Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deletion-addition of a smooth conic for free curves

Anca Macinic

TL;DR

This work investigates how adding or deleting a smooth conic affects the freeness and plus-one generated status of free CL-arrangements and, more generally, of reduced plane curves. By exploiting the module of logarithmic derivations $D_0({\mathcal{C}})$ and vector-bundle exact sequences, the authors derive precise conditions and Hilbert-series formulas that classify the resulting curve as free, plus-one generated, or neither, under quasihomogeneous assumptions; they extend these results to non-quasihomogeneous triples via an adjusted invariant $\epsilon$ and a modified count $k$. The main contributions include complete deletion/addition theorems with explicit exponents and minimal resolutions, generalized obstructions to geometric and combinatorial restrictions, and a broad extension from CL-arrangements to reduced curves. The results provide constructive methods to generate new free and plus-one generated examples and deepen the understanding of freeness in the conic–line setting, with implications for hyperplane arrangement theory and the study of logarithmic derivations.

Abstract

We describe the behaviour of a free reduced plane projective curve with respect to the deletion, respectively addition, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the geometry and combinatorics of a free reduced curve, generalizing results known a priori only for free projective line arrangements.

Deletion-addition of a smooth conic for free curves

TL;DR

This work investigates how adding or deleting a smooth conic affects the freeness and plus-one generated status of free CL-arrangements and, more generally, of reduced plane curves. By exploiting the module of logarithmic derivations and vector-bundle exact sequences, the authors derive precise conditions and Hilbert-series formulas that classify the resulting curve as free, plus-one generated, or neither, under quasihomogeneous assumptions; they extend these results to non-quasihomogeneous triples via an adjusted invariant and a modified count . The main contributions include complete deletion/addition theorems with explicit exponents and minimal resolutions, generalized obstructions to geometric and combinatorial restrictions, and a broad extension from CL-arrangements to reduced curves. The results provide constructive methods to generate new free and plus-one generated examples and deepen the understanding of freeness in the conic–line setting, with implications for hyperplane arrangement theory and the study of logarithmic derivations.

Abstract

We describe the behaviour of a free reduced plane projective curve with respect to the deletion, respectively addition, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the geometry and combinatorics of a free reduced curve, generalizing results known a priori only for free projective line arrangements.
Paper Structure (6 sections, 20 theorems, 88 equations)

This paper contains 6 sections, 20 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. From CL-arrangements to reduced curves
  5. Non-quasihomogeneous case
  6. Examples

Key Result

Proposition 2.2

A curve ${\mathcal{C}}$ is plus-one generated with exponents $(a,b)$ and level $\ell$ if and only if ${\mathcal{C}}$ has generalized exponents $(a,b,\ell)$ and $a+b=\deg({\mathcal{C}})$.

Theorems & Definitions (41)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Theorem 2.8
  • ...and 31 more