The Algebraic Fundamental Group of a Curve Singularity

Abstract

A classical theorem of Brauner [3], (also Kahler [10], Zariski [15]), gives a formula for the generators and relations of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular curve in C. These elegant formulas depend only on the characteristic pairs of a Puiseux series expansion of the curve. Zariski states this Theorem as part of his discussion on Puiseux series in the chapter on resolution of singularities in his book ”Algebraic Surfaces” [16]. Let γ be the germ of an analyically irreducible plane curve singularity at the origin. Let De be an epsilon ball centered at the origin in C , and let Se be the boundary of De. For e sufficiently small, the pair (De, γ ∩De) is homeomorphic to the pair consisting of the cone over Se and the cone over γ ∩ Se (c.f. Theorem 2.10 [11]). Thus Se − γ ∩ Se is a strong deformation retract of Be − γ ∩ Be, and the topological fundamental group of the knot is isomorphic to π 1 (Be − γ ∩Be). The arithmetic analogue of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular plane curve is thus the algebraic fundamental group π1(Spec(R) − V (f)) , where R = k[[x, y]] is a power series ring over an algebraically closed field k, and f ∈ R is irreducible. The basic theory of the algebraic fundamental group is classical, being understood for Riemann surfaces in the 19th century. The algebraic fundamental group is constructed from the finite topological covers, which are algebraic. Abhyankar extended the algebraic fundamental group to arbitrary characteristic [2] and Grothendieck [8] defined the fundamental group in general. In positive characteristic, Puiseux series expansions do not always exist (c.f section 2.1 of [5]). However, in characteristic zero, the characteristic pairs of the Puiseux series are determined by the resolution graph of a resolution of singularites of the curve germ. As such, the characteristic pairs can be defined in any characteristic from the resolution graph (c.f [5]). In this paper we prove an arithmetic analogue of Brauner’s theorem, valid in arbitrary characteristic. The generators and relations in our Theorem (Theorem 0.1