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Cylinders in canonical del Pezzo fibrations

Masatomo Sawahara

Abstract

Cylinders in projective varieties play an important role in connection with unipotent group actions on certain affine algebraic varieties. The previous work due to Dubouloz and Kishimoto deals with the condition for a del Pezzo fibration to contain a vertical cylinder. In the present work, as a generalization in the sense of singularities, we shall determine the condition under which a del Pezzo fibration with canonical singularities admits a vertical cylinder by means of degree and type of singularities found on the corresponding generic fiber.

Cylinders in canonical del Pezzo fibrations

Abstract

Cylinders in projective varieties play an important role in connection with unipotent group actions on certain affine algebraic varieties. The previous work due to Dubouloz and Kishimoto deals with the condition for a del Pezzo fibration to contain a vertical cylinder. In the present work, as a generalization in the sense of singularities, we shall determine the condition under which a del Pezzo fibration with canonical singularities admits a vertical cylinder by means of degree and type of singularities found on the corresponding generic fiber.

Paper Structure

This paper contains 20 sections, 50 theorems, 57 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Du Val singularities
  4. Types of weak del Pezzo surfaces
  5. Properties of weak del Pezzo surfaces
  6. Properties with respect to cylinders
  7. Degree $3$ or higher
  8. Classification of Du Val del Pezzo surfaces of Picard rank one
  9. Proof of Theorems \ref{['main(1-1)']} and \ref{['main(1-2)']}
  10. Divisors on weak del Pezzo surfaces
  11. Properties of $\mathbb{Q}$-divisors composed of $(-2)$-curves
  12. Construction of $(-1)$-curves on weak del Pezzo surface
  13. Proof of Proposition \ref{['ADE-prop']}(3)
  14. Degree $2$ or lower
  15. Base locus with respect to cylinder
  16. ...and 5 more sections

Key Result

Theorem 1.4

Any Du Val del Pezzo surface $S$ over $k$ with Picard rank $\rho_k(S)=1$ and of degree greater than or equal to $5$ contains a cylinder.

Theorems & Definitions (122)

  • Definition 1.1: DK18
  • Definition 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4: cf. Ura83
  • Remark 2.5
  • ...and 112 more