Birational Geometry of sextic del Pezzo surfaces

TL;DR

This paper provides a comprehensive study of degree 6 del Pezzo surfaces over arbitrary perfect fields, detailing their classification, automorphisms, birational transformations, and special properties like infinite pliability.

Contribution

It offers an explicit Galois cohomology-based classification, automorphism group descriptions, and insights into birational transformations and pliability unique to sextic del Pezzo surfaces.

Findings

Classification of sextic del Pezzo surfaces over perfect fields

Description of automorphism groups and birational models

Construction of nontrivial quotients of birational transformation groups

Abstract

We study the biregular and birational geometry of degree 6 del Pezzo surfaces with Picard number 1, defined over an arbitrary perfect field. Using Galois cohomology techniques, we obtain an explicit description of cocycles for such surfaces and describe the Severi-Brauer varieties associated with them, recovering the biregular classification of sextic del Pezzo surfaces. We then compute the automorphism groups of such surfaces, describe their closed points in general position and investigate the structure of Sarkisov links at such points and the corresponding birational models, answering a question of M. Rost. Using this description, we show that degree 6 del Pezzo surfaces are the only solid surfaces that admit infinite pliability. We also find a system of generators and relations for the groups of birational transformations of such surfaces and use it to construct nontrivial quotients of these groups, including free groups on uncountable sets.