On homological properties of conic-line arrangements with simple singularities

We study arrangements of smooth conics and lines in the complex projective plane whose singularities are limited to nodes, tacnodes, and ordinary triple points.

The first part of the paper gives numerical restrictions for plus-one generated conic arrangements with defect $\nu(C)=3$ and explains how these restrictions interact with Bézout's theorem, the Dimca--Sernesi bound for the minimal degree of a Jacobian syzygy, and Hirzebruch-type inequalities.

In particular, the possible numbers of conics are bounded, and the exceptional low-degree cases are separated from those that remain open.

The second part concerns arrangements of total degree at most $6$.

We identify the weak and strong Ziegler pairs occurring in the database recorded in the Appendix.