Plane sextics with a type $\bold{E}_8$ singular point

Abstract

We construct explicit geometric models for and compute the fundamen- tal groups of all plane sextics with simple singularities only and with at least one type E8 singular point. In particular, we discover four new sextics with nonabelian fundamental groups; two of them are irreducible. The groups of the two irreducible sextics found are finite. The principal tool used is the reduction to trigonal curves and Grothendieck's dessins d'enfants. 1.1. Principal results. This paper is a continuation of my paper (6), where we started the study of the equisingular deformation families and the fundamental groups of plane sextics (i.e., curves B ⊂ P 2 of degree six) with a distinguished triple singular point, using the representation of such sextics via trigonal curves in Hirzebruch surfaces and Grothendieck's dessins d'enfants. (All varieties in the paper are over C and are considered in their Hausdorff topology.) Recall that, in spite of the fact that the deformation classification of sextics can be reduced to a relatively simple, although tedious, arithmetical problem, see (2), the geometry of the pairs (P 2 , B) remains a terra incognita, as the construction relies upon the global Torelli theorem for K3-surfaces and is quite implicit. On the contrary, the approach suggested in (6), although not resulting in a defining equation for B, gives one a fairly good understanding of the topology of (P 2 , B); in particular, it is sufficient for the computation of the fundamental groupπ1(P 2 r B). A few other applications of this approach and more motivation can be found in (6); for a brief overview of the latest achievements on the subject, see C. Eyral, M. Oka (7). In the present paper, we deal with the case when the distinguished triple point in question is of type E8. (The case of a type E7 singular point was considered in (6), and the case of E6 is the subject of a forthcoming paper.) As in (6), a simple trick with the skeletons reduces most sextics B ⊂ P 2 to certain trigonal curves ¯ Bin �2 (instead of the original surface �3); this simplifies dramatically the classification of the sextics and the computation of their fundamental groups. It is still unclear if there is a simple geometric relation between B and ¯ B ' . Throughout the paper, we consider a plane sextic B ⊂ P 2 satisfying the following conditions: (∗) B has simple singularities only, and B has a distinguished singular point P of type E8.