Minimal compactifications of the affine plane with only star-shaped singularities

TL;DR

This work classifies minimal compactifications of the complex affine plane $\mathbb{C}^2$ with star-shaped singularities. By developing and applying a framework of weighted dual graphs, contractions, and the D-natural divisor, it reduces the problem to a finite list of dual-graph configurations (Appendix 5) for $C+D$, and analyzes when boundary divisors can be contracted to yield minimal compactifications. The authors then determine, case by case, when the canonical divisors are numerically trivial or ample, producing explicit criteria in terms of the graph data and a key integer $\ell$, and they derive constructive examples with non-log canonical singularities and ample anti-canonical divisors. The results unify and extend prior classifications in the log-canonical and quotient-singularity regimes, and provide a complete panorama of the geometry of these minimal compactifications along with their numerical invariants.

Abstract

We consider minimal compactifications of the complex affine plane. Minimal compactifications of the affine plane with at most log canonical singularities are classified. Moreover, every minimal compactification of the affine plane with at most log canonical singularities has only star-shaped singular points. In this article, we classify minimal compactifications of the affine plane with only star-shaped singular points.

Figures (4)

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Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1: Mum61Gra62
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof