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On the module of derivations of a line arrangement

Alexandru Dimca

TL;DR

This work analyzes the derivations that kill the defining polynomial of a line arrangement through the graded module $D_0(f)$ and introduces local derivations $\tilde{D}_p$ at multiplicity points. It proves that, for degrees $j\ge d-3$, these local derivations span $D_0(f)$ in a way controlled by the combinatorics via the invariant $\tau({\mathcal A})$, and it constructs associated polynomials $g_p$ that link to freeness and geometric positioning of multiple points. A pivotal exact sequence framework under addition-deletion of lines, together with Bourbaki-ideal techniques, yields precise criteria for freeness, plus-one freeness, and bounds on exponents, tying algebraic invariants to geometric configurations. The results provide both generation and obstruction criteria for $D_0(f)$, along with geometric consequences such as lower bounds on exponents and descriptions of how multiple points sit relative to unions of lines.

Abstract

To each multiple point $p$ in a line arrangement $ \mathcal A$ in the complex projective plane we associate a local derivation $\tilde D_p \in D_0( \mathcal A)$. We show first that these derivations span the graded module of derivations $D_0( \mathcal A)$ in all degrees $\geq d -3$, where $d$ is the number of lines in $ \mathcal A$, see Theorem 1.4 and Theorem 1.6. Then, to each local derivation $\tilde D_p \in D_0( \mathcal A)$ we associate a polynomial $g_p$ which seems to play a key role in the characterization of the freeness of $ \mathcal A$, see Theorem 1.10, as well as in the study of the position of the multiple points of $ \mathcal A$ with respect to unions of lines, see Corollary 1.13 and Conjecture 1.14. Corollary 1.9 gives a result of an independent interest, namely a lower bound for the maximal exponent of a plane curve having a line as an irreducible component.

On the module of derivations of a line arrangement

TL;DR

This work analyzes the derivations that kill the defining polynomial of a line arrangement through the graded module and introduces local derivations at multiplicity points. It proves that, for degrees , these local derivations span in a way controlled by the combinatorics via the invariant , and it constructs associated polynomials that link to freeness and geometric positioning of multiple points. A pivotal exact sequence framework under addition-deletion of lines, together with Bourbaki-ideal techniques, yields precise criteria for freeness, plus-one freeness, and bounds on exponents, tying algebraic invariants to geometric configurations. The results provide both generation and obstruction criteria for , along with geometric consequences such as lower bounds on exponents and descriptions of how multiple points sit relative to unions of lines.

Abstract

To each multiple point in a line arrangement in the complex projective plane we associate a local derivation . We show first that these derivations span the graded module of derivations in all degrees , where is the number of lines in , see Theorem 1.4 and Theorem 1.6. Then, to each local derivation we associate a polynomial which seems to play a key role in the characterization of the freeness of , see Theorem 1.10, as well as in the study of the position of the multiple points of with respect to unions of lines, see Corollary 1.13 and Conjecture 1.14. Corollary 1.9 gives a result of an independent interest, namely a lower bound for the maximal exponent of a plane curve having a line as an irreducible component.

Paper Structure

This paper contains 20 sections, 20 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['thmD']} and of Corollary \ref{['corD']}
  3. The proof of Theorem \ref{['thmD']}
  4. The proof of Corollary \ref{['corD']}
  5. The proofs of Theorems \ref{['thm1']} and \ref{['thm2']} and a related result
  6. The proof of Theorem \ref{['thm1']}
  7. The proof of Theorem \ref{['thm2']}
  8. The proofs of Theorems \ref{['thm1G']}, \ref{['thmG']} and of Corollary \ref{['cor1G']}
  9. Proof of Theorem \ref{['thm1G']}
  10. Proof of Corollary \ref{['cor1G']}
  11. Proof of Theorem \ref{['thmG']}
  12. The proof of Theorem \ref{['thmG']}
  13. Local derivations, Bourbaki ideals and the proof of Theorem \ref{['thm4']} and Proposition \ref{['prop40']}
  14. Proof of Theorem \ref{['thm4']}
  15. Proof of Proposition \ref{['prop40']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

For the arrangements ${\mathcal{A}}:f=0$ of $d$ lines in $\mathbb{P}^2$ one has the following.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 18 more