On the Artal-Carmona-Cogolludo construction (Q2878664)

The geometry and topology of singular complex plane projective curves of degree six has been deeply investigated during the last dozen of years (see e.g. \textit{A. Degtyarev} [Topology of algebraic curves. An approach via dessins d'enfants. Berlin: de Gruyter (2012; Zbl 1273.14065)]). The present paper bridges some of the remaining gaps about the question of the defining equations: the author derives explicit defining equations for a number of irreducible maximizing plane sextics (i. e. irreducible sextics with the maximal total Milnor number \(\mu=19\)), with double singular points only. The main tools used in the paper are the Artal-Carmona-Cogolludo construction developed in [\textit{E. Artal Bartolo} et al., in: Trends in singularities. Basel: Birkhäuser. 1--29 (2002; Zbl 1018.14009)] and Moody's results (see [\textit{E. I. Moody}, Bull. Am. Math. Soc. 49, 433--436 (1943; Zbl 0061.32705)]) concerning the Bertini involution \( {\mathbb{P}}^2 \dashrightarrow{\mathbb{P}}^2\). The defining equations obtained are used to describe the minimal fields of definition, the fundamental group of the complement (for most real curves) and to construct a few examples of the so-called arithmetic Zariski pairs (see [\textit{I. Shimada}, Adv. Geom. 8, No. 2, 205--225 (2008; Zbl 1153.14017)]).