On free arrangements of three conics

TL;DR

The paper addresses the classification of free arrangements of three smooth conics in $\mathbb{P}^2_{\mathbb{C}}$ with quasi-homogeneous singularities, using Milnor–Tjurina invariants, the Jacobian/Milnor algebra, and the Du Plessis–Wall freeness criterion. It combines a decomposition into conic pairs with a finite combinatorial analysis of singularities to derive a complete ADE-only classification, yielding six explicit, projectively distinct configurations with detailed equations. It further shows that no free arrangement contains a $J_{2,0}$ singularity, so the six ADE cases exhaust all possibilities under the quasi-homogeneous setting. These results advance the understanding of freeness for plane curve arrangements and provide concrete models for three-conic configurations while outlining a method that may extend to more components.

Abstract

We give a complete classification of free arrangement of three smooth conics on complex projective plane admitting only ${\rm ADE}$ singularities and $J_{2,0}$ singularities.

Figures (2)

• Figure 1: An example of the arrangement from Proposition \ref{['prop: a3d8d8']} ($u = \frac{1}{2}$, $Z = X + Y + 1$).
• Figure 2: An example of the arrangement from Proposition \ref{['prop: last']} ($p = 0,\ q = 1,\ Z = Y + 1$).

Theorems & Definitions (58)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Example 2.5
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof