Vector bundles and finite covers

TL;DR

This paper explores the relationship between vector bundles and finite covers, showing that every vector bundle on a smooth projective curve can be derived from a branched cover's structure sheaf, linking algebraic geometry and number theory.

Contribution

It establishes that, up to a twist, all vector bundles on a smooth projective curve originate from the direct image of a branched cover's structure sheaf, unifying concepts across multiple mathematical fields.

Findings

Every vector bundle on a smooth projective curve is, up to a twist, the direct image of a structure sheaf of a branched cover.

The results connect algebraic constructions of finite covers with vector bundle theory.

Provides a new perspective on the realization of vector bundles via branched covers.

Abstract

Motivated by the problem of finding algebraic constructions of finite coverings in commutative algebra, the Steinitz realization problem in number theory, and the study of Hurwitz spaces in algebraic geometry, we investigate the vector bundles underlying the structure sheaf of a finite flat branched covering. We prove that, up to a twist, every vector bundle on a smooth projective curve arises from the direct image of the structure sheaf of a smooth, connected branched cover.