On Irreducible Sextics with Non-abelian Fundamental Group
نویسندگان
چکیده
We calculate the fundamental groups π = π1(P rB) for all irreducible plane sextics B ⊂ P2 with simple singularities for which π is known to admit a dihedral quotient D10. All groups found are shown to be finite, two of them being of large order: 960 and 21600.
منابع مشابه
On the Artal–carmona–cogolludo Construction
We derive explicit defining equations for a number of irreducible maximizing plane sextics with double singular points only. For most real curves, we also compute the fundamental group of the complement; all groups found are abelian, which suffices to complete the computation of the groups of all non-maximizing irreducible sextics. As a by-product, examples of Zariski pairs in the strongest pos...
متن کاملOka’s Conjecture on Irreducible Plane Sextics
We partially prove and partially disprove Oka’s conjecture on the fundamental group/Alexander polynomial of an irreducible plane sextic. Among other results, we enumerate all irreducible sextics with simple singularities admitting dihedral coverings and find examples of Alexander equivalent Zariski pairs of irreducible
متن کاملIrreducible Plane Sextics with Large Fundamental Groups
We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are show...
متن کاملFundamental Group of Sextics of Torus Type
We show that the fundamental group of the complement of any irreducible tame torus sextics in P is isomorphic to Z2 ∗ Z3 except one class. The exceptional class has the configuration of the singularities {C3,9, 3A2} and the fundamental group is bigger than Z2 ∗ Z3. In fact, the Alexander polynomial is given by (t 2 − t + 1). For the proof, we first reduce the assertion to maximal curves and the...
متن کاملPlane Sextics with a Type E8 Singular Point
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 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...
متن کامل