Algebraicity of normal analytic compactifications of ℂ2 with one irreducible curve at infinity

Mondal, Pinaki

doi:10.2140/ant.2016.10.1641

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Algebraicity of normal analytic compactifications of $\mathbb{C}^2$ with one irreducible curve at infinity

Pinaki Mondal

Vol. 10 (2016), No. 8, 1641–1682

DOI: 10.2140/ant.2016.10.1641

Abstract

We present an effective criterion to determine if a normal analytic compactification of $ℂ^{2}$ with one irreducible curve at infinity is algebraic or not. As a byproduct we establish a correspondence between normal algebraic compactifications of $ℂ^{2}$ with one irreducible curve at infinity and algebraic curves contained in $ℂ^{2}$ with one place at infinity. Using our criterion we construct pairs of homeomorphic normal analytic surfaces with minimally elliptic singularities such that one of the surfaces is algebraic and the other is not. Our main technical tool is the sequence of key forms — a “global” variant of the sequence of key polynomials introduced by MacLane [1936] to study valuations in the “local” setting — which also extends the notion of approximate roots of polynomials considered by Abhyankar and Moh [19 73].

Keywords

algebraicity, compactifications, one place at infinity, valuations

Mathematical Subject Classification 2010

Primary: 14J26

Secondary: 32J05, 14E15, 14E05

Milestones

Received: 4 October 2015

Revised: 10 April 2016

Accepted: 27 June 2016

Published: 7 October 2016

Authors

Pinaki Mondal
College of The Bahamas Thompson Boulevard Nassau Bahamas

Vol. 10, No. 8, 2016