Divisor class groups of double covers over projective spaces

Abstract

In this paper, we prove that the divisor class group of a double cover of the complex projective space $\mathbb{P}^n$ is generated by divisorial sheaves whose direct images split into direct sums of two invertible sheaves on $\mathbb{P}^n$. This result shows that any locally free sheaf of rank two on $\mathbb{P}^n$ is generated by direct sums of line bundles on $\mathbb{P}^n$ via some double cover. Moreover, we give a condition for an irreducible divisor on $\mathbb{P}^n$ to be a splitting divisor for a double cover whose divisor class group is finitely generated.