Projective geometry for blueprints

TL;DR

This paper extends the Proj-construction to blue schemes, enabling the definition of projective spaces and varieties over blueprints, and explores their descent to F_1-models with explicit examples like the Grassmannian.

Contribution

It introduces a generalized Proj-construction for blue schemes, allowing projective geometry over blueprints and F_1-models, with explicit case studies.

Findings

Defined projective space and varieties over blueprints.

Demonstrated descent of subvarieties to F_1-models.

Explicit analysis of the Grassmannian Gr(2,4).

Abstract

In this note, we generalize the Proj-construction from usual schemes to blue schemes. This yields the definition of projective space and projective varieties over a blueprint. In particular, it is possible to descend closed subvarieties of a projective space to a canonical F_1-model. We discuss this explicitly in case of the Grassmannian Gr(2,4).