Galois points on singular plane quartic curves
Let \(C\) be an irreducible (possibly singular) plane curve over the complex number field \(k= \mathbb{C}\) of degree \(d=p+1\), where \(p\) is an odd prime number, and let \(k(C)\) be the rational function field of \(C\). Let \(\varepsilon \colon X \to C\) be a birational morphism from the smooth model \(X\) onto \(C\), and \(K=k(X)=k(C)\). For a point \(P\in C\) consider the projection \(\pi_P \colon X \ni x \mapsto \overline{P\varepsilon (x)} \in \mathbb{P}^1\), where \(\overline{P\varepsilon(x)}\) is the line passing through \(P\) and \(\varepsilon(x)\), \(\mathbb{P}^1\) is one-dimensional projective space of all lines in \(\mathbb{P}^2\) passing through \(P\). The projection \(\pi_P\) induces the field extension \(\pi_P^* \colon K_P := k(\mathbb{P}^1) \hookrightarrow K\). This paper continues the study of the extensions \(K/K_P\) which was started in a series of earlier papers of the author. A non-singular point \(P\in C\) is called smooth Galois point if \(K/K_P\) is a Galois extension. The main result of the paper gives the estimates for the number \(\delta(C)\) of smooth Galois points. It turns out that if \(P\) is a smooth Galois point then it is a flex of order \(d-2\) or not a flex. Thus \(\delta(C)= \delta_{d-2}(C)+\delta_0(C)\), where \(\delta_{d-2}(C)\) is the number of smooth flexes and \(\delta_0(C)\) is the number of not flexes. In this notation the estimates for \(\delta(C)\) are as follows: (i) if \(C\) has a cusp, then \(\delta(C)\leq 1\) if \(d\geq 5\), and \(\delta(C)\leq 2\) if \(d=4\); (ii) if \(C\) has no cusp, then \( (A(d,g)+1)\delta_{d-2}(C) +A(d,g)\delta_0(C)\leq 3(2g+d-2), \) where \(g\) is the genus of the smooth model of \(C\), and \(A(d,g)=(d-3)(2g+2d-4)/(d-2)\). As an application, the author studies the case of a singular curve \(C\) of degree \(d=4\) (the smooth curves were considered in an earlier paper of the author and \textit{K. Yoshihara} [J. Algebra 226, 283--294 (2000; Zbl 0983.11067)]), and presents a list of sharp estimates for \(\delta(C)\) depending on the type of curve and of its singularity. Also, the defining equations of quartics \(C\) with a Galois point are deduced in the cases where either \(C\) is rational or an elliptic quartic curve.
scientific article; zbMATH DE number 2182056