Rational curve with Galois point and extendable Galois automorphism
Let \(C\) be an irreducible plane projective curve of degree \(d \geq 3\) over an algebraically closed field \(k\) of characteristic zero, and \(k(C)\) its function field. Let \(P\) be a point of projective plane \(\mathbb{P}^2\) and \(\pi_P : \mathbb{P}^2 \dashrightarrow \mathbb{P}^1\) be the projection with center \(P\). The restriction \(\pi\) to \(C\) gives a dominant rational map \(\tilde{\pi} : C \dashrightarrow \mathbb{P}^1\) which induces a finite field extension \(k(C)/k(\mathbb{P}^1)\). \(P\) is called a Galois point for \(C\) if \(k(C)/k(\mathbb{P}^1)\) is a Galois extension. Let \(G\) be the corresponding Galois group. The author studies when a birational transformation of \(C\) induced by an element of \(G\) can be extended to a projective or birational transformation of the plane. Previously [J. Algebra 239, No. 1, 340--355 (2001; Zbl 1064.14023)], the author showed that if \(C\) is smooth and \(d\geq 4\), then \(G\) is cyclic and an element of \(G\) can be extended to a projective transformation of \(\mathbb{P}^2\). It is not so when \(C\) has a singular point. It is shown in the paper under review that in general case an element of \(G\) can not be extended to a birational transformation of \(\mathbb{P}^2\). This is done by presenting several examples. In addition, the defining equation of a rational curve with a Galois point are presented, and a special birational transformation is introduced in order to simplify the defining equation.
scientific article; zbMATH DE number 5549370
scientific article