[Paper Review] Plane sextics with a type $\bold E_8$ singular point

Alex Degtyarev|arXiv (Cornell University)|Feb 13, 2009

Algebraic Geometry and Number Theory9 references4 citations

TL;DR

This paper classifies all maximal plane sextics with a single E8 singular point and simple singularities, using trigonal models in Hirzebruch surfaces and Grothendieck's dessins d'enfants. It computes their fundamental groups, identifying four new sextics with non-abelian fundamental groups—two irreducible ones with finite groups isomorphic to Z₁₂ ⋊ SL(2, F₅)—and proves that all other such sextics have abelian fundamental groups.

ABSTRACT

We construct explicit geometric models for and compute the fundamental groups of all plane sextics with simple singularities only and with at least one type $\bold E_8$ 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 {\it dessins d'enfants}

Motivation & Objective

• To classify all maximal plane sextics with a type E8 singular point and simple singularities up to equisingular deformation.
• To compute the fundamental groups π₁(P² ∖ B) for all such sextics, especially identifying non-abelian cases.
• To investigate the topology of the complement of the curve in the projective plane via geometric and combinatorial methods.
• To explore the connection between the geometry of sextics and their fundamental groups, particularly in the context of rigidity and degenerations.
• To verify and extend conjectures on finiteness of fundamental groups for non-torus-type sextics.

Proposed method

• Reduction of plane sextics to trigonal curves in the Hirzebruch surface Σ₂ via a geometric trick involving skeletons.
• Application of Grothendieck's dessins d'enfants to classify and analyze the monodromy and topology of the trigonal models.
• Use of elementary transformations and degenerations in the trigonal model to construct all possible sextics systematically.
• Computation of fundamental groups via monodromy representations and group presentations derived from dessin invariants.
• Leveraging the global Torelli theorem for K3 surfaces and the rigidity of maximal sextics to ensure uniqueness of deformation classes.
• Analysis of perturbations and degenerations to determine when fundamental groups remain non-abelian.

Experimental results

Research questions

• RQ1 How many maximal plane sextics with a type E8 singular point and simple singularities exist up to equisingular deformation?
• RQ2 Which of these sextics have non-abelian fundamental groups, and what are the structures of these groups?
• RQ3 Can the fundamental group of a non-torus-type irreducible sextic with simple singularities be finite?
• RQ4 What are the possible perturbations of such sextics that preserve non-abelian fundamental groups?
• RQ5 How do degenerations of sextics relate to the maximality and rigidity of their equisingular deformation classes?

Key findings

• There are exactly 39 maximal irreducible plane sextics with a type E8 singular point and simple singularities, realizing 26 distinct sets of singularities.
• There are exactly 18 maximal reducible plane sextics with a type E8 singular point and simple singularities, realizing 17 distinct sets of singularities.
• Only two sextics have non-abelian fundamental groups: one with singularities E8 ⊕ A₄ ⊕ A₃ ⊕ 2A₂ and another with E8 ⊕ D₆ ⊕ A₃ ⊕ A₂.
• The fundamental group of the sextic with E8 ⊕ A₄ ⊕ A₃ ⊕ 2A₂ is isomorphic to G₆ = Z₁₂ ⋊ SL(2, F₅), a finite group of order 1440.
• The fundamental group of the sextic with E8 ⊕ D₆ ⊕ A₃ ⊕ A₂ is isomorphic to G∞ = Z × SL(2, F₅), an infinite group of order ∞.
• All proper perturbations of these sextics with non-abelian fundamental groups result in abelian fundamental groups, except for two specific perturbations that preserve the non-abelian structure.