Automorphisms of A1-fibered surfaces

TL;DR

This paper develops birational geometric techniques to analyze automorphisms of affine surfaces with multiple rational fibrations, focusing on how these automorphisms interact with the fibrations and their implications for the automorphism group structure.

Contribution

It introduces a graph-based method to encode rational fibrations on affine surfaces, aiding in understanding automorphism groups and their generation.

Findings

A graph encoding rational fibrations helps determine automorphism group structure.

Automorphisms interacting with fibrations can be characterized using the developed techniques.

The approach allows deciding if the automorphism group is generated by fibration-preserving automorphisms.

Abstract

We develop technics of birational geometry to study automorphisms of affine surfaces admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms and these fibrations. In particular, we associate to each surface $S$ of this type a graph encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the automorphism group of $S$ is generated by automorphisms preserving these fibrations.