Forms of del Pezzo surfaces of degree 5 and 6

TL;DR

This paper characterizes when del Pezzo surfaces of degrees 5 and 6 exist over a field with specific Galois group actions and computes automorphism groups for degree 5 surfaces over any field.

Contribution

It provides necessary and sufficient conditions for the existence of these surfaces with prescribed Galois actions and determines automorphism groups for degree 5 del Pezzo surfaces.

Findings

Criteria for existence of del Pezzo surfaces of degrees 5 and 6 with given Galois actions

Explicit automorphism groups for degree 5 del Pezzo surfaces over arbitrary fields

Classification of Galois actions on the graph of (-1)-curves

Abstract

In this paper we obtain necessary and sufficient condition for existence of del Pezzo surfaces of degree $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $\text{Gal} ( K^{\text{sep}}/K)$ on the graph of $(-1)$-curves. Also we compute automorphism groups of del Pezzo surfaces of degree $5$ over arbitrary fields.