On cubic-line arrangements with simple singularities (Q6624999)

In the paper under review the author studies arrangements of smooth cubics and lines in the complex projective plane admiting certain \(\mathrm{ADE}\)-singularities. His first result is a Hirzebruch-type inequality.\N\NTheorem A. Let \(\mathcal{EL} = \{\mathcal{E}_{1}, \dots, \mathcal{E}_{k}, \ell_{1}, \dots \ell_{d}\} \subset \mathbb{P}^{2}_{\mathbb{C}}\) be an arrangement consisting of \(k\geq 1\) smooth cubics and \(d \geq 1\) lines such that \(3k+d \geq 6\). Assume that \(\mathcal{EL}\) admits \(n_{2}\) nodes, \(t_{5}\) singular points of type \(A_{5}\), and \(n_{3}\) ordinary triple points. Then the following inequality holds: \N\[\N27k + n_{2} + \frac{3}{4}n_{3} \geq d + 5t_{5}.\N\]\NThen the author examines the freeness of the class of arrangements described in the above result. Recall that a reduced plane curve is free if its associated module of derivations is a free module over the polynomial ring.\N\NTheorem B. Let \(\mathcal{EL} = \{\mathcal{E}_{1}, \dots, \mathcal{E}_{k}, \ell_{1}, \dots, \ell_{d}\} \subset \mathbb{P}^{2}_{\mathbb{C}}\) be an arrangement consisting of \(k\geq 1\) smooth cubics and \(d \geq 1\) lines, and assume that \(\mathcal{EL}\) admits \(n_{2}\) nodes, \(t_{5}\) singular points of type \(A_{5}\), and \(n_{3}\) ordinary triple points. If \(\mathcal{EL}\) is a free arrangement, then \(3k+d \in \{6,7,9\}\), possibly except the case of \(3k+d=9\).