A canonical treatment of line bundles over general projective spaces

TL;DR

This paper explores the geometry and line bundle theory over projective spaces of finite-dimensional vector spaces over arbitrary fields, emphasizing basis-free methods and the role of algebraic closedness.

Contribution

It provides a basis-free, algebraically closedness-aware treatment of line bundles over general projective spaces, extending classical theory to broader fields.

Findings

Characterization of global sections of line bundles

Basis-free construction methods

Impact of field algebraic closedness

Abstract

Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of these bundles is examined. Care is taken in two directions: (1) places where algebraic closedness of the field are important are pointed out; (2) basis free constructions are used exclusively.