On the fundamental groups of the components of a class of projective plane curves

Kwai-Man Fan, Purdue University

Abstract

Let $C\subset \doubc P\sp2$ be an algebraic curve. Suppose that there is a projective line L intersecting C at a single point P, we will prove that there is a subjective homomorphism from the local fundamental group of C at P onto $\pi\sb1(\doubc P\sp2\\ C).$ If in addition the link of C at P is a torus link, then the defining relations of $\pi\sb1(\doubc P\sp2\\ C)$ correspond to the torus link has simple form. These observations, together with a study of a global property of the curve C given by the equation $F(z,w,v)=w\sp{n} - f(z,v)$ where f(z,w) is a homogeneous polynomial of degree n, allow us to compute $\pi\sb1(\doubc P\sp2\\ C).$ In the last part of this paper we will compute the fundamental group of the complement of a projective plane curve C given by a polynomial g(z,w) such that (i) the w-axis intersects C only (0,0), (ii) g is analytically irreducible at (0,0), and (iii) the Puiseux expansion w = p(z) of g at (0,0) has only one characteristic pair. We will prove that this fundamental group is abelian.

Degree

Ph.D.

Advisors

Abhyankar, Purdue University.

Subject Area

Mathematics

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