Divisor class groups of double covers over projective spaces

TL;DR

This paper investigates the structure of divisor class groups of double covers over projective spaces, revealing how they are generated and providing conditions for splitting divisors, thus advancing understanding of sheaf generation on these covers.

Contribution

It demonstrates that divisor class groups of double covers over projective spaces are generated by specific divisorial sheaves with split direct images, and establishes criteria for splitting divisors.

Findings

Divisor class groups are generated by sheaves with split direct images.

Any rank-two locally free sheaf on $ ext{P}^n$ arises from double covers.

Conditions for irreducible divisors to be splitting divisors are provided.

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.