Galois Points and Cremona Transformations

TL;DR

This paper investigates the extension of Galois groups associated with plane curve points to Cremona transformations, establishing conditions under which such extensions are possible or impossible, with specific examples illustrating these cases.

Contribution

It characterizes when Galois groups of plane curve points extend to Cremona transformations, especially for groups of order at most 3, and provides explicit counterexamples for higher degrees.

Findings

Galois groups of order ≤ 3 extend to Jonquières groups

Extensions fail for degrees ≥ 4 in general

Examples of extendable and non-extendable Galois groups provided

Abstract

In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to $\mathrm{Bir}(\mathbb{P}^2)$. If the Galois group has order at most $3$, we prove that it always extends to a subgroup of the Jonqui\`eres group associated to the point $P$. In degree at least $4$, we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonqui\`eres maps with respect to $P$. We also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.