Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Automorphism group of a rational surface: Not too big not too small

Kyounghee Kim

Abstract

This article concerns the realization problem of subgroups of Coxeter groups. We construct a subgroup $G$ of the Coxeter group $W_{15}$ such that $G$ is realized as automorphism groups of a rational surface $X$ and $G \cong Aut(X)^* \cong D_3 \times \mathbb{Z}$. We also show that there is an element $ω$ in $W_{14}$ is not realizable.

An Automorphism group of a rational surface: Not too big not too small

Abstract

This article concerns the realization problem of subgroups of Coxeter groups. We construct a subgroup of the Coxeter group such that is realized as automorphism groups of a rational surface and . We also show that there is an element in is not realizable.
Paper Structure (24 sections, 30 theorems, 75 equations, 1 figure)

This paper contains 24 sections, 30 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quadratic birational maps
  3. The Cremona Involution
  4. Two more Cremona Involutions
  5. Quadratic Birational Maps
  6. Dynamical Degree
  7. Orbit Data
  8. Lifting to an Automorphism
  9. Coxeter group $W_n$
  10. The smallest spectral radius in $W_n$.
  11. Connection to Rational surface automorphisms
  12. Not-realizable element
  13. Salem Numbers
  14. Quadratic rational surface automorphisms
  15. Quadratic birational map fixing three concurrent lines
  16. ...and 9 more sections

Key Result

Lemma 2.3

If $\check f: \mathbf{P}^2 \dasharrow \mathbf{P}^2$ is a basic quadratic birational map, then so is the inverse $\check f^{-1}$. Furthermore If $\check f$ has orbit data $n_1,n_2,n_3$ with a permutation $\sigma \in \Sigma_3$, then the orbit data for $\check f^{-1}$ are given by $n_{\sigma^{-1}(1)},n

Figures (1)

  • Figure 1: Coxeter Graph $E_n$

Theorems & Definitions (60)

  • Remark
  • Remark
  • Remark
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 50 more