Birational classification of pointless del Pezzo surfaces of degree 8

TL;DR

This paper extends the classification of pointless del Pezzo surfaces of degree 8 over perfect fields, showing they are birationally equivalent if and only if they are isomorphic, and describes minimal models in this class.

Contribution

It generalizes Colliot-Thélène's result from quadric surfaces to all degree 8 del Pezzo surfaces with Picard number 1, providing a birational classification.

Findings

Pointless del Pezzo surfaces of degree 8 are birationally equivalent iff they are isomorphic.

The paper describes minimal surfaces birationally equivalent to a given pointless del Pezzo surface.

The classification extends known results from quadric surfaces to higher degree del Pezzo surfaces.

Abstract

Let k be a perfect field. Recently J.-L. Colliot-Th\'el\`ene showed that two pointless quadric surfaces over k are birationally equivalent if and only if they are isomorphic. We show that this result holds for arbitrary del Pezzo surfaces of degree $8$ with the Picard number $1$, and describe minimal surfaces birationally equivalent to a given pointless del Pezzo surface of degree $8$.