Singular plane curves: freeness and combinatorics

Abstract

In this paper we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some challenging open problems that can point to new directions in research, for example by introducing weak Ziegler pairs of curve arrangements. Moreover, we construct new examples of different Ziegler pairs, in both the classical and the weak sense, and present new geometric approaches to construction problems of singular plane curves.

17 pages, 2 figures (right now in TikZ), Michael Cuntz joined as a co-author, to appear in Innovations in Incidence Geometry - Algebraic, Topological and Combinatorial

References in corpus (5)

• Freeness versus maximal global Tjurina number for plane curves
• On conic-line arrangements with nodes, tacnodes, and ordinary triple points
• Computing the nonfree locus of the moduli space of arrangements and Terao's freeness conjecture
• $\mathcal{Q}$-conic arrangements in the complex projective plane
• On free and nearly free arrangements of conics admitting certain $\mathrm{ADE}$ singularities

Cited by in corpus (1)

• On plus-one generated arrangements of plane conics