Zariski \(k\)-plets of rational curve arrangements and dihedral covers
A Zariski \(k\)-plet is a set of \(k\) reduced (complex projective) plane curves having all homeomorphic tubular neighborhoods but pairwise not exchanged by an homeomorphism of \(\mathbb{P}^2\). The existence of Zariski \(k-\)plets is related with the number of connected components of equisingular families of plane curves: the curves in a \(k-\)plet are equisingular but lie in different connected components of the corresponding family. In this paper the authors find new examples of Zariski \(k\)-plets for arbitrary high \(k\). More precisely, for every \(m\geq 4\), they construct a Zariski \(\left[\frac{m}{2}\right]\)-plet of curves of degree \(m+2\), all union of a smooth conic and a rational nodal curve of degree \(m\) tangent to the conic in \(m\) points. The idea is the following. Taking a smooth conic \(C_0\), the double cover of \(\mathbb{P}^2\) branched on it, say \(f\), is \(\mathbb{P}^1 \times \mathbb{P}^1\). Then the Zariski \(\left[\frac{m}{2}\right]\)-plet is obtained by taking curves that are the union of \(C_0\) with \(f(D)\), with \(D\) general rational nodal curve of bidegree \((a,b)\) with \(a+b=m\) (varying \((a,b)\) with \(b \leq a\)). They prove that two such a curve obtained taking different \((a,b)\) are not exchanged by an homeomorphism of \(\mathbb{P}^2\) since they can show (in slightly more general hypotheses) that, for \(n\geq 3\), there exists a \(D_{2n}\) Galois cover of \(\mathbb{P}^2\) branched on \(2C_0+nf(D)\) if and only if \(n\) divides \((a-b)\).
scientific article; zbMATH DE number 2117955