Automorphisms of quartic del Pezzo surfaces in characteristic zero

TL;DR

This paper classifies automorphism groups of quartic del Pezzo surfaces over fields of characteristic zero, detailing which groups act on these surfaces and providing explicit equations for such actions.

Contribution

It provides a comprehensive classification of automorphism groups for quartic del Pezzo surfaces over characteristic zero fields and constructs explicit examples for each group.

Findings

Classified automorphism groups for quartic del Pezzo surfaces.

Determined conditions for rationality, stable rationality, and unirationality.

Constructed explicit equations for surfaces with given automorphism groups.

Abstract

For each field $k$ of characteristic zero, we classify which groups act by automorphisms on a quartic del Pezzo surface over $k$. We also determine which groups act on $k$-rational, stably $k$-rational, or $k$-unirational quartic del Pezzo surfaces. For each group $G$ that is realized over $k$, we exhibit explicit equations for a quartic del Pezzo surface $X$ in $\mathbb{P}^4_k$ such that $G$ acts by automorphisms on $X$.