Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cylinders in weighted Fano varieties

Adrien Dubouloz, In-Kyun Kim, Takashi Kishimoto, Joonyeong Won

Abstract

Cylinders in Fano varieties receives a lot of attentions recently from the viewpoints of birational geometry and unipotent geometry. In this article, we provide a survey of several known et new results concerning the anti-canonically polar cylindricity of quasi-smooth, well-formed weighted Fano complete intersections in weighted projective spaces.

Cylinders in weighted Fano varieties

Abstract

Cylinders in Fano varieties receives a lot of attentions recently from the viewpoints of birational geometry and unipotent geometry. In this article, we provide a survey of several known et new results concerning the anti-canonically polar cylindricity of quasi-smooth, well-formed weighted Fano complete intersections in weighted projective spaces.
Paper Structure (21 sections, 31 theorems, 64 equations, 3 tables)

This paper contains 21 sections, 31 theorems, 64 equations, 3 tables.

Table of Contents

  1. Cylindrical projective varieties: basic examples and obstructions
  2. Normal projective varieties with unipotent group actions are cylindrical
  3. Obstructions to cylindricity
  4. Log-resolution of cylinders
  5. Pseudo-effectiveness of canonical divisors and cylindricity
  6. Cylindricity and $\alpha$-invariant of Fano varieties
  7. $K$-stability and anti-canonical polar cylindricity of Fano varieties
  8. Weighted complete intersections in weighted projective spaces
  9. Weighted projective spaces: definition and basic properties
  10. Cylindricity of weighted projective spaces
  11. Weighted complete intersections: well-formedness and quasi-smoothnes
  12. Well-formedness
  13. Quasi-smoothness
  14. Cylindricity of weighted hypersurfaces
  15. A class of cylindrical Fano weighted hypersurfaces
  16. ...and 6 more sections

Key Result

Proposition 1.2

A normal projective variety $X$ endowed with a non-trivial action of a unipotent group contains an $H$-polar cylinder with respect to every ample Weil $\mathbb{Q}$-divisor class $H$.

Theorems & Definitions (67)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • Remark 1.4
  • Proposition 1.5
  • proof
  • Corollary 1.6
  • proof
  • Remark 1.7
  • ...and 57 more