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.

There is an error in the proof of Lemma 3.1. The effectivity of the Cartier divisor in page 7 line 6 from below does not always hold because D may be non-normal

References in corpus (1)

• Dihedral Galois covers of algebraic varieties and the simple cases