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Addition-deletion for conic-line arrangements with split Chern polynomial

Anca Măcinic, Jean Vallès

TL;DR

The work investigates obstructions arising from factoring the Chern polynomial $c_{\mathcal{T}_{\mathcal{C}}}(t)$ of the logarithmic vector-field bundle for reduced plane curves, extending known results for line arrangements to conic-line configurations. Central to the approach are addition-deletion exact sequences for lines and smooth conics that connect singularity incidence and the curve's freeness to the splitting type of rank-2 bundles, with a conic-restriction framework akin to Yoshinaga-type criteria. The authors establish precise bounds on intersection counts encoded by $k$ (and $k/2$ parameters) that govern when a curve is free with exponents $(a,b)$ or not, and they provide a new conic-based splitting criterion for rank-2 bundles. They also introduce a novel addition-deletion perspective for conics, yielding a formula relating Milnor and Tjurina-like invariants to a new parameter $k_0$, and demonstrate how these invariants control the global structure of ${\mathcal{C}}$ under conic addition. Overall, the paper significantly broadens obstruction-theoretic tools from line arrangements to conics within the Chern-polynomial framework, with implications for the study of freeness and vector-bundle splitting on plane curves.

Abstract

We present combinatorial/geometric obstructions induced by the factorization over the integers of the Chern polynomial of the bundle of logarithmic vector fields associated to a complex projective plane curve. Our results generalize at the same time similar results on projective lines arrangements whose characteristic polynomial factors over the integers and results on free curves. We give a splitting criterion for a rank 2 vector bundle, in terms of restrictions to smooth conics.

Addition-deletion for conic-line arrangements with split Chern polynomial

TL;DR

The work investigates obstructions arising from factoring the Chern polynomial of the logarithmic vector-field bundle for reduced plane curves, extending known results for line arrangements to conic-line configurations. Central to the approach are addition-deletion exact sequences for lines and smooth conics that connect singularity incidence and the curve's freeness to the splitting type of rank-2 bundles, with a conic-restriction framework akin to Yoshinaga-type criteria. The authors establish precise bounds on intersection counts encoded by (and parameters) that govern when a curve is free with exponents or not, and they provide a new conic-based splitting criterion for rank-2 bundles. They also introduce a novel addition-deletion perspective for conics, yielding a formula relating Milnor and Tjurina-like invariants to a new parameter , and demonstrate how these invariants control the global structure of under conic addition. Overall, the paper significantly broadens obstruction-theoretic tools from line arrangements to conics within the Chern-polynomial framework, with implications for the study of freeness and vector-bundle splitting on plane curves.

Abstract

We present combinatorial/geometric obstructions induced by the factorization over the integers of the Chern polynomial of the bundle of logarithmic vector fields associated to a complex projective plane curve. Our results generalize at the same time similar results on projective lines arrangements whose characteristic polynomial factors over the integers and results on free curves. We give a splitting criterion for a rank 2 vector bundle, in terms of restrictions to smooth conics.

Paper Structure

This paper contains 10 sections, 11 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On the number of singularities of ${\mathcal{C}}$ situated on a line
  4. Proof of Theorem \ref{['thm:curve_line_comb_chernA']}
  5. On the number of singularities of ${\mathcal{C}}$ situated on a smooth conic
  6. Proof of Theorem \ref{['thm:conic_restr']}
  7. Proof of Theorem \ref{['thm:conic_criterion']}
  8. Proof of Theorem \ref{['thm:curve_conic_comb_chern']}
  9. Proof of Theorem \ref{['thm:curve_conic_comb_chern_add']}
  10. The addition-deletion of a smooth conic to a curve, revisited

Key Result

Theorem 1.1

Let ${\pazocal{A}}$ be an arrangement of projective lines such that, for some $a,b \in \mathbb{Z}_{>0}, a \leq b$, we have (i) For $L \in {\pazocal{A}}$, denote ${\pazocal{A}}':= {\pazocal{A}} \setminus \{L\}$. Then: (ii) For any line $L \subset \mathbb{P}^2, \; L \notin {\pazocal{A}}$, either $|{\pazocal{A}} \cap L| \leq a+1$ or $|{\pazocal{A}} \cap L| \geq b+1$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 3.1
  • Remark 3.2
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • ...and 12 more