Equivariant birational geometry of quintic del Pezzo surface

TL;DR

This paper classifies the $G$-minimal surfaces birational to the quintic del Pezzo surface under a specific group action, identifying exactly two such surfaces: the surface itself and a product of projective lines.

Contribution

It provides a complete classification of $G$-birational models of the quintic del Pezzo surface for a particular group action, revealing the equivariant birational geometry structure.

Findings

Exactly two $G$-minimal surfaces are $G$-birational to the quintic del Pezzo surface.

The two surfaces are the quintic del Pezzo surface and $ ext{P}^1 imes ext{P}^1$.

The group acting is isomorphic to $C_5 times C_4$.

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 $G \cong C_5 \rtimes C_4$. These surfaces are the quintic del Pezzo surface itself and the surface $\mathbb{P}^1 \times \mathbb{P}^1$.