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On the Castelnuovo-Mumford regularity of curve arrangements

Alexandru Dimca

Abstract

The Castelnuovo-Mumford regularity of the Jacobian algebra and of the graded module of derivations associated to a general curve arrangement in the complex projective plane are studied. The key result is an addition-deletion type result, similar to results obtained by H. Schenck, H. Terao, S. Tohaneanu and M. Yoshinaga, but in which no quasi homogeneity assumption is needed.

On the Castelnuovo-Mumford regularity of curve arrangements

Abstract

The Castelnuovo-Mumford regularity of the Jacobian algebra and of the graded module of derivations associated to a general curve arrangement in the complex projective plane are studied. The key result is an addition-deletion type result, similar to results obtained by H. Schenck, H. Terao, S. Tohaneanu and M. Yoshinaga, but in which no quasi homogeneity assumption is needed.
Paper Structure (6 sections, 11 theorems, 71 equations)

This paper contains 6 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. The proofs of the main results
  4. Proof of Theorem \ref{['thm1']}
  5. Proof of Corollary \ref{['corT1']}
  6. Proof of Theorem \ref{['thm2']} and Corollary \ref{['corT2']}

Key Result

Theorem 1.1

With the above notation, assume that $s>1$ and $C_s$ is a smooth curve. Then there is an exact sequence of sheaves on $\mathbb{P}^2$ given by where $i_s: C_s \to \mathbb{P}^2$ is the inclusion and ${\mathcal{F}}={\mathcal{O}}_{C_s}(D)$ a line bundle on ${C_s}$ such that where $g_s$ is the genus of the smooth curve $C_s$ and $r$ is the number of points in the reduced scheme of $C' \cap C_s$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 7 more