Projection of a nonsingular plane quintic curve and the dihedral group of order eight

Abstract

Let $C$ be a nonsingular plane quintic curve over the complex number field $\mathbb{C}$, and let ${\pi }_P:C\to {\mathbb{P}}^1$ be a projection from $P\in C$. Let $L_P$ be the Galois closure of the field extension $\mathbb{C}(C)/\mathbb{C}({\mathbb{P}}^1)$ induced by ${\pi }_P$, where $\mathbb{C}(C)$ and $\mathbb{C}({\mathbb{P}}^1)$ are the rational function fields of $C$ and ${\mathbb{P}}^1$, respectively. We call the point $P$ a $D_4$-point if the Galois group of $L_P/\mathbb{C}({\mathbb{P}}^1)$ is isomorphic to the dihedral group $D_4$ of order eight. In this paper, we prove that the number of $D_4$-points for $C$ equals $0$, $1$, $3$, $5$, or $15$, and show that the curve with $15$ $D_4$-points is projectively equivalent to the Fermat quintic curve.