold:

Theorems & Definitions (33)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.10
  • Remark 1.11
  • ...and 23 more