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Fundamental Group for Some Cuspidal Curves
Author(s) -
Cogolludo José Ignacio
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609398005323
Subject(s) - mathematics , conic section , gravitational singularity , abelian group , family of curves , group (periodic table) , pure mathematics , polynomial , mathematics subject classification , fundamental group , combinatorics , geometry , mathematical analysis , organic chemistry , chemistry
In [ 1 ], Hirano gives a method for constructing families of curves with a large number of singularities. The idea is to consider an abelian covering of P 2 ramified along three lines in general position, and to take the pull‐back of a curve C intersecting the lines non‐generically. Similar constructions are used by Shimada in [ 10 ] and Oka in [ 8 ]. We apply this method for the case where C is a conic, constructing a family of curves with the following asymptotic behaviour (see [ 9 ]):lim n → ∞( ∑ p ∈ Sing C ˜ nμ ( p )) /( deg (C ˜ n ) ) 2 = 3 4 .The goal of this paper is to calculate the fundamental group for the curves in this family as well as their Alexander polynomial. 1991 Mathematics Subject Classification 14H20, 14H30, 14E20.