The decomposition group of a line in the plane

TL;DR

This paper characterizes the decomposition group of a line in the plane, showing it is generated by degree 1 elements and one degree 2 element, and that it cannot be decomposed as a non-trivial amalgamated product.

Contribution

It provides a detailed structural description of the decomposition group of a line in the plane, including its generators and algebraic decomposition properties.

Findings

Generated by degree 1 elements and one degree 2 element

Does not decompose as a non-trivial amalgamated product

Structural insight into plane birational transformations

Abstract

We show that the decomposition group of a line $L$ in the plane, i.e. the subgroup of plane birational transformations that send $L$ to itself birationally, is generated by its elements of degree 1 and one element of degree 2, and that it does not decompose as a non-trivial amalgamated product.