The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a $K3$ surface with a non symplectic action of $\Z_4$. In this paper we study the geometry of the one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes.

Keywords:

genus three curves, $K3$ surfaces genus three curves, $K3$ surfaces

MSC Classifications:

14J28, 14J50, 14J10 show english descriptions $K3$ surfaces and Enriques surfaces
Automorphisms of surfaces and higher-dimensional varieties
Families, moduli, classification: algebraic theory 14J28 - $K3$ surfaces and Enriques surfaces
14J50 - Automorphisms of surfaces and higher-dimensional varieties
14J10 - Families, moduli, classification: algebraic theory

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