Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Del Pezzo surfaces of Picard number one admitting a torus action

Daniel Haettig, Beatrice Hafner, Juergen Hausen, Justus Springer

TL;DR

The paper tackles the problem of classifying log del Pezzo surfaces with a torus action of Picard number one for a given Gorenstein index ${\iota}$, providing explicit, computable classification algorithms. It encodes candidate surfaces via fake weighted projective spaces ${Z(P)}$ and ${\mathbb K}^*$-surfaces ${X(P,\lambda)}$ inside them, exploiting unit-fraction decompositions of $1/\iota$ to generate weight data and rigid equivalence on defining matrices. The main contributions are the algorithms and exhaustive catalogs up to ${\iota \le 200}$, yielding ${117{,}065}$ toric and ${154{,}138}$ non-toric families (271{,}203 total), along with detailed singularity analyses and del Pezzo verifications; the results provide a comprehensive atlas for ${\mathbb K}^*$-surface geometry under torus actions. This serves as a concrete, computable resource for researchers studying toric and complexity-one torus actions on rational surfaces and can guide future extensions to higher dimensions or indices.

Abstract

We present efficient classification algorithms for log del Pezzo surfaces with torus action of Picard number one and given Gorenstein index. Explicit results are obtained up to Gorenstein index 200.

Del Pezzo surfaces of Picard number one admitting a torus action

TL;DR

The paper tackles the problem of classifying log del Pezzo surfaces with a torus action of Picard number one for a given Gorenstein index , providing explicit, computable classification algorithms. It encodes candidate surfaces via fake weighted projective spaces and -surfaces inside them, exploiting unit-fraction decompositions of to generate weight data and rigid equivalence on defining matrices. The main contributions are the algorithms and exhaustive catalogs up to , yielding toric and non-toric families (271{,}203 total), along with detailed singularity analyses and del Pezzo verifications; the results provide a comprehensive atlas for -surface geometry under torus actions. This serves as a concrete, computable resource for researchers studying toric and complexity-one torus actions on rational surfaces and can guide future extensions to higher dimensions or indices.

Abstract

We present efficient classification algorithms for log del Pezzo surfaces with torus action of Picard number one and given Gorenstein index. Explicit results are obtained up to Gorenstein index 200.
Paper Structure (7 sections, 42 theorems, 146 equations, 3 algorithms)

This paper contains 7 sections, 42 theorems, 146 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Basics on fake weighted projective spaces
  3. Classifying fake weighted projective planes
  4. Constructing ${\mathbb K}^*$-surfaces of Picard number one
  5. Singularities of ${\mathbb K}^*$-surfaces of Picard number one
  6. Classifying ${\mathbb K}^*$-surfaces of Picard number one
  7. Tables for the classification

Key Result

Theorem 1.1

There are 271.203 families of log del Pezzo surfaces with torus action of Picard number one and Gorenstein index at most 200. The numbers of families for given Gorenstein index develop as follows:

Theorems & Definitions (102)

  • Theorem 1.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Example 2.9
  • Remark 2.10
  • ...and 92 more