Double covers and vector bundles of rank two

TL;DR

This paper explores the relationship between double covers and rank two vector bundles, detailing the group structure of associated vector bundle sets and proposing new methods for studying Picard groups and constructing vector bundles.

Contribution

It describes the group structure of vector bundle sets related to double covers using transition functions, extending the correspondence with Picard groups.

Findings

Group structure of vector bundle sets derived from Picard groups

Transition functions characterize the vector bundle set structure

Proposes new approaches to studying Picard groups and constructing vector bundles

Abstract

In 2017, Catanese--Perroni gave a natural correspondence between the Picard group of a double cover and a set of pairs of a vector bundle of rank two and a certain morphism of vector bundles on the base space. In this paper, we describe the group structure of the latter set induced from the Picard group in terms of transition functions of vector bundles of rank two. This study is derived from the study of the embedded topology of plane curves. It also proposes approaches to the study of Picard groups of double covers, and to the construction of vector bundles of rank two.