Equivariant birational geometry of quintic del Pezzo surface

Abstract

In this paper we prove that there are exactly two $G$-minimal surfaces which are $G$-birational to the quintic del Pezzo surface, where $G\cong C_5⋊C_4$. These surfaces are the quintic del Pezzo surface itself and the surface ${\mathbb{P}}^1\times {\mathbb{P}}^1$.