On free and nearly free arrangements of conics admitting certain ADE singularities (Q6592694)

The paper is a continuation of papers about arrangements of plane curves and their properties, especially freeness and nearly freeness.\N\NThe first main result of the paper is a fact that an arrangement of at least 2 smooth conics with only nodes and ordinary triple points as singularities is never nearly free.\N\NRecall that an arrangement \(C\) if nearly free if it is \(3\)-syzygy, \(d_1+d_2=d\), \(d_2=d_3\) and \(e_1=d+d_2\), where \(m\)-syzygy means that \(M(f)=S/J_f\) has the following minimal graded free resolution \N\[0\longrightarrow \bigoplus_{i=1}^{m-2}S(-e_i)\longrightarrow \bigoplus_{i=1}^{m}S(1-d-d_i)\longrightarrow S^3(1-d)\longrightarrow S\] \Nwith \(e_1\leq e_2\leq\ldots\leq e_{m-2}\), \(1\leq d_1\leq\ldots \leq d_m\), \(d\) is the degree of a homogeneous polynomial \(f\) describing \(C\) and \(S=\mathbb{C}[x,y,z]\).\N\NThe second result of the paper is that if an arrangement of \(k\geq 2\) smooth connics with only nodes, ordinary triple points, tacnodes, and points of type \(A_5\) and \(A_7\) as singularities is free, that \(k\in\{2,3,4\}\). Recall that an arrangement is free if it is \(2\)-syzygy and \(d_1+d_2=d-1\). Moreover the author constructs free arrangements of \(k\) conics with the above assumptions.\N\NThe next part of the paper is focused on constructions of new examples of conic arrangements with prescribed singularities that are nearly free. The paper also contains examples of interesting arrangements of plane curves which are free and nearly free.