Finite abelian groups of K3 surfaces with smooth quotient

Abstract: The quotient space of a $K3$ surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, we will study finite groups which act faithfully on $K3$ surfaces such that the quotient space are smooth. In particular, we will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a $K3$ surface to a Hirzebrunch surface such that the branch divisor is that effective divisor. Furthermore, we will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface. Subsequently, we study the same theme for Enriques surfaces.