Classical Zariski pairs (Q2890239)

A classical Zariski pair is a pair of irreducible plane sextics that share the same combinatorial type of singularities but differ by their Alexander polynomial \((t^ 2-t+1)^d,d\geq0\).NEWLINENEWLINEFollowing \textit{A. Degtyarev} [J. Knot Theory Ramifications 3, No. 4, 439--454 (1994; Zbl 0848.57006), and J. Lond. Math. Soc., II. Ser. 78, No. 2, 329--351 (2008; Zbl 1158.14026)], the common set of singularities of curves constituting a classical Zariski pair is of the form \(\Sigma=e{\mathbf E}_6\oplus{\bigoplus}^6_{i=1}a_i{\mathbf A}_{3i-1}\oplus n{\mathbf A}_1\) with \(2e+\Sigma ia_i=6\) and, within each pair, one may have \(d=1\) (abundant curve) or \(d=0\) (non-abundant curve).NEWLINENEWLINEIn this paper the author enumerates and classifies up to equisingular deformation all irreducible plane sextics constituting a classical Zariski pair.NEWLINENEWLINEHis main result is that each of the four sets of singularities \(6{\mathbf A}_2\oplus4{\mathbf A}_1, 2{\mathbf A}_2\oplus2{\mathbf E}_6\oplus2{\mathbf A}_1, 4{\mathbf A}_2\oplus{\mathbf E}_6\oplus3{\mathbf A}_1, 3{\mathbf E}_6\oplus{\mathbf A}_1\) is realized by abundant sextics only; the set of singularities \({\mathbf A}_{11}\oplus{\mathbf E}_6\oplus{\mathbf A}_1\) is realized by three families, one abundant and two complex conjugate non-abundant and this exceptional set of singularities has submaximal total Milnor number 18; any other set of singularities is realized by two deformation families, one abundant and one non-abundant.