Extendable birational transformations belonging to Galois points

TL;DR

This paper investigates when birational transformations from Galois points on plane curves can be extended to Cremona transformations, proving that cyclic groups of order three yield transformations expressible as de Jonquières transformations.

Contribution

It demonstrates that for Galois groups isomorphic to cyclic groups of order three, the associated birational transformations can be extended to Cremona transformations as de Jonquières transformations.

Findings

Transformations from cyclic Galois groups of order three are extendable to Cremona transformations.

Such transformations can be explicitly expressed as de Jonquières transformations.

The paper provides conditions under which Galois point-induced transformations are extendable.

Abstract

We study birational transformations belonging to Galois points. Let $P$ be a Galois point for a plane curve $C$ and $G_P$ be a Galois group at $P$. Then an element of $G_P$ induces a birational transformation of $C$. In general, it is difficult to determine when this birational transformations can be extended to a Cremona (or projective) transformation. In this article, we shall prove that if the Galois group is isomorphic to the cyclic group of order three, then any element of the Galois group has an expression as a de Jonqui\`{e}res transformation. In particular, they can be extended to Cremona transformations.