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Cylinders in Du Val del Pezzo surfaces of degree one with Picard rank two

Jaehyun Kim, Dae-Won Lee, Masatomo Sawahara

TL;DR

The paper investigates Du Val del Pezzo surfaces of degree one with Picard rank two, proving that the existence of an anticanonical polar cylinder implies the ample polar cylindricity, i.e., $Amp^{\mathrm{cyl}}(S)=Amp(S)$. It introduces and exploits a polarity-cone framework, comparing polarity cones of cylinders with the ample cone, and uses explicit cone computations on minimal resolutions. For thirteen singularity types that admit an anticanonical polar cylinder, the authors show that the ample cone is covered by the polarity cones of a finite collection of cylinders; for several types additional cylinders arising from $\mathbb{P}^1$-fibrations are constructed to complete the coverings. The results extend the understanding of the cylindrical ample cone to degree-one, rank-two cases and provide a method potentially adaptable to higher Picard ranks, with comprehensive data in the Appendix.

Abstract

We prove that for Du Val del Pezzo surfaces of degree one with Picard rank two, the existence of an anticanonical polar cylinder implies the ample polar cylindricity.

Cylinders in Du Val del Pezzo surfaces of degree one with Picard rank two

TL;DR

The paper investigates Du Val del Pezzo surfaces of degree one with Picard rank two, proving that the existence of an anticanonical polar cylinder implies the ample polar cylindricity, i.e., . It introduces and exploits a polarity-cone framework, comparing polarity cones of cylinders with the ample cone, and uses explicit cone computations on minimal resolutions. For thirteen singularity types that admit an anticanonical polar cylinder, the authors show that the ample cone is covered by the polarity cones of a finite collection of cylinders; for several types additional cylinders arising from -fibrations are constructed to complete the coverings. The results extend the understanding of the cylindrical ample cone to degree-one, rank-two cases and provide a method potentially adaptable to higher Picard ranks, with comprehensive data in the Appendix.

Abstract

We prove that for Du Val del Pezzo surfaces of degree one with Picard rank two, the existence of an anticanonical polar cylinder implies the ample polar cylindricity.

Paper Structure

This paper contains 5 sections, 6 theorems, 48 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Polarity of Cylinders
  4. $\mathbb{P}^1$-Fibrations on minimal resolutions
  5. Appendix

Key Result

Theorem 1.2

Let $S_d$ be a Du Val del Pezzo surface of degree $d$. Then $S_d$ admits an anticanonical polar cylinder if and only if it does not fall into one of the following cases:

Figures (23)

  • Figure 1: $\mathop{\mathrm{Amp}}\nolimits(S)$ and $\mathop{\mathrm{Pol}}\nolimits(U_{0})$ for type $\mathsf{A_5+A_2}$
  • Figure 2: $\mathop{\mathrm{Amp}}\nolimits(S)$ and $\mathop{\mathrm{Pol}}\nolimits(U_{0})$ for type $\mathsf{A_7'}$
  • Figure 3: $\mathop{\mathrm{Amp}}\nolimits(S)$ and $\mathop{\mathrm{Pol}}\nolimits(U_{0})$ for type $\mathsf{D_6+A_1}$
  • Figure 4: $\mathop{\mathrm{Amp}}\nolimits(S)$ and $\mathop{\mathrm{Pol}}\nolimits(U_{0})$ for type $\mathsf{E_7}$
  • Figure 5: The region covered by $\mathop{\mathrm{Pol}}\nolimits(U_{0})$ for type $\mathsf{A_5+2A_1}$
  • ...and 18 more figures

Theorems & Definitions (13)

  • Conjecture 1.1: Cheltsov2017
  • Theorem 1.2: Cheltsov2016s
  • Theorem 1.3: Sawahara2024Sawahara2025
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5: Cheltsov2016sSawahara2024Sawahara2025
  • Theorem 3.1
  • proof
  • ...and 3 more