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Polarized cylinders in Du Val del Pezzo surfaces of degree two

Masatomo Sawahara

TL;DR

The paper studies $H$-polar cylinders on Du Val del Pezzo surfaces of degree at least $2$, connecting the existence of $(-K_S)$-polar cylinders to the ability to realize any ample $ ext{Q}$-divisor as the complement of a divisor giving an $H$-polar cylinder. The authors develop a framework built on $ ext{P}^1$-fibrations arising from special $(-1)$-curves and detailed divisor calculus to construct polarized cylinders across all singularity types (captured by Dynkin classifications) for degree $2$ surfaces and extend these methods to higher degrees via their weak del Pezzo resolutions. The main contributions are (i) a comprehensive construction showing Amp$^{ ext{cyl}}(S)= ext{Amp}(S)$ under broad Dynkin-type conditions and (ii) the key theorem that $-K_S ext{ in Amp}^{ ext{cyl}}(S)$ is equivalent to Amp$^{ ext{cyl}}(S)= ext{Amp}(S)$ for all Du Val del Pezzo surfaces of degree $ ext{d} \, ext{ge}\, 2$, which partially confirms the conjectural landscape for polarized cylinders on log del Pezzo surfaces. The findings have implications for $ ext{G}_a$-actions on affine cones and the broader geometry of flexible affine cones associated with polarized cylinders.

Abstract

Let $S$ be a del Pezzo surface with at worst Du Val singularities of degree $2$ such that $S$ admits an $(-K_S)$-polar cylinder. In this article, we construct an $H$-polar cylinder for any ample $\mathbb{Q}$-divisor $H$ on $S$.

Polarized cylinders in Du Val del Pezzo surfaces of degree two

TL;DR

The paper studies -polar cylinders on Du Val del Pezzo surfaces of degree at least , connecting the existence of -polar cylinders to the ability to realize any ample -divisor as the complement of a divisor giving an -polar cylinder. The authors develop a framework built on -fibrations arising from special -curves and detailed divisor calculus to construct polarized cylinders across all singularity types (captured by Dynkin classifications) for degree surfaces and extend these methods to higher degrees via their weak del Pezzo resolutions. The main contributions are (i) a comprehensive construction showing Amp under broad Dynkin-type conditions and (ii) the key theorem that is equivalent to Amp for all Du Val del Pezzo surfaces of degree , which partially confirms the conjectural landscape for polarized cylinders on log del Pezzo surfaces. The findings have implications for -actions on affine cones and the broader geometry of flexible affine cones associated with polarized cylinders.

Abstract

Let be a del Pezzo surface with at worst Du Val singularities of degree such that admits an -polar cylinder. In this article, we construct an -polar cylinder for any ample -divisor on .

Paper Structure

This paper contains 15 sections, 41 theorems, 108 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic results
  4. Du Val del Pezzo surfaces and Weak del Pezzo surfaces
  5. Singularity types of Du Val del Pezzo surfaces
  6. $(-1)$-curves on Du Val del Pezzo surfaces of degree two
  7. Construction of cylinders in rational surfaces
  8. Notation on some curves
  9. Type $\textsf{D} _n$ $(n \not = 5)$ or $\textsf{E} _n$ case
  10. Type $\textsf{D} _5$ case
  11. Type $(\textsf{A} _5)'$ case
  12. Type $(\textsf{A} _3+\textsf{A} _1)'$ case
  13. Type $\textsf{A} _n$ $(n \ge 3)$ case
  14. Type $\textsf{A}_2$ case
  15. Proof of Theorem \ref{['main(1)']}

Key Result

Theorem 1.1

Let $X$ be a normal projective variety, and let $H$ be an ample $\mathbb{Q}$-divisor on $X$. Then the following affine variety: admits a non-trivial $\mathbb{G} _a$-action if and only if $X$ contains an $H$-polar cylinder.

Figures (9)

  • Figure 1: The weighted dual graphs in Proposition \ref{['prop(3-0)']}
  • Figure 2: The configuration of $\widetilde{C}$ in Lemma \ref{['lem(3-1-1)']}
  • Figure 3: The configuration of $g : \widetilde{S} \to \mathbb{P} ^1_{\Bbbk}$ in Subsection \ref{['4-3']}.
  • Figure 4: The configuration of $g : \widetilde{S} \to \mathbb{P} ^1_{\Bbbk}$ in Subsection \ref{['4-4']}.
  • Figure 5: The configuration of $g : \widetilde{S} \to \mathbb{P} ^1_{\Bbbk}$ in Subsection \ref{['4-5']}.
  • ...and 4 more figures

Theorems & Definitions (78)

  • Theorem 1.1: KPZ11KPZ14
  • Definition 1.2
  • Conjecture 1.3: CPW17
  • Theorem 1.4: KPZ11KPZ14CPW16a
  • Theorem 1.5: CPW17
  • Remark 1.6
  • Theorem 1.7: CPW16b
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 68 more