On plane conic arrangements with nodes and tacnodes (Q2152183)

An arrangement of \(k\) regular conics in the complex projective plane can be viewed as a reducible algebraic curve of degree \(2k\) and, as such, has singularities. Under the general assumption that these are either nodes or tacnodes, the authors provide the estimate \(t \le \frac{1}{4}n + 5k\) for the difference of the number \(n\) of nodes and the number \(t\) of tacnodes. As a corollary, they obtain the bound \(t \le \frac{1}{3}k^2+3k\) which, for sufficiently large \(k\), improves upon the upper bound \(t \le \frac{4}{9}k^2+\frac{4}{3}k\) by [\textit{Y. Miyaoka}, Math. Ann. 268, 159--171 (1984; Zbl 0521.14013)]. Moreover, the authors show that the conic arrangements in question are not free for \(k \ge 2\) and nearly free if and only if \(k \in \{2,3,4\}\) and \(t = k(k-1)\). The article ends with the open question whether for an ordinary singularity \(p\) with multiplicity \(m \ge 5\) of the conic arrangement the difference of Milnor number and Tjurina number at \(p\) is either zero or \(m - 4\)?