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

TL;DR

This paper explores the realization of specific subgroups of Coxeter groups as automorphism groups of rational surfaces, demonstrating a particular subgroup is realizable and identifying an element that is not.

Contribution

It constructs a subgroup of a Coxeter group that can be realized as an automorphism group of a rational surface and shows an element that cannot be realized.

Findings

Constructed a subgroup G of W_{15} isomorphic to D_3 x Z as automorphisms of a rational surface.

Proved that a certain element in W_{14} is not realizable as an automorphism.

Provided insights into the realization problem of Coxeter group subgroups in algebraic geometry.

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 $\omega$ in $W_{14}$ is not realizable.