Notes on minimal compactifications of the affine plane
type: Item
instance of: scholarly article
title: Notes on minimal compactifications of the affine plane
zbMATH Open document ID: 1203.14041
author: Hideo Kojima
author: Takeshi Takahashi
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2009-10-02T00:00:00Z
review text: The authors consider normal compact complex surfaces \(X\) which contain a closed irreducible analytic curve \(\Gamma\) such that \(X-\Gamma\) is biholomorphic to \(\mathbb{C}^2\). Such \(X\) can be obtained by taking a smooth projective compactification \(\widetilde X\) of \(\mathbb{C}^2\) and contracting suitable all but one irreducible components of \(\widetilde X- \mathbb{C}^2\) to finitely many normal singular points. With the assumption that \(X\) is not smooth and \(X\) has at most log canonical singular points, it is proved that \(X\) is a numerical del Pezzo surface of rank 1 and every singular point of \(X\) is rational. Letting \(\widetilde X-\mathbb{C}^2\) to be a minimal resolution of singularities, the authors classify the possible dual graphs of \(\widetilde X- \mathbb{C}^2\). C. P. Ramanujan has proved that any smooth projective compactification \(\widetilde X\) of \(\mathbb{C}^2\) has the property if \(\widetilde X-\mathbb{C}^2\) is not a linear chain then there is some \((-1)\)-curve in \(\widetilde X- \mathbb{C}^2\) such that after contracting it we still get a normal crossing divisor.
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 5610620
zbMATH Keywords: affine 2-space
zbMATH Keywords: compactification
zbMATH Keywords: log canonical singularity
reviewed by: Rajendra Gurjar
MaRDI profile type: Publication
full work available at URL: s10231-008-0069-2
OpenAlex ID: W2026260214
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DOI: 10.1007/S10231-008-0069-2
description: scientific article; zbMATH DE number 5610620