On plus-one generated arrangements of plane conics
Artur Bromboszcz, Bartosz Jarosławski, Piotr Pokora
TL;DR
This work advances the study of plus-one generated plane curve arrangements by focusing on conics in the complex projective plane with quasi-homogeneous singularities. It introduces a Poincaré-type polynomial and Hirzebruch-type inequality to relate combinatorial data to homological invariants, and provides a partial weak-combinatorial classification for plus-one generated conic arrangements with nodes and tacnodes. The authors construct explicit examples and families illustrating+1 generation, including a degree-8 strong Ziegler pair of conic-line arrangements, and pose several open problems regarding syzygy degrees and geometric realizability. Overall, the paper deepens the understanding of the interplay between combinatorics, singularity theory, and the algebra of Jacobian ideals in conic arrangements, while highlighting computational approaches via SINGULAR.
Abstract
In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi-homogeneous singularities. First, we introduce a new tool that enables us to characterize the property of being plus-one generated within the class of conic arrangements with some naturally chosen quasi-homogeneous singularities. Next, we present a classification result on plus-one generated conic arrangements admitting only nodes and tacnodes as singularities. Building on results regarding conic arrangements with nodes and tacnodes, we present new examples of strong Ziegler pairs of conic-line arrangements -- that is, arrangements having the same strong combinatorics but distinct derivation modules.