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Divisorial contractions to codimension three orbits

Samuel Boissière, Enrica Floris

Abstract

Let $G$ be a connected algebraic group. We study $G$-equivariant extremal contractions whose centre is a codimension three $G$-simply connected orbit. In the spirit of an important result by Kawakita in 2001, we prove that those contractions are weighted blow-ups.

Divisorial contractions to codimension three orbits

Abstract

Let be a connected algebraic group. We study -equivariant extremal contractions whose centre is a codimension three -simply connected orbit. In the spirit of an important result by Kawakita in 2001, we prove that those contractions are weighted blow-ups.

Paper Structure

This paper contains 16 sections, 15 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The setup
  3. Convention
  4. Divisorial contractions
  5. Equivariant divisorial contractions
  6. Preliminaries on the MMP with scaling
  7. Preliminaries on equivariant birational transformations
  8. Equivariant blow-ups
  9. Equivariant weighted blow-ups
  10. Equivariant resolution of singularities
  11. Equivariant MMP
  12. The tower
  13. Construction of the tower
  14. Characterisation of the tower as a weighted blow-up
  15. Proof of Theorem \ref{['th:main']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $f\colon Y\to X$ be $3$-dimensional divisorial contraction, which contracts its exceptional divisor to a smooth point. Then $f$ is a weighted blow-up.

Figures (2)

  • Figure 1: The tower construction
  • Figure 2: The counter-example

Theorems & Definitions (41)

  • Theorem 1.1: Kawakita, Kawakita
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 31 more