Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimal compactifications of the affine plane with nef canonical divisors

Masatomo Sawahara

Abstract

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. We show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.

Minimal compactifications of the affine plane with nef canonical divisors

Abstract

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. We show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.
Paper Structure (15 sections, 17 theorems, 49 equations)

This paper contains 15 sections, 17 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contractility of divisors on smooth projective surfaces
  4. Some notations and properties of compactifications of the affine plane
  5. Determinants of weighted dual graphs
  6. Boundaries of minimal compactifications of the affine plane
  7. Birational transformations of minimal compactifications of the affine plane
  8. Case(1)
  9. Case(2)
  10. Case(3)
  11. Proof of Theorem \ref{['main']}
  12. Proof of Theorem \ref{['main(2)']} (1)
  13. Proof of Theorem \ref{['main(2)']} (2)
  14. Further problems
  15. A supplement to Example \ref{['eg(6-1)']}

Key Result

Theorem 1.1

Let $(X,\Gamma )$ be a minimal compactification of $\mathbb{C} ^2$. Assume that $X$ has at worse log canonical singularities and $\mathrm{Sing} (X) \not= \emptyset$. Then:

Theorems & Definitions (49)

  • Theorem 1.1: KT09
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 39 more