Families of Galois closure curves for plane quartic curves

Abstract

For a smooth quartic plane curve $C$ we show an existence of a family of Galois closure curves $\phi : S \longrightarrow C$, where $S$ is a nonsingular projective surface and $\phi ^{-1}(P)$ is isomorphic to the Galois closure curve $C_{P}$ for a general point $P \in C$. Moreover we determine the types of singular fibers. As a corollary we can say that $C_{P}$ is not isomorphic to $C_{Q}$ if $P$ is close to $Q$.