Non-symplectic involutions of normal K3 surfaces associated with Hirzebruch surfaces

For a non-symplectic involution g of a K3 surface X, the fixed point set of g and the quotient space \(X/\langle g\rangle \) $X/⟨g⟩$ are well studied. In this paper, we study a non-symplectic involution f of a normal K3 surface Y such that the smooth model of Y is a double cover K3 surface of a Hirzebruch surface and f is induced by its covering involution. We determine the fixed point set of f, the singular points of Y, and the quotient space \(Y/\langle f\rangle \) $Y/⟨f⟩$ .